Theory HOL-Complex_Analysis.Complex_Singularities
theory Complex_Singularities
imports Conformal_Mappings
begin
subsection ‹Non-essential singular points›
definition is_pole ::
"('a::topological_space ⇒ 'b::real_normed_vector) ⇒ 'a ⇒ bool" where
"is_pole f a = (LIM x (at a). f x :> at_infinity)"
lemma is_pole_cong:
assumes "eventually (λx. f x = g x) (at a)" "a=b"
shows "is_pole f a ⟷ is_pole g b"
unfolding is_pole_def using assms by (intro filterlim_cong,auto)
lemma is_pole_transform:
assumes "is_pole f a" "eventually (λx. f x = g x) (at a)" "a=b"
shows "is_pole g b"
using is_pole_cong assms by auto
lemma is_pole_shift_iff:
fixes f :: "('a::real_normed_vector ⇒ 'b::real_normed_vector)"
shows "is_pole (f ∘ (+) d) a = is_pole f (a + d)"
by (metis add_diff_cancel_right' filterlim_shift_iff is_pole_def)
lemma is_pole_tendsto:
fixes f::"('a::topological_space ⇒ 'b::real_normed_div_algebra)"
shows "is_pole f x ⟹ ((inverse o f) ⤏ 0) (at x)"
unfolding is_pole_def
by (auto simp add:filterlim_inverse_at_iff[symmetric] comp_def filterlim_at)
lemma is_pole_shift_0:
fixes f :: "('a::real_normed_vector ⇒ 'b::real_normed_vector)"
shows "is_pole f z ⟷ is_pole (λx. f (z + x)) 0"
unfolding is_pole_def by (subst at_to_0) (auto simp: filterlim_filtermap add_ac)
lemma is_pole_shift_0':
fixes f :: "('a::real_normed_vector ⇒ 'b::real_normed_vector)"
shows "NO_MATCH 0 z ⟹ is_pole f z ⟷ is_pole (λx. f (z + x)) 0"
by (metis is_pole_shift_0)
lemma is_pole_compose_iff:
assumes "filtermap g (at x) = (at y)"
shows "is_pole (f ∘ g) x ⟷ is_pole f y"
unfolding is_pole_def filterlim_def filtermap_compose assms ..
lemma is_pole_inverse_holomorphic:
assumes "open s"
and f_holo:"f holomorphic_on (s-{z})"
and pole:"is_pole f z"
and non_z:"∀x∈s-{z}. f x≠0"
shows "(λx. if x=z then 0 else inverse (f x)) holomorphic_on s"
proof -
define g where "g ≡ λx. if x=z then 0 else inverse (f x)"
have "isCont g z" unfolding isCont_def using is_pole_tendsto[OF pole]
by (simp add: g_def cong: LIM_cong)
moreover have "continuous_on (s-{z}) f" using f_holo holomorphic_on_imp_continuous_on by auto
hence "continuous_on (s-{z}) (inverse o f)" unfolding comp_def
by (auto elim!:continuous_on_inverse simp add:non_z)
hence "continuous_on (s-{z}) g" unfolding g_def
using continuous_on_eq by fastforce
ultimately have "continuous_on s g" using open_delete[OF ‹open s›] ‹open s›
by (auto simp add:continuous_on_eq_continuous_at)
moreover have "(inverse o f) holomorphic_on (s-{z})"
unfolding comp_def using f_holo
by (auto elim!:holomorphic_on_inverse simp add:non_z)
hence "g holomorphic_on (s-{z})"
using g_def holomorphic_transform by force
ultimately show ?thesis unfolding g_def using ‹open s›
by (auto elim!: no_isolated_singularity)
qed
lemma not_is_pole_holomorphic:
assumes "open A" "x ∈ A" "f holomorphic_on A"
shows "¬is_pole f x"
proof -
have "continuous_on A f"
by (intro holomorphic_on_imp_continuous_on) fact
with assms have "isCont f x"
by (simp add: continuous_on_eq_continuous_at)
hence "f ─x→ f x"
by (simp add: isCont_def)
thus "¬is_pole f x"
unfolding is_pole_def
using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto
qed
lemma is_pole_inverse_power: "n > 0 ⟹ is_pole (λz::complex. 1 / (z - a) ^ n) a"
unfolding is_pole_def inverse_eq_divide [symmetric]
by (intro filterlim_compose[OF filterlim_inverse_at_infinity] tendsto_intros)
(auto simp: filterlim_at eventually_at intro!: exI[of _ 1] tendsto_eq_intros)
lemma is_pole_cmult_iff [simp]:
assumes "c ≠ 0"
shows "is_pole (λz. c * f z :: 'a :: real_normed_field) z ⟷ is_pole f z"
proof
assume "is_pole (λz. c * f z) z"
with ‹c≠0› have "is_pole (λz. inverse c * (c * f z)) z"
unfolding is_pole_def
by (force intro: tendsto_mult_filterlim_at_infinity)
thus "is_pole f z"
using ‹c≠0› by (simp add: field_simps)
next
assume "is_pole f z"
with ‹c≠0› show "is_pole (λz. c * f z) z"
by (auto intro!: tendsto_mult_filterlim_at_infinity simp: is_pole_def)
qed
lemma is_pole_uminus_iff [simp]: "is_pole (λz. -f z :: 'a :: real_normed_field) z ⟷ is_pole f z"
using is_pole_cmult_iff[of "-1" f] by (simp del: is_pole_cmult_iff)
lemma is_pole_inverse: "is_pole (λz::complex. 1 / (z - a)) a"
using is_pole_inverse_power[of 1 a] by simp
lemma is_pole_divide:
fixes f :: "'a :: t2_space ⇒ 'b :: real_normed_field"
assumes "isCont f z" "filterlim g (at 0) (at z)" "f z ≠ 0"
shows "is_pole (λz. f z / g z) z"
proof -
have "filterlim (λz. f z * inverse (g z)) at_infinity (at z)"
using assms filterlim_compose filterlim_inverse_at_infinity isCont_def
tendsto_mult_filterlim_at_infinity by blast
thus ?thesis by (simp add: field_split_simps is_pole_def)
qed
lemma is_pole_basic:
assumes "f holomorphic_on A" "open A" "z ∈ A" "f z ≠ 0" "n > 0"
shows "is_pole (λw. f w / (w - z) ^ n) z"
proof (rule is_pole_divide)
have "continuous_on A f" by (rule holomorphic_on_imp_continuous_on) fact
with assms show "isCont f z" by (auto simp: continuous_on_eq_continuous_at)
have "filterlim (λw. (w - z) ^ n) (nhds 0) (at z)"
using assms by (auto intro!: tendsto_eq_intros)
thus "filterlim (λw. (w - z) ^ n) (at 0) (at z)"
by (intro filterlim_atI tendsto_eq_intros)
(insert assms, auto simp: eventually_at_filter)
qed fact+
lemma is_pole_basic':
assumes "f holomorphic_on A" "open A" "0 ∈ A" "f 0 ≠ 0" "n > 0"
shows "is_pole (λw. f w / w ^ n) 0"
using is_pole_basic[of f A 0] assms by simp
lemma is_pole_compose:
assumes "is_pole f w" "g ─z→ w" "eventually (λz. g z ≠ w) (at z)"
shows "is_pole (λx. f (g x)) z"
using assms(1) unfolding is_pole_def
by (rule filterlim_compose) (use assms in ‹auto simp: filterlim_at›)
lemma is_pole_plus_const_iff:
"is_pole f z ⟷ is_pole (λx. f x + c) z"
proof
assume "is_pole f z"
then have "filterlim f at_infinity (at z)" unfolding is_pole_def .
moreover have "((λ_. c) ⤏ c) (at z)" by auto
ultimately have " LIM x (at z). f x + c :> at_infinity"
using tendsto_add_filterlim_at_infinity'[of f "at z"] by auto
then show "is_pole (λx. f x + c) z" unfolding is_pole_def .
next
assume "is_pole (λx. f x + c) z"
then have "filterlim (λx. f x + c) at_infinity (at z)"
unfolding is_pole_def .
moreover have "((λ_. -c) ⤏ -c) (at z)" by auto
ultimately have " LIM x (at z). f x :> at_infinity"
using tendsto_add_filterlim_at_infinity'[of "(λx. f x + c)"
"at z" "(λ_. - c)" "-c"]
by auto
then show "is_pole f z" unfolding is_pole_def .
qed
lemma is_pole_minus_const_iff:
"is_pole (λx. f x - c) z ⟷ is_pole f z"
using is_pole_plus_const_iff [of f z "-c"] by simp
lemma is_pole_alt:
"is_pole f x = (∀B>0. ∃U. open U ∧ x∈U ∧ (∀y∈U. y≠x ⟶ norm (f y)≥B))"
unfolding is_pole_def
unfolding filterlim_at_infinity[of 0,simplified] eventually_at_topological
by auto
lemma is_pole_mult_analytic_nonzero1:
assumes "is_pole g x" "f analytic_on {x}" "f x ≠ 0"
shows "is_pole (λx. f x * g x) x"
unfolding is_pole_def
proof (rule tendsto_mult_filterlim_at_infinity)
show "f ─x→ f x"
using assms by (simp add: analytic_at_imp_isCont isContD)
qed (use assms in ‹auto simp: is_pole_def›)
lemma is_pole_mult_analytic_nonzero2:
assumes "is_pole f x" "g analytic_on {x}" "g x ≠ 0"
shows "is_pole (λx. f x * g x) x"
proof -
have g: "g analytic_on {x}"
using assms by auto
show ?thesis
using is_pole_mult_analytic_nonzero1 [OF ‹is_pole f x› g] ‹g x ≠ 0›
by (simp add: mult.commute)
qed
lemma is_pole_mult_analytic_nonzero1_iff:
assumes "f analytic_on {x}" "f x ≠ 0"
shows "is_pole (λx. f x * g x) x ⟷ is_pole g x"
proof
assume "is_pole g x"
thus "is_pole (λx. f x * g x) x"
by (intro is_pole_mult_analytic_nonzero1 assms)
next
assume "is_pole (λx. f x * g x) x"
hence "is_pole (λx. inverse (f x) * (f x * g x)) x"
by (rule is_pole_mult_analytic_nonzero1)
(use assms in ‹auto intro!: analytic_intros›)
also have "?this ⟷ is_pole g x"
proof (rule is_pole_cong)
have "eventually (λx. f x ≠ 0) (at x)"
using assms by (simp add: analytic_at_neq_imp_eventually_neq)
thus "eventually (λx. inverse (f x) * (f x * g x) = g x) (at x)"
by eventually_elim auto
qed auto
finally show "is_pole g x" .
qed
lemma is_pole_mult_analytic_nonzero2_iff:
assumes "g analytic_on {x}" "g x ≠ 0"
shows "is_pole (λx. f x * g x) x ⟷ is_pole f x"
by (subst mult.commute, rule is_pole_mult_analytic_nonzero1_iff) (fact assms)+
lemma frequently_const_imp_not_is_pole:
fixes z :: "'a::first_countable_topology"
assumes "frequently (λw. f w = c) (at z)"
shows "¬ is_pole f z"
proof
assume "is_pole f z"
from assms have "z islimpt {w. f w = c}"
by (simp add: islimpt_conv_frequently_at)
then obtain g where g: "⋀n. g n ∈ {w. f w = c} - {z}" "g ⇢ z"
unfolding islimpt_sequential by blast
then have "(f ∘ g) ⇢ c"
by (simp add: tendsto_eventually)
moreover have *: "filterlim g (at z) sequentially"
using g by (auto simp: filterlim_at)
have "filterlim (f ∘ g) at_infinity sequentially"
unfolding o_def by (rule filterlim_compose [OF _ *])
(use ‹is_pole f z› in ‹simp add: is_pole_def›)
ultimately show False
using not_tendsto_and_filterlim_at_infinity trivial_limit_sequentially by blast
qed
text ‹The proposition
\<^term>‹∃x. ((f::complex⇒complex) ⤏ x) (at z) ∨ is_pole f z›
can be interpreted as the complex function \<^term>‹f› has a non-essential singularity at \<^term>‹z›
(i.e. the singularity is either removable or a pole).›
definition not_essential::"[complex ⇒ complex, complex] ⇒ bool" where
"not_essential f z = (∃x. f─z→x ∨ is_pole f z)"
definition isolated_singularity_at::"[complex ⇒ complex, complex] ⇒ bool" where
"isolated_singularity_at f z = (∃r>0. f analytic_on ball z r-{z})"
lemma not_essential_cong:
assumes "eventually (λx. f x = g x) (at z)" "z = z'"
shows "not_essential f z ⟷ not_essential g z'"
unfolding not_essential_def using assms filterlim_cong is_pole_cong by fastforce
lemma not_essential_compose_iff:
assumes "filtermap g (at z) = at z'"
shows "not_essential (f ∘ g) z = not_essential f z'"
unfolding not_essential_def filterlim_def filtermap_compose assms is_pole_compose_iff[OF assms]
by blast
lemma isolated_singularity_at_cong:
assumes "eventually (λx. f x = g x) (at z)" "z = z'"
shows "isolated_singularity_at f z ⟷ isolated_singularity_at g z'"
proof -
have "isolated_singularity_at g z"
if "isolated_singularity_at f z" "eventually (λx. f x = g x) (at z)" for f g
proof -
from that(1) obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
by (auto simp: isolated_singularity_at_def)
from that(2) obtain r' where r': "r' > 0" "∀x∈ball z r'-{z}. f x = g x"
unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_commute)
have "f holomorphic_on ball z r - {z}"
using r(2) by (subst (asm) analytic_on_open) auto
hence "f holomorphic_on ball z (min r r') - {z}"
by (rule holomorphic_on_subset) auto
also have "?this ⟷ g holomorphic_on ball z (min r r') - {z}"
using r' by (intro holomorphic_cong) auto
also have "… ⟷ g analytic_on ball z (min r r') - {z}"
by (subst analytic_on_open) auto
finally show ?thesis
unfolding isolated_singularity_at_def
by (intro exI[of _ "min r r'"]) (use ‹r > 0› ‹r' > 0› in auto)
qed
from this[of f g] this[of g f] assms show ?thesis
by (auto simp: eq_commute)
qed
lemma removable_singularity:
assumes "f holomorphic_on A - {x}" "open A"
assumes "f ─x→ c"
shows "(λy. if y = x then c else f y) holomorphic_on A" (is "?g holomorphic_on _")
proof -
have "continuous_on A ?g"
unfolding continuous_on_def
proof
fix y assume y: "y ∈ A"
show "(?g ⤏ ?g y) (at y within A)"
proof (cases "y = x")
case False
have "continuous_on (A - {x}) f"
using assms(1) by (meson holomorphic_on_imp_continuous_on)
hence "(f ⤏ ?g y) (at y within A - {x})"
using False y by (auto simp: continuous_on_def)
also have "?this ⟷ (?g ⤏ ?g y) (at y within A - {x})"
by (intro filterlim_cong refl) (auto simp: eventually_at_filter)
also have "at y within A - {x} = at y within A"
using y assms False
by (intro at_within_nhd[of _ "A - {x}"]) auto
finally show ?thesis .
next
case [simp]: True
have "f ─x→ c"
by fact
also have "?this ⟷ (?g ⤏ ?g y) (at y)"
by (intro filterlim_cong) (auto simp: eventually_at_filter)
finally show ?thesis
by (meson Lim_at_imp_Lim_at_within)
qed
qed
moreover {
have "?g holomorphic_on A - {x}"
using assms(1) holomorphic_transform by fastforce
}
ultimately show ?thesis
by (rule no_isolated_singularity) (use assms in auto)
qed
lemma removable_singularity':
assumes "isolated_singularity_at f z"
assumes "f ─z→ c"
shows "(λy. if y = z then c else f y) analytic_on {z}"
proof -
from assms obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
by (auto simp: isolated_singularity_at_def)
have "(λy. if y = z then c else f y) holomorphic_on ball z r"
proof (rule removable_singularity)
show "f holomorphic_on ball z r - {z}"
using r(2) by (subst (asm) analytic_on_open) auto
qed (use assms in auto)
thus ?thesis
using r(1) unfolding analytic_at by (intro exI[of _ "ball z r"]) auto
qed
named_theorems singularity_intros "introduction rules for singularities"
lemma holomorphic_factor_unique:
fixes f::"complex ⇒ complex" and z::complex and r::real and m n::int
assumes "r>0" "g z≠0" "h z≠0"
and asm:"∀w∈ball z r-{z}. f w = g w * (w-z) powi n ∧ g w≠0 ∧ f w = h w * (w - z) powi m ∧ h w≠0"
and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r"
shows "n=m"
proof -
have [simp]:"at z within ball z r ≠ bot" using ‹r>0›
by (auto simp add:at_within_ball_bot_iff)
have False when "n>m"
proof -
have "(h ⤏ 0) (at z within ball z r)"
proof (rule Lim_transform_within[OF _ ‹r>0›, where f="λw. (w - z) powi (n - m) * g w"])
have "∀w∈ball z r-{z}. h w = (w-z)powi(n-m) * g w"
using ‹n>m› asm ‹r>0› by (simp add: field_simps power_int_diff) force
then show "⟦x' ∈ ball z r; 0 < dist x' z;dist x' z < r⟧
⟹ (x' - z) powi (n - m) * g x' = h x'" for x' by auto
next
define F where "F ≡ at z within ball z r"
define f' where "f' ≡ λx. (x - z) powi (n-m)"
have "f' z=0" using ‹n>m› unfolding f'_def by auto
moreover have "continuous F f'" unfolding f'_def F_def continuous_def
using ‹n>m›
by (auto simp add: Lim_ident_at intro!:tendsto_powr_complex_0 tendsto_eq_intros)
ultimately have "(f' ⤏ 0) F" unfolding F_def
by (simp add: continuous_within)
moreover have "(g ⤏ g z) F"
unfolding F_def
using ‹r>0› centre_in_ball continuous_on_def g_holo holomorphic_on_imp_continuous_on by blast
ultimately show " ((λw. f' w * g w) ⤏ 0) F" using tendsto_mult by fastforce
qed
moreover have "(h ⤏ h z) (at z within ball z r)"
using holomorphic_on_imp_continuous_on[OF h_holo]
by (auto simp add:continuous_on_def ‹r>0›)
ultimately have "h z=0" by (auto intro!: tendsto_unique)
thus False using ‹h z≠0› by auto
qed
moreover have False when "m>n"
proof -
have "(g ⤏ 0) (at z within ball z r)"
proof (rule Lim_transform_within[OF _ ‹r>0›, where f="λw. (w - z) powi (m - n) * h w"])
have "∀w∈ball z r -{z}. g w = (w-z) powi (m-n) * h w" using ‹m>n› asm
by (simp add:field_simps power_int_diff) force
then show "⟦x' ∈ ball z r; 0 < dist x' z;dist x' z < r⟧
⟹ (x' - z) powi (m - n) * h x' = g x'" for x' by auto
next
define F where "F ≡ at z within ball z r"
define f' where "f' ≡λx. (x - z) powi (m-n)"
have "f' z=0" using ‹m>n› unfolding f'_def by auto
moreover have "continuous F f'" unfolding f'_def F_def continuous_def
using ‹m>n›
by (auto simp: Lim_ident_at intro!:tendsto_powr_complex_0 tendsto_eq_intros)
ultimately have "(f' ⤏ 0) F" unfolding F_def
by (simp add: continuous_within)
moreover have "(h ⤏ h z) F"
using holomorphic_on_imp_continuous_on[OF h_holo,unfolded continuous_on_def] ‹r>0›
unfolding F_def by auto
ultimately show " ((λw. f' w * h w) ⤏ 0) F" using tendsto_mult by fastforce
qed
moreover have "(g ⤏ g z) (at z within ball z r)"
using holomorphic_on_imp_continuous_on[OF g_holo]
by (auto simp add:continuous_on_def ‹r>0›)
ultimately have "g z=0" by (auto intro!: tendsto_unique)
thus False using ‹g z≠0› by auto
qed
ultimately show "n=m" by fastforce
qed
lemma holomorphic_factor_puncture:
assumes f_iso:"isolated_singularity_at f z"
and "not_essential f z"
and non_zero:"∃⇩Fw in (at z). f w≠0"
shows "∃!n::int. ∃g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z≠0
∧ (∀w∈cball z r-{z}. f w = g w * (w-z) powi n ∧ g w≠0)"
proof -
define P where "P = (λf n g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z≠0
∧ (∀w∈cball z r - {z}. f w = g w * (w-z) powi n ∧ g w≠0))"
have imp_unique:"∃!n::int. ∃g r. P f n g r" when "∃n g r. P f n g r"
proof (rule ex_ex1I[OF that])
fix n1 n2 :: int
assume g1_asm:"∃g1 r1. P f n1 g1 r1" and g2_asm:"∃g2 r2. P f n2 g2 r2"
define fac where "fac ≡ λn g r. ∀w∈cball z r-{z}. f w = g w * (w - z) powi n ∧ g w ≠ 0"
obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z≠0"
and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto
obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z≠0"
and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto
define r where "r ≡ min r1 r2"
have "r>0" using ‹r1>0› ‹r2>0› unfolding r_def by auto
moreover have "∀w∈ball z r-{z}. f w = g1 w * (w-z) powi n1 ∧ g1 w≠0
∧ f w = g2 w * (w - z) powi n2 ∧ g2 w≠0"
using ‹fac n1 g1 r1› ‹fac n2 g2 r2› unfolding fac_def r_def
by fastforce
ultimately show "n1=n2"
using g1_holo g2_holo ‹g1 z≠0› ‹g2 z≠0›
apply (elim holomorphic_factor_unique)
by (auto simp add:r_def)
qed
have P_exist:"∃ n g r. P h n g r" when
"∃z'. (h ⤏ z') (at z)" "isolated_singularity_at h z" "∃⇩Fw in (at z). h w≠0"
for h
proof -
from that(2) obtain r where "r>0" and r: "h analytic_on ball z r - {z}"
unfolding isolated_singularity_at_def by auto
obtain z' where "(h ⤏ z') (at z)" using ‹∃z'. (h ⤏ z') (at z)› by auto
define h' where "h'=(λx. if x=z then z' else h x)"
have "h' holomorphic_on ball z r"
proof (rule no_isolated_singularity'[of "{z}"])
show "⋀w. w ∈ {z} ⟹ (h' ⤏ h' w) (at w within ball z r)"
by (simp add: LIM_cong Lim_at_imp_Lim_at_within ‹h ─z→ z'› h'_def)
show "h' holomorphic_on ball z r - {z}"
using r analytic_imp_holomorphic h'_def holomorphic_transform by fastforce
qed auto
have ?thesis when "z'=0"
proof -
have "h' z=0" using that unfolding h'_def by auto
moreover have "¬ h' constant_on ball z r"
using ‹∃⇩Fw in (at z). h w≠0› unfolding constant_on_def frequently_def eventually_at h'_def
by (metis ‹0 < r› centre_in_ball dist_commute mem_ball that)
moreover note ‹h' holomorphic_on ball z r›
ultimately obtain g r1 n where "0 < n" "0 < r1" "ball z r1 ⊆ ball z r" and
g:"g holomorphic_on ball z r1"
"⋀w. w ∈ ball z r1 ⟹ h' w = (w - z) ^ n * g w"
"⋀w. w ∈ ball z r1 ⟹ g w ≠ 0"
using holomorphic_factor_zero_nonconstant[of _ "ball z r" z thesis,simplified,
OF ‹h' holomorphic_on ball z r› ‹r>0› ‹h' z=0› ‹¬ h' constant_on ball z r›]
by (auto simp add:dist_commute)
define rr where "rr=r1/2"
have "P h' n g rr"
unfolding P_def rr_def
using ‹n>0› ‹r1>0› g by (auto simp add:powr_nat)
then have "P h n g rr"
unfolding h'_def P_def by auto
then show ?thesis unfolding P_def by blast
qed
moreover have ?thesis when "z'≠0"
proof -
have "h' z≠0" using that unfolding h'_def by auto
obtain r1 where "r1>0" "cball z r1 ⊆ ball z r" "∀x∈cball z r1. h' x≠0"
proof -
have "isCont h' z" "h' z≠0"
by (auto simp add: Lim_cong_within ‹h ─z→ z'› ‹z'≠0› continuous_at h'_def)
then obtain r2 where r2:"r2>0" "∀x∈ball z r2. h' x≠0"
using continuous_at_avoid[of z h' 0 ] unfolding ball_def by auto
define r1 where "r1=min r2 r / 2"
have "0 < r1" "cball z r1 ⊆ ball z r"
using ‹r2>0› ‹r>0› unfolding r1_def by auto
moreover have "∀x∈cball z r1. h' x ≠ 0"
using r2 unfolding r1_def by simp
ultimately show ?thesis using that by auto
qed
then have "P h' 0 h' r1" using ‹h' holomorphic_on ball z r› unfolding P_def by auto
then have "P h 0 h' r1" unfolding P_def h'_def by auto
then show ?thesis unfolding P_def by blast
qed
ultimately show ?thesis by auto
qed
have ?thesis when "∃x. (f ⤏ x) (at z)"
apply (rule_tac imp_unique[unfolded P_def])
using P_exist[OF that(1) f_iso non_zero] unfolding P_def .
moreover have ?thesis when "is_pole f z"
proof (rule imp_unique[unfolded P_def])
obtain e where [simp]:"e>0" and e_holo:"f holomorphic_on ball z e - {z}" and e_nz: "∀x∈ball z e-{z}. f x≠0"
proof -
have "∀⇩F z in at z. f z ≠ 0"
using ‹is_pole f z› filterlim_at_infinity_imp_eventually_ne unfolding is_pole_def
by auto
then obtain e1 where e1:"e1>0" "∀x∈ball z e1-{z}. f x≠0"
using that eventually_at[of "λx. f x≠0" z UNIV,simplified] by (auto simp add:dist_commute)
obtain e2 where e2:"e2>0" "f holomorphic_on ball z e2 - {z}"
using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by auto
show ?thesis
using e1 e2 by (force intro: that[of "min e1 e2"])
qed
define h where "h ≡ λx. inverse (f x)"
have "∃n g r. P h n g r"
proof -
have "(λx. inverse (f x)) analytic_on ball z e - {z}"
by (metis e_holo e_nz open_ball analytic_on_open holomorphic_on_inverse open_delete)
moreover have "h ─z→ 0"
using Lim_transform_within_open assms(2) h_def is_pole_tendsto that by fastforce
moreover have "∃⇩Fw in (at z). h w≠0"
using non_zero by (simp add: h_def)
ultimately show ?thesis
using P_exist[of h] ‹e > 0›
unfolding isolated_singularity_at_def h_def
by blast
qed
then obtain n g r
where "0 < r" and
g_holo:"g holomorphic_on cball z r" and "g z≠0" and
g_fac:"(∀w∈cball z r-{z}. h w = g w * (w - z) powi n ∧ g w ≠ 0)"
unfolding P_def by auto
have "P f (-n) (inverse o g) r"
proof -
have "f w = inverse (g w) * (w - z) powi (- n)" when "w∈cball z r - {z}" for w
by (metis g_fac h_def inverse_inverse_eq inverse_mult_distrib power_int_minus that)
then show ?thesis
unfolding P_def comp_def
using ‹r>0› g_holo g_fac ‹g z≠0› by (auto intro:holomorphic_intros)
qed
then show "∃x g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z ≠ 0
∧ (∀w∈cball z r - {z}. f w = g w * (w - z) powi x ∧ g w ≠ 0)"
unfolding P_def by blast
qed
ultimately show ?thesis using ‹not_essential f z› unfolding not_essential_def by presburger
qed
lemma not_essential_transform:
assumes "not_essential g z"
assumes "∀⇩F w in (at z). g w = f w"
shows "not_essential f z"
using assms unfolding not_essential_def
by (simp add: filterlim_cong is_pole_cong)
lemma isolated_singularity_at_transform:
assumes "isolated_singularity_at g z"
assumes "∀⇩F w in (at z). g w = f w"
shows "isolated_singularity_at f z"
using assms isolated_singularity_at_cong by blast
lemma not_essential_powr[singularity_intros]:
assumes "LIM w (at z). f w :> (at x)"
shows "not_essential (λw. (f w) powi n) z"
proof -
define fp where "fp=(λw. (f w) powi n)"
have ?thesis when "n>0"
proof -
have "(λw. (f w) ^ (nat n)) ─z→ x ^ nat n"
using that assms unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
then have "fp ─z→ x ^ nat n" unfolding fp_def
by (smt (verit) LIM_cong power_int_def that)
then show ?thesis unfolding not_essential_def fp_def by auto
qed
moreover have ?thesis when "n=0"
proof -
have "∀⇩F x in at z. fp x = 1"
using that filterlim_at_within_not_equal[OF assms] by (auto simp: fp_def)
then have "fp ─z→ 1"
by (simp add: tendsto_eventually)
then show ?thesis unfolding fp_def not_essential_def by auto
qed
moreover have ?thesis when "n<0"
proof (cases "x=0")
case True
have "(λx. f x ^ nat (- n)) ─z→ 0"
using assms True that unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
moreover have "∀⇩F x in at z. f x ^ nat (- n) ≠ 0"
by (smt (verit) True assms eventually_at_topological filterlim_at power_eq_0_iff)
ultimately have "LIM w (at z). inverse ((f w) ^ (nat (-n))) :> at_infinity"
by (metis filterlim_atI filterlim_compose filterlim_inverse_at_infinity)
then have "LIM w (at z). fp w :> at_infinity"
proof (elim filterlim_mono_eventually)
show "∀⇩F x in at z. inverse (f x ^ nat (- n)) = fp x"
using filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def
by (smt (verit) eventuallyI power_int_def power_inverse that)
qed auto
then show ?thesis unfolding fp_def not_essential_def is_pole_def by auto
next
case False
let ?xx= "inverse (x ^ (nat (-n)))"
have "(λw. inverse ((f w) ^ (nat (-n)))) ─z→?xx"
using assms False unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
then have "fp ─z→ ?xx"
by (smt (verit, best) LIM_cong fp_def power_int_def power_inverse that)
then show ?thesis unfolding fp_def not_essential_def by auto
qed
ultimately show ?thesis by linarith
qed
lemma isolated_singularity_at_powr[singularity_intros]:
assumes "isolated_singularity_at f z" "∀⇩F w in (at z). f w≠0"
shows "isolated_singularity_at (λw. (f w) powi n) z"
proof -
obtain r1 where "r1>0" "f analytic_on ball z r1 - {z}"
using assms(1) unfolding isolated_singularity_at_def by auto
then have r1:"f holomorphic_on ball z r1 - {z}"
using analytic_on_open[of "ball z r1-{z}" f] by blast
obtain r2 where "r2>0" and r2:"∀w. w ≠ z ∧ dist w z < r2 ⟶ f w ≠ 0"
using assms(2) unfolding eventually_at by auto
define r3 where "r3=min r1 r2"
have "(λw. (f w) powi n) holomorphic_on ball z r3 - {z}"
by (intro holomorphic_on_power_int) (use r1 r2 in ‹auto simp: dist_commute r3_def›)
moreover have "r3>0" unfolding r3_def using ‹0 < r1› ‹0 < r2› by linarith
ultimately show ?thesis
by (meson open_ball analytic_on_open isolated_singularity_at_def open_delete)
qed
lemma non_zero_neighbour:
assumes f_iso:"isolated_singularity_at f z"
and f_ness:"not_essential f z"
and f_nconst:"∃⇩Fw in (at z). f w≠0"
shows "∀⇩F w in (at z). f w≠0"
proof -
obtain fn fp fr where [simp]:"fp z ≠ 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
"∀w∈cball z fr - {z}. f w = fp w * (w - z) powi fn ∧ fp w ≠ 0"
using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
have "f w ≠ 0" when " w ≠ z" "dist w z < fr" for w
proof -
have "f w = fp w * (w - z) powi fn" "fp w ≠ 0"
using fr(2)[rule_format, of w] using that by (auto simp add:dist_commute)
moreover have "(w - z) powi fn ≠0"
unfolding powr_eq_0_iff using ‹w≠z› by auto
ultimately show ?thesis by auto
qed
then show ?thesis using ‹fr>0› unfolding eventually_at by auto
qed
lemma non_zero_neighbour_pole:
assumes "is_pole f z"
shows "∀⇩F w in (at z). f w≠0"
using assms filterlim_at_infinity_imp_eventually_ne[of f "at z" 0]
unfolding is_pole_def by auto
lemma non_zero_neighbour_alt:
assumes holo: "f holomorphic_on S"
and "open S" "connected S" "z ∈ S" "β ∈ S" "f β ≠ 0"
shows "∀⇩F w in (at z). f w≠0 ∧ w∈S"
proof (cases "f z = 0")
case True
from isolated_zeros[OF holo ‹open S› ‹connected S› ‹z ∈ S› True ‹β ∈ S› ‹f β ≠ 0›]
obtain r where "0 < r" "ball z r ⊆ S" "∀w ∈ ball z r - {z}.f w ≠ 0" by metis
then show ?thesis
by (smt (verit) open_ball centre_in_ball eventually_at_topological insertE insert_Diff subsetD)
next
case False
obtain r1 where r1:"r1>0" "∀y. dist z y < r1 ⟶ f y ≠ 0"
using continuous_at_avoid[of z f, OF _ False] assms(2,4) continuous_on_eq_continuous_at
holo holomorphic_on_imp_continuous_on by blast
obtain r2 where r2:"r2>0" "ball z r2 ⊆ S"
using assms openE by blast
show ?thesis unfolding eventually_at
by (metis (no_types) dist_commute dual_order.strict_trans linorder_less_linear mem_ball r1 r2 subsetD)
qed
lemma not_essential_times[singularity_intros]:
assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
shows "not_essential (λw. f w * g w) z"
proof -
define fg where "fg = (λw. f w * g w)"
have ?thesis when "¬ ((∃⇩Fw in (at z). f w≠0) ∧ (∃⇩Fw in (at z). g w≠0))"
proof -
have "∀⇩Fw in (at z). fg w=0"
using fg_def frequently_elim1 not_eventually that by fastforce
from tendsto_cong[OF this] have "fg ─z→0" by auto
then show ?thesis unfolding not_essential_def fg_def by auto
qed
moreover have ?thesis when f_nconst:"∃⇩Fw in (at z). f w≠0" and g_nconst:"∃⇩Fw in (at z). g w≠0"
proof -
obtain fn fp fr where [simp]:"fp z ≠ 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
"∀w∈cball z fr - {z}. f w = fp w * (w - z) powi fn ∧ fp w ≠ 0"
using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
obtain gn gp gr where [simp]:"gp z ≠ 0" and "gr > 0"
and gr: "gp holomorphic_on cball z gr"
"∀w∈cball z gr - {z}. g w = gp w * (w - z) powi gn ∧ gp w ≠ 0"
using holomorphic_factor_puncture[OF g_iso g_ness g_nconst,THEN ex1_implies_ex] by auto
define r1 where "r1=(min fr gr)"
have "r1>0" unfolding r1_def using ‹fr>0› ‹gr>0› by auto
have fg_times:"fg w = (fp w * gp w) * (w - z) powi (fn+gn)" and fgp_nz:"fp w*gp w≠0"
when "w∈ball z r1 - {z}" for w
proof -
have "f w = fp w * (w - z) powi fn" "fp w≠0"
using fr(2)[rule_format,of w] that unfolding r1_def by auto
moreover have "g w = gp w * (w - z) powi gn" "gp w ≠ 0"
using gr(2)[rule_format, of w] that unfolding r1_def by auto
ultimately show "fg w = (fp w * gp w) * (w - z) powi (fn+gn)" "fp w*gp w≠0"
using that unfolding fg_def by (auto simp add:power_int_add)
qed
have [intro]: "fp ─z→fp z" "gp ─z→gp z"
using fr(1) ‹fr>0› gr(1) ‹gr>0›
by (meson open_ball ball_subset_cball centre_in_ball
continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on
holomorphic_on_subset)+
have ?thesis when "fn+gn>0"
proof -
have "(λw. (fp w * gp w) * (w - z) ^ (nat (fn+gn))) ─z→0"
using that by (auto intro!:tendsto_eq_intros)
then have "fg ─z→ 0"
apply (elim Lim_transform_within[OF _ ‹r1>0›])
by (smt (verit, best) Diff_iff dist_commute fg_times mem_ball power_int_def singletonD that zero_less_dist_iff)
then show ?thesis unfolding not_essential_def fg_def by auto
qed
moreover have ?thesis when "fn+gn=0"
proof -
have "(λw. fp w * gp w) ─z→fp z*gp z"
using that by (auto intro!:tendsto_eq_intros)
then have "fg ─z→ fp z*gp z"
apply (elim Lim_transform_within[OF _ ‹r1>0›])
apply (subst fg_times)
by (auto simp add:dist_commute that)
then show ?thesis unfolding not_essential_def fg_def by auto
qed
moreover have ?thesis when "fn+gn<0"
proof -
have "LIM x at z. (x - z) ^ nat (- (fn + gn)) :> at 0"
using eventually_at_topological that
by (force intro!: tendsto_eq_intros filterlim_atI)
moreover have "∃c. (λc. fp c * gp c) ─z→ c ∧ 0 ≠ c"
using ‹fp ─z→ fp z› ‹gp ─z→ gp z› tendsto_mult by fastforce
ultimately have "LIM w (at z). fp w * gp w / (w-z)^nat (-(fn+gn)) :> at_infinity"
using filterlim_divide_at_infinity by blast
then have "is_pole fg z" unfolding is_pole_def
apply (elim filterlim_transform_within[OF _ _ ‹r1>0›])
using that
by (simp_all add: dist_commute fg_times of_int_of_nat divide_simps power_int_def del: minus_add_distrib)
then show ?thesis unfolding not_essential_def fg_def by auto
qed
ultimately show ?thesis unfolding not_essential_def fg_def by fastforce
qed
ultimately show ?thesis by auto
qed
lemma not_essential_inverse[singularity_intros]:
assumes f_ness:"not_essential f z"
assumes f_iso:"isolated_singularity_at f z"
shows "not_essential (λw. inverse (f w)) z"
proof -
define vf where "vf = (λw. inverse (f w))"
have ?thesis when "¬(∃⇩Fw in (at z). f w≠0)"
proof -
have "∀⇩Fw in (at z). f w=0"
using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
then have "vf ─z→0"
unfolding vf_def by (simp add: tendsto_eventually)
then show ?thesis unfolding not_essential_def vf_def by auto
qed
moreover have ?thesis when "is_pole f z"
proof -
have "vf ─z→0"
using that filterlim_at filterlim_inverse_at_iff unfolding is_pole_def vf_def by blast
then show ?thesis unfolding not_essential_def vf_def by auto
qed
moreover have ?thesis when "∃x. f─z→x " and f_nconst:"∃⇩Fw in (at z). f w≠0"
proof -
from that obtain fz where fz:"f─z→fz" by auto
have ?thesis when "fz=0"
proof -
have "(λw. inverse (vf w)) ─z→0"
using fz that unfolding vf_def by auto
moreover have "∀⇩F w in at z. inverse (vf w) ≠ 0"
using non_zero_neighbour[OF f_iso f_ness f_nconst]
unfolding vf_def by auto
ultimately show ?thesis unfolding not_essential_def vf_def
using filterlim_atI filterlim_inverse_at_iff is_pole_def by blast
qed
moreover have ?thesis when "fz≠0"
using fz not_essential_def tendsto_inverse that by blast
ultimately show ?thesis by auto
qed
ultimately show ?thesis using f_ness unfolding not_essential_def by auto
qed
lemma isolated_singularity_at_inverse[singularity_intros]:
assumes f_iso:"isolated_singularity_at f z"
and f_ness:"not_essential f z"
shows "isolated_singularity_at (λw. inverse (f w)) z"
proof -
define vf where "vf = (λw. inverse (f w))"
have ?thesis when "¬(∃⇩Fw in (at z). f w≠0)"
proof -
have "∀⇩Fw in (at z). f w=0"
using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
then have "∀⇩Fw in (at z). vf w=0"
unfolding vf_def by auto
then obtain d1 where "d1>0" and d1:"∀x. x ≠ z ∧ dist x z < d1 ⟶ vf x = 0"
unfolding eventually_at by auto
then have "vf holomorphic_on ball z d1-{z}"
apply (rule_tac holomorphic_transform[of "λ_. 0"])
by (auto simp add:dist_commute)
then have "vf analytic_on ball z d1 - {z}"
by (simp add: analytic_on_open open_delete)
then show ?thesis using ‹d1>0› unfolding isolated_singularity_at_def vf_def by auto
qed
moreover have ?thesis when f_nconst:"∃⇩Fw in (at z). f w≠0"
proof -
have "∀⇩F w in at z. f w ≠ 0" using non_zero_neighbour[OF f_iso f_ness f_nconst] .
then obtain d1 where d1:"d1>0" "∀x. x ≠ z ∧ dist x z < d1 ⟶ f x ≠ 0"
unfolding eventually_at by auto
obtain d2 where "d2>0" and d2:"f analytic_on ball z d2 - {z}"
using f_iso unfolding isolated_singularity_at_def by auto
define d3 where "d3=min d1 d2"
have "d3>0" unfolding d3_def using ‹d1>0› ‹d2>0› by auto
moreover
have "f analytic_on ball z d3 - {z}"
by (smt (verit, best) Diff_iff analytic_on_analytic_at d2 d3_def mem_ball)
then have "vf analytic_on ball z d3 - {z}"
unfolding vf_def
by (intro analytic_on_inverse; simp add: d1(2) d3_def dist_commute)
ultimately show ?thesis unfolding isolated_singularity_at_def vf_def by auto
qed
ultimately show ?thesis by auto
qed
lemma not_essential_divide[singularity_intros]:
assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
shows "not_essential (λw. f w / g w) z"
proof -
have "not_essential (λw. f w * inverse (g w)) z"
by (simp add: f_iso f_ness g_iso g_ness isolated_singularity_at_inverse not_essential_inverse not_essential_times)
then show ?thesis by (simp add:field_simps)
qed
lemma
assumes f_iso:"isolated_singularity_at f z"
and g_iso:"isolated_singularity_at g z"
shows isolated_singularity_at_times[singularity_intros]:
"isolated_singularity_at (λw. f w * g w) z"
and isolated_singularity_at_add[singularity_intros]:
"isolated_singularity_at (λw. f w + g w) z"
proof -
obtain d1 d2 where "d1>0" "d2>0"
and d1:"f analytic_on ball z d1 - {z}" and d2:"g analytic_on ball z d2 - {z}"
using f_iso g_iso unfolding isolated_singularity_at_def by auto
define d3 where "d3=min d1 d2"
have "d3>0" unfolding d3_def using ‹d1>0› ‹d2>0› by auto
have fan: "f analytic_on ball z d3 - {z}"
by (smt (verit, best) Diff_iff analytic_on_analytic_at d1 d3_def mem_ball)
have gan: "g analytic_on ball z d3 - {z}"
by (smt (verit, best) Diff_iff analytic_on_analytic_at d2 d3_def mem_ball)
have "(λw. f w * g w) analytic_on ball z d3 - {z}"
using analytic_on_mult fan gan by blast
then show "isolated_singularity_at (λw. f w * g w) z"
using ‹d3>0› unfolding isolated_singularity_at_def by auto
have "(λw. f w + g w) analytic_on ball z d3 - {z}"
using analytic_on_add fan gan by blast
then show "isolated_singularity_at (λw. f w + g w) z"
using ‹d3>0› unfolding isolated_singularity_at_def by auto
qed
lemma isolated_singularity_at_uminus[singularity_intros]:
assumes f_iso:"isolated_singularity_at f z"
shows "isolated_singularity_at (λw. - f w) z"
using assms unfolding isolated_singularity_at_def using analytic_on_neg by blast
lemma isolated_singularity_at_id[singularity_intros]:
"isolated_singularity_at (λw. w) z"
unfolding isolated_singularity_at_def by (simp add: gt_ex)
lemma isolated_singularity_at_minus[singularity_intros]:
assumes "isolated_singularity_at f z" and "isolated_singularity_at g z"
shows "isolated_singularity_at (λw. f w - g w) z"
unfolding diff_conv_add_uminus
using assms isolated_singularity_at_add isolated_singularity_at_uminus by blast
lemma isolated_singularity_at_divide[singularity_intros]:
assumes "isolated_singularity_at f z"
and "isolated_singularity_at g z"
and "not_essential g z"
shows "isolated_singularity_at (λw. f w / g w) z"
unfolding divide_inverse
by (simp add: assms isolated_singularity_at_inverse isolated_singularity_at_times)
lemma isolated_singularity_at_const[singularity_intros]:
"isolated_singularity_at (λw. c) z"
unfolding isolated_singularity_at_def by (simp add: gt_ex)
lemma isolated_singularity_at_holomorphic:
assumes "f holomorphic_on s-{z}" "open s" "z∈s"
shows "isolated_singularity_at f z"
using assms unfolding isolated_singularity_at_def
by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
lemma isolated_singularity_at_altdef:
"isolated_singularity_at f z ⟷ eventually (λz. f analytic_on {z}) (at z)"
proof
assume "isolated_singularity_at f z"
then obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
unfolding isolated_singularity_at_def by blast
have "eventually (λw. w ∈ ball z r - {z}) (at z)"
using r(1) by (intro eventually_at_in_open) auto
thus "eventually (λz. f analytic_on {z}) (at z)"
by eventually_elim (use r analytic_on_subset in auto)
next
assume "eventually (λz. f analytic_on {z}) (at z)"
then obtain A where A: "open A" "z ∈ A" "⋀w. w ∈ A - {z} ⟹ f analytic_on {w}"
unfolding eventually_at_topological by blast
then show "isolated_singularity_at f z"
by (meson analytic_imp_holomorphic analytic_on_analytic_at isolated_singularity_at_holomorphic)
qed
lemma isolated_singularity_at_shift:
assumes "isolated_singularity_at (λx. f (x + w)) z"
shows "isolated_singularity_at f (z + w)"
proof -
from assms obtain r where r: "r > 0" and ana: "(λx. f (x + w)) analytic_on ball z r - {z}"
unfolding isolated_singularity_at_def by blast
have "((λx. f (x + w)) ∘ (λx. x - w)) analytic_on (ball (z + w) r - {z + w})"
by (rule analytic_on_compose_gen[OF _ ana])
(auto simp: dist_norm algebra_simps intro!: analytic_intros)
hence "f analytic_on (ball (z + w) r - {z + w})"
by (simp add: o_def)
thus ?thesis using r
unfolding isolated_singularity_at_def by blast
qed
lemma isolated_singularity_at_shift_iff:
"isolated_singularity_at f (z + w) ⟷ isolated_singularity_at (λx. f (x + w)) z"
using isolated_singularity_at_shift[of f w z]
isolated_singularity_at_shift[of "λx. f (x + w)" "-w" "w + z"]
by (auto simp: algebra_simps)
lemma isolated_singularity_at_shift_0:
"NO_MATCH 0 z ⟹ isolated_singularity_at f z ⟷ isolated_singularity_at (λx. f (z + x)) 0"
using isolated_singularity_at_shift_iff[of f 0 z] by (simp add: add_ac)
lemma not_essential_shift:
assumes "not_essential (λx. f (x + w)) z"
shows "not_essential f (z + w)"
proof -
from assms consider c where "(λx. f (x + w)) ─z→ c" | "is_pole (λx. f (x + w)) z"
unfolding not_essential_def by blast
thus ?thesis
proof cases
case (1 c)
hence "f ─z + w→ c"
by (smt (verit, ccfv_SIG) LIM_cong add.assoc filterlim_at_to_0)
thus ?thesis
by (auto simp: not_essential_def)
next
case 2
hence "is_pole f (z + w)"
by (subst is_pole_shift_iff [symmetric]) (auto simp: o_def add_ac)
thus ?thesis
by (auto simp: not_essential_def)
qed
qed
lemma not_essential_shift_iff: "not_essential f (z + w) ⟷ not_essential (λx. f (x + w)) z"
using not_essential_shift[of f w z]
not_essential_shift[of "λx. f (x + w)" "-w" "w + z"]
by (auto simp: algebra_simps)
lemma not_essential_shift_0:
"NO_MATCH 0 z ⟹ not_essential f z ⟷ not_essential (λx. f (z + x)) 0"
using not_essential_shift_iff[of f 0 z] by (simp add: add_ac)
lemma not_essential_holomorphic:
assumes "f holomorphic_on A" "x ∈ A" "open A"
shows "not_essential f x"
by (metis assms at_within_open continuous_on holomorphic_on_imp_continuous_on not_essential_def)
lemma not_essential_analytic:
assumes "f analytic_on {z}"
shows "not_essential f z"
using analytic_at assms not_essential_holomorphic by blast
lemma not_essential_id [singularity_intros]: "not_essential (λw. w) z"
by (simp add: not_essential_analytic)
lemma is_pole_imp_not_essential [intro]: "is_pole f z ⟹ not_essential f z"
by (auto simp: not_essential_def)
lemma tendsto_imp_not_essential [intro]: "f ─z→ c ⟹ not_essential f z"
by (auto simp: not_essential_def)
lemma eventually_not_pole:
assumes "isolated_singularity_at f z"
shows "eventually (λw. ¬is_pole f w) (at z)"
proof -
from assms obtain r where "r > 0" and r: "f analytic_on ball z r - {z}"
by (auto simp: isolated_singularity_at_def)
then have "eventually (λw. w ∈ ball z r - {z}) (at z)"
by (intro eventually_at_in_open) auto
thus "eventually (λw. ¬is_pole f w) (at z)"
by (metis (no_types, lifting) analytic_at analytic_on_analytic_at eventually_mono not_is_pole_holomorphic r)
qed
lemma not_islimpt_poles:
assumes "isolated_singularity_at f z"
shows "¬z islimpt {w. is_pole f w}"
using eventually_not_pole [OF assms]
by (auto simp: islimpt_conv_frequently_at frequently_def)
lemma analytic_at_imp_no_pole: "f analytic_on {z} ⟹ ¬is_pole f z"
using analytic_at not_is_pole_holomorphic by blast
lemma not_essential_const [singularity_intros]: "not_essential (λ_. c) z"
by blast
lemma not_essential_uminus [singularity_intros]:
assumes f_ness: "not_essential f z"
assumes f_iso:"isolated_singularity_at f z"
shows "not_essential (λw. -f w) z"
proof -
have "not_essential (λw. -1 * f w) z"
by (intro assms singularity_intros)
thus ?thesis by simp
qed
lemma isolated_singularity_at_analytic:
assumes "f analytic_on {z}"
shows "isolated_singularity_at f z"
by (meson Diff_subset analytic_at assms holomorphic_on_subset isolated_singularity_at_holomorphic)
subsection ‹The order of non-essential singularities (i.e. removable singularities or poles)›
definition zorder :: "(complex ⇒ complex) ⇒ complex ⇒ int" where
"zorder f z = (THE n. (∃h r. r>0 ∧ h holomorphic_on cball z r ∧ h z≠0
∧ (∀w∈cball z r - {z}. f w = h w * (w-z) powi n
∧ h w ≠0)))"
definition zor_poly
::"[complex ⇒ complex, complex] ⇒ complex ⇒ complex" where
"zor_poly f z = (SOME h. ∃r. r > 0 ∧ h holomorphic_on cball z r ∧ h z ≠ 0
∧ (∀w∈cball z r - {z}. f w = h w * (w - z) powi (zorder f z)
∧ h w ≠0))"
lemma zorder_exist:
fixes f::"complex ⇒ complex" and z::complex
defines "n ≡ zorder f z" and "g ≡ zor_poly f z"
assumes f_iso:"isolated_singularity_at f z"
and f_ness:"not_essential f z"
and f_nconst:"∃⇩Fw in (at z). f w≠0"
shows "g z≠0 ∧ (∃r. r>0 ∧ g holomorphic_on cball z r
∧ (∀w∈cball z r - {z}. f w = g w * (w-z) powi n ∧ g w ≠0))"
proof -
define P where "P = (λn g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z≠0
∧ (∀w∈cball z r - {z}. f w = g w * (w-z) powi n ∧ g w≠0))"
have "∃!k. ∃g r. P k g r"
using holomorphic_factor_puncture[OF assms(3-)] unfolding P_def by auto
then have "∃g r. P n g r"
unfolding n_def P_def zorder_def
by (drule_tac theI',argo)
then have "∃r. P n g r"
unfolding P_def zor_poly_def g_def n_def
by (drule_tac someI_ex,argo)
then obtain r1 where "P n g r1" by auto
then show ?thesis unfolding P_def by auto
qed
lemma zorder_shift:
shows "zorder f z = zorder (λu. f (u + z)) 0"
unfolding zorder_def
apply (rule arg_cong [of concl: The])
apply (auto simp: fun_eq_iff Ball_def dist_norm)
subgoal for x h r
apply (rule_tac x="h o (+)z" in exI)
apply (rule_tac x="r" in exI)
apply (intro conjI holomorphic_on_compose holomorphic_intros)
apply (simp_all flip: cball_translation)
apply (simp add: add.commute)
done
subgoal for x h r
apply (rule_tac x="h o (λu. u-z)" in exI)
apply (rule_tac x="r" in exI)
apply (intro conjI holomorphic_on_compose holomorphic_intros)
apply (simp_all add: flip: cball_translation_subtract)
by (metis diff_add_cancel eq_iff_diff_eq_0 norm_minus_commute)
done
lemma zorder_shift': "NO_MATCH 0 z ⟹ zorder f z = zorder (λu. f (u + z)) 0"
by (rule zorder_shift)
lemma
fixes f::"complex ⇒ complex" and z::complex
assumes f_iso:"isolated_singularity_at f z"
and f_ness:"not_essential f z"
and f_nconst:"∃⇩Fw in (at z). f w≠0"
shows zorder_inverse: "zorder (λw. inverse (f w)) z = - zorder f z"
and zor_poly_inverse: "∀⇩Fw in (at z). zor_poly (λw. inverse (f w)) z w
= inverse (zor_poly f z w)"
proof -
define vf where "vf = (λw. inverse (f w))"
define fn vfn where
"fn = zorder f z" and "vfn = zorder vf z"
define fp vfp where
"fp = zor_poly f z" and "vfp = zor_poly vf z"
obtain fr where [simp]:"fp z ≠ 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
"∀w∈cball z fr - {z}. f w = fp w * (w - z) powi fn ∧ fp w ≠ 0"
using zorder_exist[OF f_iso f_ness f_nconst,folded fn_def fp_def]
by auto
have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powi (-fn)"
and fr_nz: "inverse (fp w) ≠ 0"
when "w∈ball z fr - {z}" for w
proof -
have "f w = fp w * (w - z) powi fn" "fp w ≠ 0"
using fr(2) that by auto
then show "vf w = (inverse (fp w)) * (w - z) powi (-fn)" "inverse (fp w)≠0"
by (simp_all add: power_int_minus vf_def)
qed
obtain vfr where [simp]:"vfp z ≠ 0" and "vfr>0" and vfr:"vfp holomorphic_on cball z vfr"
"(∀w∈cball z vfr - {z}. vf w = vfp w * (w - z) powi vfn ∧ vfp w ≠ 0)"
proof -
have "isolated_singularity_at vf z"
using isolated_singularity_at_inverse[OF f_iso f_ness] unfolding vf_def .
moreover have "not_essential vf z"
using not_essential_inverse[OF f_ness f_iso] unfolding vf_def .
moreover have "∃⇩F w in at z. vf w ≠ 0"
using f_nconst unfolding vf_def by (auto elim:frequently_elim1)
ultimately show ?thesis using zorder_exist[of vf z, folded vfn_def vfp_def] that by auto
qed
define r1 where "r1 = min fr vfr"
have "r1>0" using ‹fr>0› ‹vfr>0› unfolding r1_def by simp
show "vfn = - fn"
proof (rule holomorphic_factor_unique)
have §: "⋀w. ⟦fp w = 0; dist z w < fr⟧ ⟹ False"
using fr_nz by force
then show "∀w∈ball z r1 - {z}.
vf w = vfp w * (w - z) powi vfn ∧
vfp w ≠ 0 ∧ vf w = inverse (fp w) * (w - z) powi (- fn) ∧
inverse (fp w) ≠ 0"
using fr_inverse r1_def vfr(2)
by (smt (verit) Diff_iff inverse_nonzero_iff_nonzero mem_ball mem_cball)
show "vfp holomorphic_on ball z r1"
using r1_def vfr(1) by auto
show "(λw. inverse (fp w)) holomorphic_on ball z r1"
by (metis § ball_subset_cball fr(1) holomorphic_on_inverse holomorphic_on_subset mem_ball min.cobounded2 min.commute r1_def subset_ball)
qed (use ‹r1>0› in auto)
have "vfp w = inverse (fp w)" when "w∈ball z r1-{z}" for w
proof -
have "w ∈ ball z fr - {z}" "w ∈ cball z vfr - {z}" "w≠z" using that unfolding r1_def by auto
from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)] ‹vfn = - fn› ‹w≠z›
show ?thesis by auto
qed
then show "∀⇩Fw in (at z). vfp w = inverse (fp w)"
unfolding eventually_at using ‹r1>0›
by (metis DiffI dist_commute mem_ball singletonD)
qed
lemma zor_poly_shift:
assumes iso1: "isolated_singularity_at f z"
and ness1: "not_essential f z"
and nzero1: "∃⇩F w in at z. f w ≠ 0"
shows "∀⇩F w in nhds z. zor_poly f z w = zor_poly (λu. f (z + u)) 0 (w-z)"
proof -
obtain r1 where "r1>0" "zor_poly f z z ≠ 0" and
holo1:"zor_poly f z holomorphic_on cball z r1" and
rball1:"∀w∈cball z r1 - {z}.
f w = zor_poly f z w * (w - z) powi (zorder f z) ∧
zor_poly f z w ≠ 0"
using zorder_exist[OF iso1 ness1 nzero1] by blast
define ff where "ff=(λu. f (z + u))"
have "isolated_singularity_at ff 0"
unfolding ff_def
using iso1 isolated_singularity_at_shift_iff[of f 0 z]
by (simp add:algebra_simps)
moreover have "not_essential ff 0"
unfolding ff_def
using ness1 not_essential_shift_iff[of f 0 z]
by (simp add:algebra_simps)
moreover have "∃⇩F w in at 0. ff w ≠ 0"
unfolding ff_def using nzero1
by (smt (verit, ccfv_SIG) add.commute eventually_at_to_0
eventually_mono not_frequently)
ultimately obtain r2 where "r2>0" "zor_poly ff 0 0 ≠ 0" and
holo2:"zor_poly ff 0 holomorphic_on cball 0 r2" and
rball2:"∀w∈cball 0 r2 - {0}.
ff w = zor_poly ff 0 w * w powi (zorder ff 0) ∧
zor_poly ff 0 w ≠ 0"
using zorder_exist[of ff 0] by auto
define r where "r=min r1 r2"
have "r>0" using ‹r1>0› ‹r2>0› unfolding r_def by auto
have "zor_poly f z w = zor_poly ff 0 (w - z)"
if "w∈ball z r - {z}" for w
proof -
define n where "n ≡ zorder f z"
have "f w = zor_poly f z w * (w - z) powi n"
using n_def r_def rball1 that by auto
moreover have "f w = zor_poly ff 0 (w - z) * (w - z) powi n"
proof -
have "w-z∈cball 0 r2 - {0}"
using r_def that by (auto simp:dist_complex_def)
from rball2[rule_format, OF this]
have "ff (w - z) = zor_poly ff 0 (w - z) * (w - z)
powi (zorder ff 0)"
by simp
moreover have "of_int (zorder ff 0) = n"
unfolding n_def ff_def by (simp add:zorder_shift' add.commute)
ultimately show ?thesis unfolding ff_def by auto
qed
ultimately have "zor_poly f z w * (w - z) powi n
= zor_poly ff 0 (w - z) * (w - z) powi n"
by auto
moreover have "(w - z) powi n ≠0"
using that by auto
ultimately show ?thesis
using mult_cancel_right by blast
qed
then have "∀⇩F w in at z. zor_poly f z w
= zor_poly ff 0 (w - z)"
unfolding eventually_at
by (metis DiffI ‹0 < r› dist_commute mem_ball singletonD)
moreover have "isCont (zor_poly f z) z"
using holo1[THEN holomorphic_on_imp_continuous_on]
by (simp add: ‹0 < r1› continuous_on_interior)
moreover
have "isCont (zor_poly ff 0) 0"
using ‹0 < r2› centre_in_ball continuous_on_interior holo2 holomorphic_on_imp_continuous_on interior_cball by blast
then have "isCont (λw. zor_poly ff 0 (w - z)) z"
unfolding isCont_iff by simp
ultimately show "∀⇩F w in nhds z. zor_poly f z w = zor_poly ff 0 (w - z)"
by (elim at_within_isCont_imp_nhds;auto)
qed
lemma
fixes f g::"complex ⇒ complex" and z::complex
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
and f_ness:"not_essential f z" and g_ness:"not_essential g z"
and fg_nconst: "∃⇩Fw in (at z). f w * g w≠ 0"
shows zorder_times:"zorder (λw. f w * g w) z = zorder f z + zorder g z" and
zor_poly_times:"∀⇩Fw in (at z). zor_poly (λw. f w * g w) z w
= zor_poly f z w *zor_poly g z w"
proof -
define fg where "fg = (λw. f w * g w)"
define fn gn fgn where
"fn = zorder f z" and "gn = zorder g z" and "fgn = zorder fg z"
define fp gp fgp where
"fp = zor_poly f z" and "gp = zor_poly g z" and "fgp = zor_poly fg z"
have f_nconst:"∃⇩Fw in (at z). f w ≠ 0" and g_nconst:"∃⇩Fw in (at z).g w≠ 0"
using fg_nconst by (auto elim!:frequently_elim1)
obtain fr where [simp]:"fp z ≠ 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
"∀w∈cball z fr - {z}. f w = fp w * (w - z) powi fn ∧ fp w ≠ 0"
using zorder_exist[OF f_iso f_ness f_nconst,folded fp_def fn_def] by auto
obtain gr where [simp]:"gp z ≠ 0" and "gr > 0"
and gr: "gp holomorphic_on cball z gr"
"∀w∈cball z gr - {z}. g w = gp w * (w - z) powi gn ∧ gp w ≠ 0"
using zorder_exist[OF g_iso g_ness g_nconst,folded gn_def gp_def] by auto
define r1 where "r1=min fr gr"
have "r1>0" unfolding r1_def using ‹fr>0› ‹gr>0› by auto
have fg_times:"fg w = (fp w * gp w) * (w - z) powi (fn+gn)" and fgp_nz:"fp w*gp w≠0"
when "w∈ball z r1 - {z}" for w
proof -
have "f w = fp w * (w - z) powi fn" "fp w ≠ 0"
using fr(2)[rule_format,of w] that unfolding r1_def by auto
moreover have "g w = gp w * (w - z) powi gn" "gp w ≠ 0"
using gr(2) that unfolding r1_def by auto
ultimately show "fg w = (fp w * gp w) * (w - z) powi (fn+gn)" "fp w*gp w≠0"
using that unfolding fg_def by (auto simp add:power_int_add)
qed
obtain fgr where [simp]:"fgp z ≠ 0" and "fgr > 0"
and fgr: "fgp holomorphic_on cball z fgr"
"∀w∈cball z fgr - {z}. fg w = fgp w * (w - z) powi fgn ∧ fgp w ≠ 0"
proof -
have "isolated_singularity_at fg z"
unfolding fg_def using isolated_singularity_at_times[OF f_iso g_iso] .
moreover have "not_essential fg z"
by (simp add: f_iso f_ness fg_def g_iso g_ness not_essential_times)
moreover have "∃⇩F w in at z. fg w ≠ 0"
using fg_def fg_nconst by blast
ultimately show ?thesis
using that zorder_exist[of fg z] fgn_def fgp_def by fastforce
qed
define r2 where "r2 = min fgr r1"
have "r2>0" using ‹r1>0› ‹fgr>0› unfolding r2_def by simp
show "fgn = fn + gn "
apply (rule holomorphic_factor_unique[of r2 fgp z "λw. fp w * gp w" fg])
subgoal using ‹r2>0› by simp
subgoal by simp
subgoal by simp
subgoal
proof (rule ballI)
fix w assume "w ∈ ball z r2 - {z}"
then have "w ∈ ball z r1 - {z}" "w ∈ cball z fgr - {z}" unfolding r2_def by auto
then show "fg w = fgp w * (w - z) powi fgn ∧ fgp w ≠ 0
∧ fg w = fp w * gp w * (w - z) powi (fn + gn) ∧ fp w * gp w ≠ 0"
using fg_times fgp_nz fgr(2) by blast
qed
subgoal using fgr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
subgoal using fr(1) gr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
done
have "fgp w = fp w *gp w" when "w∈ball z r2-{z}" for w
proof -
have "w ∈ ball z r1 - {z}" "w ∈ cball z fgr - {z}" "w≠z" using that unfolding r2_def by auto
from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] ‹fgn = fn + gn› ‹w≠z›
show ?thesis by auto
qed
then show "∀⇩Fw in (at z). fgp w = fp w * gp w"
using ‹r2>0› unfolding eventually_at by (auto simp add:dist_commute)
qed
lemma
fixes f g::"complex ⇒ complex" and z::complex
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
and f_ness:"not_essential f z" and g_ness:"not_essential g z"
and fg_nconst: "∃⇩Fw in (at z). f w * g w≠ 0"
shows zorder_divide:"zorder (λw. f w / g w) z = zorder f z - zorder g z" and
zor_poly_divide:"∀⇩Fw in (at z). zor_poly (λw. f w / g w) z w
= zor_poly f z w / zor_poly g z w"
proof -
have f_nconst:"∃⇩Fw in (at z). f w ≠ 0" and g_nconst:"∃⇩Fw in (at z).g w≠ 0"
using fg_nconst by (auto elim!:frequently_elim1)
define vg where "vg=(λw. inverse (g w))"
have 1: "isolated_singularity_at vg z"
by (simp add: g_iso g_ness isolated_singularity_at_inverse vg_def)
moreover have 2: "not_essential vg z"
by (simp add: g_iso g_ness not_essential_inverse vg_def)
moreover have 3: "∃⇩F w in at z. f w * vg w ≠ 0"
using fg_nconst vg_def by auto
ultimately have "zorder (λw. f w * vg w) z = zorder f z + zorder vg z"
using zorder_times[OF f_iso _ f_ness] by blast
then show "zorder (λw. f w / g w) z = zorder f z - zorder g z"
using zorder_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
by (auto simp add:field_simps)
have "∀⇩F w in at z. zor_poly (λw. f w * vg w) z w = zor_poly f z w * zor_poly vg z w"
using zor_poly_times[OF f_iso _ f_ness,of vg] 1 2 3 by blast
then show "∀⇩Fw in (at z). zor_poly (λw. f w / g w) z w = zor_poly f z w / zor_poly g z w"
using zor_poly_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
by eventually_elim (auto simp add:field_simps)
qed
lemma zorder_exist_zero:
fixes f::"complex ⇒ complex" and z::complex
defines "n≡zorder f z" and "g≡zor_poly f z"
assumes holo: "f holomorphic_on s" and
"open s" "connected s" "z∈s"
and non_const: "∃w∈s. f w ≠ 0"
shows "(if f z=0 then n > 0 else n=0) ∧ (∃r. r>0 ∧ cball z r ⊆ s ∧ g holomorphic_on cball z r
∧ (∀w∈cball z r. f w = g w * (w-z) ^ nat n ∧ g w ≠0))"
proof -
obtain r where "g z ≠ 0" and r: "r>0" "cball z r ⊆ s" "g holomorphic_on cball z r"
"(∀w∈cball z r - {z}. f w = g w * (w - z) powi n ∧ g w ≠ 0)"
proof -
have "g z ≠ 0 ∧ (∃r>0. g holomorphic_on cball z r
∧ (∀w∈cball z r - {z}. f w = g w * (w - z) powi n ∧ g w ≠ 0))"
proof (rule zorder_exist[of f z,folded g_def n_def])
show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
using holo assms(4,6)
by (meson Diff_subset open_ball analytic_on_holomorphic holomorphic_on_subset openE)
show "not_essential f z" unfolding not_essential_def
using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
by fastforce
have "∀⇩F w in at z. f w ≠ 0 ∧ w∈s"
proof -
obtain w where "w∈s" "f w≠0" using non_const by auto
then show ?thesis
by (rule non_zero_neighbour_alt[OF holo ‹open s› ‹connected s› ‹z∈s›])
qed
then show "∃⇩F w in at z. f w ≠ 0"
by (auto elim: eventually_frequentlyE)
qed
then obtain r1 where "g z ≠ 0" "r1>0" and r1:"g holomorphic_on cball z r1"
"(∀w∈cball z r1 - {z}. f w = g w * (w - z) powi n ∧ g w ≠ 0)"
by auto
obtain r2 where r2: "r2>0" "cball z r2 ⊆ s"
using assms(4,6) open_contains_cball_eq by blast
define r3 where "r3 ≡ min r1 r2"
have "r3>0" "cball z r3 ⊆ s" using ‹r1>0› r2 unfolding r3_def by auto
moreover have "g holomorphic_on cball z r3"
using r1(1) unfolding r3_def by auto
moreover have "(∀w∈cball z r3 - {z}. f w = g w * (w - z) powi n ∧ g w ≠ 0)"
using r1(2) unfolding r3_def by auto
ultimately show ?thesis using that[of r3] ‹g z≠0› by auto
qed
have fz_lim: "f─ z → f z"
by (metis assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on)
have gz_lim: "g ─z→g z"
using r
by (meson Elementary_Metric_Spaces.open_ball analytic_at analytic_at_imp_isCont
ball_subset_cball centre_in_ball holomorphic_on_subset isContD)
have if_0:"if f z=0 then n > 0 else n=0"
proof -
have "(λw. g w * (w - z) powi n) ─z→ f z"
using fz_lim Lim_transform_within_open[where s="ball z r"] r by fastforce
then have "(λw. (g w * (w - z) powi n) / g w) ─z→ f z/g z"
using gz_lim ‹g z ≠ 0› tendsto_divide by blast
then have powi_tendsto:"(λw. (w - z) powi n) ─z→ f z/g z"
using Lim_transform_within_open[where s="ball z r"] r by fastforce
have ?thesis when "n≥0" "f z=0"
proof -
have "(λw. (w - z) ^ nat n) ─z→ f z/g z"
using Lim_transform_within[OF powi_tendsto, where d=r]
by (meson power_int_def r(1) that(1))
then have *:"(λw. (w - z) ^ nat n) ─z→ 0" using ‹f z=0› by simp
moreover have False when "n=0"
proof -
have "(λw. (w - z) ^ nat n) ─z→ 1"
using ‹n=0› by auto
then show False using * using LIM_unique zero_neq_one by blast
qed
ultimately show ?thesis using that by fastforce
qed
moreover have ?thesis when "n≥0" "f z≠0"
proof -
have False when "n>0"
proof -
have "(λw. (w - z) ^ nat n) ─z→ f z/g z"
using Lim_transform_within[OF powi_tendsto, where d=r]
by (meson ‹0 ≤ n› power_int_def r(1))
moreover have "(λw. (w - z) ^ nat n) ─z→ 0"
using ‹n>0› by (auto intro!:tendsto_eq_intros)
ultimately show False using ‹f z≠0› ‹g z≠0› using LIM_unique divide_eq_0_iff by blast
qed
then show ?thesis using that by force
qed
moreover have False when "n<0"
proof -
have "(λw. inverse ((w - z) ^ nat (- n))) ─z→ f z/g z"
by (smt (verit) LIM_cong power_int_def power_inverse powi_tendsto that)
moreover
have "(λw.((w - z) ^ nat (- n))) ─z→ 0"
using that by (auto intro!:tendsto_eq_intros)
ultimately
have "(λx. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) ─z→ 0"
using tendsto_mult by fastforce
then have "(λx. 1::complex) ─z→ 0"
using Lim_transform_within_open by fastforce
then show False using LIM_const_eq by fastforce
qed
ultimately show ?thesis by fastforce
qed
moreover have "f w = g w * (w-z) ^ nat n ∧ g w ≠0" when "w∈cball z r" for w
proof (cases "w=z")
case True
then have "f ─z→f w"
using fz_lim by blast
then have "(λw. g w * (w-z) ^ nat n) ─z→f w"
proof (elim Lim_transform_within[OF _ ‹r>0›])
fix x assume "0 < dist x z" "dist x z < r"
then have "x ∈ cball z r - {z}" "x≠z"
unfolding cball_def by (auto simp add: dist_commute)
then have "f x = g x * (x - z) powi n"
using r(4)[rule_format,of x] by simp
also have "... = g x * (x - z) ^ nat n"
by (smt (verit, best) if_0 int_nat_eq power_int_of_nat)
finally show "f x = g x * (x - z) ^ nat n" .
qed
moreover have "(λw. g w * (w-z) ^ nat n) ─z→ g w * (w-z) ^ nat n"
using True by (auto intro!:tendsto_eq_intros gz_lim)
ultimately have "f w = g w * (w-z) ^ nat n" using LIM_unique by blast
then show ?thesis using ‹g z≠0› True by auto
next
case False
then have "f w = g w * (w - z) powi n ∧ g w ≠ 0"
using r(4) that by auto
then show ?thesis
by (smt (verit, best) False if_0 int_nat_eq power_int_of_nat)
qed
ultimately show ?thesis using r by auto
qed
lemma zorder_exist_pole:
fixes f::"complex ⇒ complex" and z::complex
defines "n≡zorder f z" and "g≡zor_poly f z"
assumes holo: "f holomorphic_on S-{z}" and "open S" "z∈S" and "is_pole f z"
shows "n < 0 ∧ g z≠0 ∧ (∃r. r>0 ∧ cball z r ⊆ S ∧ g holomorphic_on cball z r
∧ (∀w∈cball z r - {z}. f w = g w / (w-z) ^ nat (- n) ∧ g w ≠0))"
proof -
obtain r where "g z ≠ 0" and r: "r>0" "cball z r ⊆ S" "g holomorphic_on cball z r"
"(∀w∈cball z r - {z}. f w = g w * (w - z) powi n ∧ g w ≠ 0)"
proof -
have "g z ≠ 0 ∧ (∃r>0. g holomorphic_on cball z r
∧ (∀w∈cball z r - {z}. f w = g w * (w - z) powi n ∧ g w ≠ 0))"
proof (rule zorder_exist[of f z,folded g_def n_def])
show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
using holo assms(4,5)
by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
show "not_essential f z" unfolding not_essential_def
using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
by fastforce
from non_zero_neighbour_pole[OF ‹is_pole f z›] show "∃⇩F w in at z. f w ≠ 0"
by (auto elim: eventually_frequentlyE)
qed
then obtain r1 where "g z ≠ 0" "r1>0" and r1:"g holomorphic_on cball z r1"
"(∀w∈cball z r1 - {z}. f w = g w * (w - z) powi n ∧ g w ≠ 0)"
by auto
obtain r2 where r2: "r2>0" "cball z r2 ⊆ S"
using assms(4,5) open_contains_cball_eq by metis
define r3 where "r3=min r1 r2"
have "r3>0" "cball z r3 ⊆ S" using ‹r1>0› r2 unfolding r3_def by auto
moreover have "g holomorphic_on cball z r3"
using r1(1) unfolding r3_def by auto
moreover have "(∀w∈cball z r3 - {z}. f w = g w * (w - z) powi n ∧ g w ≠ 0)"
using r1(2) unfolding r3_def by auto
ultimately show ?thesis using that[of r3] ‹g z≠0› by auto
qed
have "n<0"
proof (rule ccontr)
assume " ¬ n < 0"
define c where "c=(if n=0 then g z else 0)"
have [simp]:"g ─z→ g z"
by (metis open_ball at_within_open ball_subset_cball centre_in_ball
continuous_on holomorphic_on_imp_continuous_on holomorphic_on_subset r(1) r(3) )
have "∀⇩F x in at z. f x = g x * (x - z) ^ nat n"
unfolding eventually_at_topological
apply (rule_tac exI[where x="ball z r"])
by (simp add: ‹¬ n < 0› linorder_not_le power_int_def r(1) r(4))
moreover have "(λx. g x * (x - z) ^ nat n) ─z→ c"
proof (cases "n=0")
case True
then show ?thesis unfolding c_def by simp
next
case False
then have "(λx. (x - z) ^ nat n) ─z→ 0" using ‹¬ n < 0›
by (auto intro!:tendsto_eq_intros)
from tendsto_mult[OF _ this,of g "g z",simplified]
show ?thesis unfolding c_def using False by simp
qed
ultimately have "f ─z→c" using tendsto_cong by fast
then show False using ‹is_pole f z› at_neq_bot not_tendsto_and_filterlim_at_infinity
unfolding is_pole_def by blast
qed
moreover have "∀w∈cball z r - {z}. f w = g w / (w-z) ^ nat (- n) ∧ g w ≠0"
using r(4) ‹n<0›
by (smt (verit) inverse_eq_divide mult.right_neutral power_int_def power_inverse times_divide_eq_right)
ultimately show ?thesis using r(1-3) ‹g z≠0› by auto
qed
lemma zorder_eqI:
assumes "open S" "z ∈ S" "g holomorphic_on S" "g z ≠ 0"
assumes fg_eq:"⋀w. ⟦w ∈ S;w≠z⟧ ⟹ f w = g w * (w - z) powi n"
shows "zorder f z = n"
proof -
have "continuous_on S g" by (rule holomorphic_on_imp_continuous_on) fact
moreover have "open (-{0::complex})" by auto
ultimately have "open ((g -` (-{0})) ∩ S)"
unfolding continuous_on_open_vimage[OF ‹open S›] by blast
moreover from assms have "z ∈ (g -` (-{0})) ∩ S" by auto
ultimately obtain r where r: "r > 0" "cball z r ⊆ S ∩ (g -` (-{0}))"
unfolding open_contains_cball by blast
let ?gg= "(λw. g w * (w - z) powi n)"
define P where "P = (λn g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z≠0
∧ (∀w∈cball z r - {z}. f w = g w * (w-z) powi n ∧ g w≠0))"
have "P n g r"
unfolding P_def using r assms(3,4,5) by auto
then have "∃g r. P n g r" by auto
moreover have unique: "∃!n. ∃g r. P n g r" unfolding P_def
proof (rule holomorphic_factor_puncture)
have "ball z r-{z} ⊆ S" using r using ball_subset_cball by blast
then have "?gg holomorphic_on ball z r-{z}"
using ‹g holomorphic_on S› r by (auto intro!: holomorphic_intros)
then have "f holomorphic_on ball z r - {z}"
by (smt (verit, best) DiffD2 ‹ball z r-{z} ⊆ S› fg_eq holomorphic_cong singleton_iff subset_iff)
then show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
using analytic_on_open open_delete r(1) by blast
next
have "not_essential ?gg z"
proof (intro singularity_intros)
show "not_essential g z"
by (meson ‹continuous_on S g› assms continuous_on_eq_continuous_at
isCont_def not_essential_def)
show " ∀⇩F w in at z. w - z ≠ 0" by (simp add: eventually_at_filter)
then show "LIM w at z. w - z :> at 0"
unfolding filterlim_at by (auto intro:tendsto_eq_intros)
show "isolated_singularity_at g z"
by (meson Diff_subset open_ball analytic_on_holomorphic
assms holomorphic_on_subset isolated_singularity_at_def openE)
qed
moreover
have "∀⇩F w in at z. g w * (w - z) powi n = f w"
unfolding eventually_at_topological using assms fg_eq by force
ultimately show "not_essential f z"
using not_essential_transform by blast
show "∃⇩F w in at z. f w ≠ 0" unfolding frequently_at
proof (intro strip)
fix d::real assume "0 < d"
define z' where "z' ≡ z+min d r / 2"
have "z' ≠ z" " dist z' z < d "
unfolding z'_def using ‹d>0› ‹r>0› by (auto simp add:dist_norm)
moreover have "f z' ≠ 0"
proof (subst fg_eq[OF _ ‹z'≠z›])
have "z' ∈ cball z r"
unfolding z'_def using ‹r>0› ‹d>0› by (auto simp add:dist_norm)
then show " z' ∈ S" using r(2) by blast
show "g z' * (z' - z) powi n ≠ 0"
using P_def ‹P n g r› ‹z' ∈ cball z r› ‹z' ≠ z› by auto
qed
ultimately show "∃x∈UNIV. x ≠ z ∧ dist x z < d ∧ f x ≠ 0" by auto
qed
qed
ultimately have "(THE n. ∃g r. P n g r) = n"
by (rule_tac the1_equality)
then show ?thesis unfolding zorder_def P_def by blast
qed
lemma simple_zeroI:
assumes "open S" "z ∈ S" "g holomorphic_on S" "g z ≠ 0"
assumes "⋀w. w ∈ S ⟹ f w = g w * (w - z)"
shows "zorder f z = 1"
using assms zorder_eqI by force
lemma higher_deriv_power:
shows "(deriv ^^ j) (λw. (w - z) ^ n) w =
pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)"
proof (induction j arbitrary: w)
case 0
thus ?case by auto
next
case (Suc j w)
have "(deriv ^^ Suc j) (λw. (w - z) ^ n) w = deriv ((deriv ^^ j) (λw. (w - z) ^ n)) w"
by simp
also have "(deriv ^^ j) (λw. (w - z) ^ n) =
(λw. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))"
using Suc by (intro Suc.IH ext)
also {
have "(… has_field_derivative of_nat (n - j) *
pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)"
using Suc.prems by (auto intro!: derivative_eq_intros)
also have "of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j =
pochhammer (of_nat (Suc n - Suc j)) (Suc j)"
by (cases "Suc j ≤ n", subst pochhammer_rec)
(insert Suc.prems, simp_all add: algebra_simps Suc_diff_le pochhammer_0_left)
finally have "deriv (λw. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w =
… * (w - z) ^ (n - Suc j)"
by (rule DERIV_imp_deriv)
}
finally show ?case .
qed
lemma zorder_zero_eqI:
assumes f_holo:"f holomorphic_on S" and "open S" "z ∈ S"
assumes zero: "⋀i. i < nat n ⟹ (deriv ^^ i) f z = 0"
assumes nz: "(deriv ^^ nat n) f z ≠ 0" and "n≥0"
shows "zorder f z = n"
proof -
obtain r where [simp]:"r>0" and "ball z r ⊆ S"
using ‹open S› ‹z∈S› openE by blast
have nz':"∃w∈ball z r. f w ≠ 0"
proof (rule ccontr)
assume "¬ (∃w∈ball z r. f w ≠ 0)"
then have "eventually (λu. f u = 0) (nhds z)"
using open_ball ‹0 < r› centre_in_ball eventually_nhds by blast
then have "(deriv ^^ nat n) f z = (deriv ^^ nat n) (λ_. 0) z"
by (intro higher_deriv_cong_ev) auto
also have "(deriv ^^ nat n) (λ_. 0) z = 0"
by (induction n) simp_all
finally show False using nz by contradiction
qed
define zn g where "zn = zorder f z" and "g = zor_poly f z"
obtain e where e_if: "if f z = 0 then 0 < zn else zn = 0" and
[simp]: "e>0" and "cball z e ⊆ ball z r" and
g_holo: "g holomorphic_on cball z e" and
e_fac: "(∀w∈cball z e. f w = g w * (w - z) ^ nat zn ∧ g w ≠ 0)"
proof -
have "f holomorphic_on ball z r"
using f_holo ‹ball z r ⊆ S› by auto
from that zorder_exist_zero[of f "ball z r" z,simplified,OF this nz',folded zn_def g_def]
show thesis by blast
qed
then obtain "zn ≥ 0" "g z ≠ 0"
by (metis centre_in_cball less_le_not_le order_refl)
define A where "A ≡ (λi. of_nat (i choose (nat zn)) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z)"
have deriv_A:"(deriv ^^ i) f z = (if zn ≤ int i then A i else 0)" for i
proof -
have "eventually (λw. w ∈ ball z e) (nhds z)"
using ‹cball z e ⊆ ball z r› ‹e>0› by (intro eventually_nhds_in_open) auto
hence "eventually (λw. f w = (w - z) ^ (nat zn) * g w) (nhds z)"
using e_fac eventually_mono by fastforce
hence "(deriv ^^ i) f z = (deriv ^^ i) (λw. (w - z) ^ nat zn * g w) z"
by (intro higher_deriv_cong_ev) auto
also have "… = (∑j=0..i. of_nat (i choose j) *
(deriv ^^ j) (λw. (w - z) ^ nat zn) z * (deriv ^^ (i - j)) g z)"
using g_holo ‹e>0›
by (intro higher_deriv_mult[of _ "ball z e"]) (auto intro!: holomorphic_intros)
also have "… = (∑j=0..i. if j = nat zn then
of_nat (i choose nat zn) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z else 0)"
proof (intro sum.cong refl, goal_cases)
case (1 j)
have "(deriv ^^ j) (λw. (w - z) ^ nat zn) z =
pochhammer (of_nat (Suc (nat zn) - j)) j * 0 ^ (nat zn - j)"
by (subst higher_deriv_power) auto
also have "… = (if j = nat zn then fact j else 0)"
by (auto simp: not_less pochhammer_0_left pochhammer_fact)
also have "of_nat (i choose j) * … * (deriv ^^ (i - j)) g z =
(if j = nat zn then of_nat (i choose (nat zn)) * fact (nat zn)
* (deriv ^^ (i - nat zn)) g z else 0)"
by simp
finally show ?case .
qed
also have "… = (if i ≥ zn then A i else 0)"
by (auto simp: A_def)
finally show "(deriv ^^ i) f z = …" .
qed
have False when "n<zn"
proof -
have "(deriv ^^ nat n) f z = 0"
using deriv_A[of "nat n"] that ‹n≥0› by auto
with nz show False by auto
qed
moreover have "n≤zn"
proof -
have "g z ≠ 0" using e_fac[rule_format,of z] ‹e>0› by simp
then have "(deriv ^^ nat zn) f z ≠ 0"
using deriv_A[of "nat zn"] by(auto simp add:A_def)
then have "nat zn ≥ nat n" using zero[of "nat zn"] by linarith
moreover have "zn≥0" using e_if by (auto split:if_splits)
ultimately show ?thesis using nat_le_eq_zle by blast
qed
ultimately show ?thesis unfolding zn_def by fastforce
qed
lemma
assumes "eventually (λz. f z = g z) (at z)" "z = z'"
shows zorder_cong:"zorder f z = zorder g z'" and zor_poly_cong:"zor_poly f z = zor_poly g z'"
proof -
define P where "P = (λff n h r. 0 < r ∧ h holomorphic_on cball z r ∧ h z≠0
∧ (∀w∈cball z r - {z}. ff w = h w * (w-z) powi n ∧ h w≠0))"
have "(∃r. P f n h r) = (∃r. P g n h r)" for n h
proof -
have *: "∃r. P g n h r" if "∃r. P f n h r" and "eventually (λx. f x = g x) (at z)" for f g
proof -
from that(1) obtain r1 where r1_P:"P f n h r1" by auto
from that(2) obtain r2 where "r2>0" and r2_dist:"∀x. x ≠ z ∧ dist x z ≤ r2 ⟶ f x = g x"
unfolding eventually_at_le by auto
define r where "r=min r1 r2"
have "r>0" "h z≠0" using r1_P ‹r2>0› unfolding r_def P_def by auto
moreover have "h holomorphic_on cball z r"
using r1_P unfolding P_def r_def by auto
moreover have "g w = h w * (w - z) powi n ∧ h w ≠ 0" when "w∈cball z r - {z}" for w
proof -
have "f w = h w * (w - z) powi n ∧ h w ≠ 0"
using r1_P that unfolding P_def r_def by auto
moreover have "f w=g w" using r2_dist[rule_format,of w] that unfolding r_def
by (simp add: dist_commute)
ultimately show ?thesis by simp
qed
ultimately show ?thesis unfolding P_def by auto
qed
from assms have eq': "eventually (λz. g z = f z) (at z)"
by (simp add: eq_commute)
show ?thesis
by (rule iffI[OF *[OF _ assms(1)] *[OF _ eq']])
qed
then show "zorder f z = zorder g z'" "zor_poly f z = zor_poly g z'"
using ‹z=z'› unfolding P_def zorder_def zor_poly_def by auto
qed
lemma zorder_times_analytic':
assumes "isolated_singularity_at f z" "not_essential f z"
assumes "g analytic_on {z}" "frequently (λz. f z * g z ≠ 0) (at z)"
shows "zorder (λx. f x * g x) z = zorder f z + zorder g z"
using assms isolated_singularity_at_analytic not_essential_analytic zorder_times by blast
lemma zorder_cmult:
assumes "c ≠ 0"
shows "zorder (λz. c * f z) z = zorder f z"
proof -
define P where
"P = (λf n h r. 0 < r ∧ h holomorphic_on cball z r ∧
h z ≠ 0 ∧ (∀w∈cball z r - {z}. f w = h w * (w - z) powi n ∧ h w ≠ 0))"
have *: "P (λx. c * f x) n (λx. c * h x) r" if "P f n h r" "c ≠ 0" for f n h r c
using that unfolding P_def by (auto intro!: holomorphic_intros)
have "(∃h r. P (λx. c * f x) n h r) ⟷ (∃h r. P f n h r)" for n
using *[of f n _ _ c] *[of "λx. c * f x" n _ _ "inverse c"] ‹c ≠ 0›
by (fastforce simp: field_simps)
hence "(THE n. ∃h r. P (λx. c * f x) n h r) = (THE n. ∃h r. P f n h r)"
by simp
thus ?thesis
by (simp add: zorder_def P_def)
qed
lemma zorder_uminus [simp]: "zorder (λz. -f z) z = zorder f z"
using zorder_cmult[of "-1" f] by simp
lemma zorder_nonzero_div_power:
assumes sz: "open S" "z ∈ S" "f holomorphic_on S" "f z ≠ 0" and "n > 0"
shows "zorder (λw. f w / (w - z) ^ n) z = - n"
by (intro zorder_eqI [OF sz]) (simp add: inverse_eq_divide power_int_minus)
lemma zor_poly_eq:
assumes "isolated_singularity_at f z" "not_essential f z" "∃⇩F w in at z. f w ≠ 0"
shows "eventually (λw. zor_poly f z w = f w * (w - z) powi - zorder f z) (at z)"
proof -
obtain r where r:"r>0"
"(∀w∈cball z r - {z}. f w = zor_poly f z w * (w - z) powi (zorder f z))"
using zorder_exist[OF assms] by blast
then have *: "∀w∈ball z r - {z}. zor_poly f z w = f w * (w - z) powi - zorder f z"
by (auto simp: field_simps power_int_minus)
have "eventually (λw. w ∈ ball z r - {z}) (at z)"
using r eventually_at_ball'[of r z UNIV] by auto
thus ?thesis by eventually_elim (insert *, auto)
qed
lemma zor_poly_zero_eq:
assumes "f holomorphic_on S" "open S" "connected S" "z ∈ S" "∃w∈S. f w ≠ 0"
shows "eventually (λw. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)) (at z)"
proof -
obtain r where r:"r>0"
"(∀w∈cball z r - {z}. f w = zor_poly f z w * (w - z) ^ nat (zorder f z))"
using zorder_exist_zero[OF assms] by auto
then have *: "∀w∈ball z r - {z}. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)"
by (auto simp: field_simps powr_minus)
have "eventually (λw. w ∈ ball z r - {z}) (at z)"
using r eventually_at_ball'[of r z UNIV] by auto
thus ?thesis by eventually_elim (insert *, auto)
qed
lemma zor_poly_pole_eq:
assumes f_iso:"isolated_singularity_at f z" "is_pole f z"
shows "eventually (λw. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)) (at z)"
proof -
obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
obtain r where r:"r>0"
"(∀w∈cball z r - {z}. f w = zor_poly f z w / (w - z) ^ nat (- zorder f z))"
using zorder_exist_pole[OF f_holo,simplified,OF ‹is_pole f z›] by auto
then have *: "∀w∈ball z r - {z}. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)"
by (auto simp: field_simps)
have "eventually (λw. w ∈ ball z r - {z}) (at z)"
using r eventually_at_ball'[of r z UNIV] by auto
thus ?thesis by eventually_elim (insert *, auto)
qed
lemma zor_poly_eqI:
fixes f :: "complex ⇒ complex" and z0 :: complex
defines "n ≡ zorder f z0"
assumes "isolated_singularity_at f z0" "not_essential f z0" "∃⇩F w in at z0. f w ≠ 0"
assumes lim: "((λx. f (g x) * (g x - z0) powi - n) ⤏ c) F"
assumes g: "filterlim g (at z0) F" and "F ≠ bot"
shows "zor_poly f z0 z0 = c"
proof -
from zorder_exist[OF assms(2-4)] obtain r where
r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
"⋀w. w ∈ cball z0 r - {z0} ⟹ f w = zor_poly f z0 w * (w - z0) powi n"
unfolding n_def by blast
from r(1) have "eventually (λw. w ∈ ball z0 r ∧ w ≠ z0) (at z0)"
using eventually_at_ball'[of r z0 UNIV] by auto
hence "eventually (λw. zor_poly f z0 w = f w * (w - z0) powi - n) (at z0)"
by eventually_elim (insert r, auto simp: field_simps power_int_minus)
moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
using r by (intro holomorphic_on_imp_continuous_on) auto
with r(1,2) have "isCont (zor_poly f z0) z0"
by (auto simp: continuous_on_eq_continuous_at)
hence "(zor_poly f z0 ⤏ zor_poly f z0 z0) (at z0)"
unfolding isCont_def .
ultimately have "((λw. f w * (w - z0) powi - n) ⤏ zor_poly f z0 z0) (at z0)"
by (blast intro: Lim_transform_eventually)
hence "((λx. f (g x) * (g x - z0) powi - n) ⤏ zor_poly f z0 z0) F"
by (rule filterlim_compose[OF _ g])
from tendsto_unique[OF ‹F ≠ bot› this lim] show ?thesis .
qed
lemma zor_poly_zero_eqI:
fixes f :: "complex ⇒ complex" and z0 :: complex
defines "n ≡ zorder f z0"
assumes "f holomorphic_on A" "open A" "connected A" "z0 ∈ A" "∃z∈A. f z ≠ 0"
assumes lim: "((λx. f (g x) / (g x - z0) ^ nat n) ⤏ c) F"
assumes g: "filterlim g (at z0) F" and "F ≠ bot"
shows "zor_poly f z0 z0 = c"
proof -
from zorder_exist_zero[OF assms(2-6)] obtain r where
r: "r > 0" "cball z0 r ⊆ A" "zor_poly f z0 holomorphic_on cball z0 r"
"⋀w. w ∈ cball z0 r ⟹ f w = zor_poly f z0 w * (w - z0) ^ nat n"
unfolding n_def by blast
from r(1) have "eventually (λw. w ∈ ball z0 r ∧ w ≠ z0) (at z0)"
using eventually_at_ball'[of r z0 UNIV] by auto
hence "eventually (λw. zor_poly f z0 w = f w / (w - z0) ^ nat n) (at z0)"
by eventually_elim (insert r, auto simp: field_simps)
moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
using r by (intro holomorphic_on_imp_continuous_on) auto
with r(1,2) have "isCont (zor_poly f z0) z0"
by (auto simp: continuous_on_eq_continuous_at)
hence "(zor_poly f z0 ⤏ zor_poly f z0 z0) (at z0)"
unfolding isCont_def .
ultimately have "((λw. f w / (w - z0) ^ nat n) ⤏ zor_poly f z0 z0) (at z0)"
by (blast intro: Lim_transform_eventually)
hence "((λx. f (g x) / (g x - z0) ^ nat n) ⤏ zor_poly f z0 z0) F"
by (rule filterlim_compose[OF _ g])
from tendsto_unique[OF ‹F ≠ bot› this lim] show ?thesis .
qed
lemma zor_poly_pole_eqI:
fixes f :: "complex ⇒ complex" and z0 :: complex
defines "n ≡ zorder f z0"
assumes f_iso:"isolated_singularity_at f z0" and "is_pole f z0"
assumes lim: "((λx. f (g x) * (g x - z0) ^ nat (-n)) ⤏ c) F"
assumes g: "filterlim g (at z0) F" and "F ≠ bot"
shows "zor_poly f z0 z0 = c"
proof -
obtain r where r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
proof -
have "∃⇩F w in at z0. f w ≠ 0"
using non_zero_neighbour_pole[OF ‹is_pole f z0›] by (auto elim:eventually_frequentlyE)
moreover have "not_essential f z0" unfolding not_essential_def using ‹is_pole f z0› by simp
ultimately show ?thesis using that zorder_exist[OF f_iso,folded n_def] by auto
qed
from r(1) have "eventually (λw. w ∈ ball z0 r ∧ w ≠ z0) (at z0)"
using eventually_at_ball'[of r z0 UNIV] by auto
have "eventually (λw. zor_poly f z0 w = f w * (w - z0) ^ nat (-n)) (at z0)"
using zor_poly_pole_eq[OF f_iso ‹is_pole f z0›] unfolding n_def .
moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
using r by (intro holomorphic_on_imp_continuous_on) auto
with r(1,2) have "isCont (zor_poly f z0) z0"
by (auto simp: continuous_on_eq_continuous_at)
hence "(zor_poly f z0 ⤏ zor_poly f z0 z0) (at z0)"
unfolding isCont_def .
ultimately have "((λw. f w * (w - z0) ^ nat (-n)) ⤏ zor_poly f z0 z0) (at z0)"
by (blast intro: Lim_transform_eventually)
hence "((λx. f (g x) * (g x - z0) ^ nat (-n)) ⤏ zor_poly f z0 z0) F"
by (rule filterlim_compose[OF _ g])
from tendsto_unique[OF ‹F ≠ bot› this lim] show ?thesis .
qed
lemma
assumes "is_pole f (x :: complex)" "open A" "x ∈ A"
assumes "⋀y. y ∈ A - {x} ⟹ (f has_field_derivative f' y) (at y)"
shows is_pole_deriv': "is_pole f' x"
and zorder_deriv': "zorder f' x = zorder f x - 1"
proof -
have holo: "f holomorphic_on A - {x}"
using assms by (subst holomorphic_on_open) auto
obtain r where r: "r > 0" "ball x r ⊆ A"
using assms(2,3) openE by blast
moreover have "open (ball x r - {x})"
by auto
ultimately have "isolated_singularity_at f x"
by (auto simp: isolated_singularity_at_def analytic_on_open
intro!: exI[of _ r] holomorphic_on_subset[OF holo])
hence ev: "∀⇩F w in at x. zor_poly f x w = f w * (w - x) ^ nat (- zorder f x)"
using ‹is_pole f x› zor_poly_pole_eq by blast
define P where "P = zor_poly f x"
define n where "n = nat (-zorder f x)"
obtain r where r: "r > 0" "cball x r ⊆ A" "P holomorphic_on cball x r" "zorder f x < 0" "P x ≠ 0"
"∀w∈cball x r - {x}. f w = P w / (w - x) ^ n ∧ P w ≠ 0"
unfolding P_def n_def using zorder_exist_pole[OF holo assms(2,3,1)] by blast
have n: "n > 0"
using r(4) by (auto simp: n_def)
have [derivative_intros]: "(P has_field_derivative deriv P w) (at w)"
if "w ∈ ball x r" for w
using that by (intro holomorphic_derivI[OF holomorphic_on_subset[OF r(3), of "ball x r"]]) auto
define D where "D = (λw. (deriv P w * (w - x) - of_nat n * P w) / (w - x) ^ (n + 1))"
define n' where "n' = n - 1"
have n': "n = Suc n'"
using n by (simp add: n'_def)
have "eventually (λw. w ∈ ball x r) (nhds x)"
using ‹r > 0› by (intro eventually_nhds_in_open) auto
hence ev'': "eventually (λw. w ∈ ball x r - {x}) (at x)"
by (auto simp: eventually_at_filter elim: eventually_mono)
{
fix w assume w: "w ∈ ball x r - {x}"
have ev': "eventually (λw. w ∈ ball x r - {x}) (nhds w)"
using w by (intro eventually_nhds_in_open) auto
have "((λw. P w / (w - x) ^ n) has_field_derivative D w) (at w)"
apply (rule derivative_eq_intros refl | use w in force)+
using w
apply (simp add: divide_simps D_def)
apply (simp add: n' algebra_simps)
done
also have "?this ⟷ (f has_field_derivative D w) (at w)"
using r by (intro has_field_derivative_cong_ev refl eventually_mono[OF ev']) auto
finally have "(f has_field_derivative D w) (at w)" .
moreover have "(f has_field_derivative f' w) (at w)"
using w r by (intro assms) auto
ultimately have "D w = f' w"
using DERIV_unique by blast
} note D_eq = this
have "is_pole D x"
unfolding D_def using n ‹r > 0› ‹P x ≠ 0›
by (intro is_pole_basic[where A = "ball x r"] holomorphic_intros holomorphic_on_subset[OF r(3)]) auto
also have "?this ⟷ is_pole f' x"
by (intro is_pole_cong eventually_mono[OF ev''] D_eq) auto
finally show "is_pole f' x" .
have "zorder f' x = -int (Suc n)"
proof (rule zorder_eqI)
show "open (ball x r)" "x ∈ ball x r"
using ‹r > 0› by auto
show "f' w = (deriv P w * (w - x) - of_nat n * P w) * (w - x) powi (- int (Suc n))"
if "w ∈ ball x r" "w ≠ x" for w
using that D_eq[of w] n by (auto simp: D_def power_int_diff power_int_minus powr_nat' divide_simps)
qed (use r n in ‹auto intro!: holomorphic_intros›)
thus "zorder f' x = zorder f x - 1"
using n by (simp add: n_def)
qed
lemma
assumes "is_pole f (x :: complex)" "isolated_singularity_at f x"
shows is_pole_deriv: "is_pole (deriv f) x"
and zorder_deriv: "zorder (deriv f) x = zorder f x - 1"
proof -
from assms(2) obtain r where r: "r > 0" "f analytic_on ball x r - {x}"
by (auto simp: isolated_singularity_at_def)
hence holo: "f holomorphic_on ball x r - {x}"
by (subst (asm) analytic_on_open) auto
have *: "x ∈ ball x r" "open (ball x r)" "open (ball x r - {x})"
using ‹r > 0› by auto
show "is_pole (deriv f) x" "zorder (deriv f) x = zorder f x - 1"
by (meson "*" assms(1) holo holomorphic_derivI is_pole_deriv' zorder_deriv')+
qed
lemma removable_singularity_deriv':
assumes "f ─x→ c" "x ∈ A" "open (A :: complex set)"
assumes "⋀y. y ∈ A - {x} ⟹ (f has_field_derivative f' y) (at y)"
shows "∃c. f' ─x→ c"
proof -
have holo: "f holomorphic_on A - {x}"
using assms by (subst holomorphic_on_open) auto
define g where "g = (λy. if y = x then c else f y)"
have deriv_g_eq: "deriv g y = f' y" if "y ∈ A - {x}" for y
proof -
have ev: "eventually (λy. y ∈ A - {x}) (nhds y)"
using that assms by (intro eventually_nhds_in_open) auto
have "(f has_field_derivative f' y) (at y)"
using assms that by auto
also have "?this ⟷ (g has_field_derivative f' y) (at y)"
by (intro has_field_derivative_cong_ev refl eventually_mono[OF ev]) (auto simp: g_def)
finally show ?thesis
by (intro DERIV_imp_deriv assms)
qed
have "g holomorphic_on A"
unfolding g_def using assms assms(1) holo by (intro removable_singularity) auto
hence "deriv g holomorphic_on A"
by (intro holomorphic_deriv assms)
hence "continuous_on A (deriv g)"
by (meson holomorphic_on_imp_continuous_on)
hence "(deriv g ⤏ deriv g x) (at x within A)"
using assms by (auto simp: continuous_on_def)
also have "?this ⟷ (f' ⤏ deriv g x) (at x within A)"
by (intro filterlim_cong refl) (auto simp: eventually_at_filter deriv_g_eq)
finally have "f' ─x→ deriv g x"
using ‹open A› ‹x ∈ A› by (meson tendsto_within_open)
thus ?thesis
by blast
qed
lemma removable_singularity_deriv:
assumes "f ─x→ c" "isolated_singularity_at f x"
shows "∃c. deriv f ─x→ c"
proof -
from assms(2) obtain r where r: "r > 0" "f analytic_on ball x r - {x}"
by (auto simp: isolated_singularity_at_def)
hence holo: "f holomorphic_on ball x r - {x}"
using analytic_imp_holomorphic by blast
show ?thesis
using assms(1)
proof (rule removable_singularity_deriv')
show "x ∈ ball x r" "open (ball x r)"
using ‹r > 0› by auto
qed (auto intro!: holomorphic_derivI[OF holo])
qed
lemma not_essential_deriv':
assumes "not_essential f x" "x ∈ A" "open A"
assumes "⋀y. y ∈ A - {x} ⟹ (f has_field_derivative f' y) (at y)"
shows "not_essential f' x"
proof -
have holo: "f holomorphic_on A - {x}"
using assms by (subst holomorphic_on_open) auto
from assms consider "is_pole f x" | c where "f ─x→ c"
by (auto simp: not_essential_def)
thus ?thesis
proof cases
case 1
hence "is_pole f' x"
using is_pole_deriv' assms by blast
thus ?thesis by (auto simp: not_essential_def)
next
case (2 c)
from 2 have "∃c. f' ─x→ c"
by (rule removable_singularity_deriv'[OF _ assms(2-4)])
thus ?thesis
by (auto simp: not_essential_def)
qed
qed
lemma not_essential_deriv[singularity_intros]:
assumes "not_essential f x" "isolated_singularity_at f x"
shows "not_essential (deriv f) x"
proof -
from assms(2) obtain r where r: "r > 0" "f analytic_on ball x r - {x}"
by (auto simp: isolated_singularity_at_def)
hence holo: "f holomorphic_on ball x r - {x}"
by (subst (asm) analytic_on_open) auto
show ?thesis
using assms(1)
proof (rule not_essential_deriv')
show "x ∈ ball x r" "open (ball x r)"
using ‹r > 0› by auto
qed (auto intro!: holomorphic_derivI[OF holo])
qed
lemma not_essential_frequently_0_imp_tendsto_0:
fixes f :: "complex ⇒ complex"
assumes sing: "isolated_singularity_at f z" "not_essential f z"
assumes freq: "frequently (λz. f z = 0) (at z)"
shows "f ─z→ 0"
proof -
from freq obtain g :: "nat ⇒ complex" where g: "filterlim g (at z) at_top" "⋀n. f (g n) = 0"
using frequently_atE by blast
have "eventually (λx. f (g x) = 0) sequentially"
using g by auto
hence fg: "(λx. f (g x)) ⇢ 0"
by (simp add: tendsto_eventually)
from assms(2) consider c where "f ─z→ c" | "is_pole f z"
unfolding not_essential_def by blast
thus ?thesis
proof cases
case (1 c)
have "(λx. f (g x)) ⇢ c"
by (rule filterlim_compose[OF 1 g(1)])
with fg have "c = 0"
using LIMSEQ_unique by blast
with 1 show ?thesis by simp
next
case 2
have "filterlim (λx. f (g x)) at_infinity sequentially"
by (rule filterlim_compose[OF _ g(1)]) (use 2 in ‹auto simp: is_pole_def›)
with fg have False
by (meson not_tendsto_and_filterlim_at_infinity sequentially_bot)
thus ?thesis ..
qed
qed
lemma not_essential_frequently_0_imp_eventually_0:
fixes f :: "complex ⇒ complex"
assumes sing: "isolated_singularity_at f z" "not_essential f z"
assumes freq: "frequently (λz. f z = 0) (at z)"
shows "eventually (λz. f z = 0) (at z)"
proof -
from sing obtain r where r: "r > 0" and "f analytic_on ball z r - {z}"
by (auto simp: isolated_singularity_at_def)
hence holo: "f holomorphic_on ball z r - {z}"
by (subst (asm) analytic_on_open) auto
have "eventually (λw. w ∈ ball z r - {z}) (at z)"
using r by (intro eventually_at_in_open) auto
from freq and this have "frequently (λw. f w = 0 ∧ w ∈ ball z r - {z}) (at z)"
using frequently_eventually_frequently by blast
hence "frequently (λw. w ∈ {w∈ball z r - {z}. f w = 0}) (at z)"
by (simp add: conj_commute)
hence limpt: "z islimpt {w∈ball z r - {z}. f w = 0}"
using islimpt_conv_frequently_at by blast
define g where "g = (λw. if w = z then 0 else f w)"
have "f ─z→ 0"
by (intro not_essential_frequently_0_imp_tendsto_0 assms)
hence g_holo: "g holomorphic_on ball z r"
unfolding g_def by (intro removable_singularity holo) auto
have g_eq_0: "g w = 0" if "w ∈ ball z r" for w
proof (rule analytic_continuation[where f = g])
show "open (ball z r)" "connected (ball z r)"
using r by auto
show "z islimpt {w∈ball z r - {z}. f w = 0}"
by fact
show "g w = 0" if "w ∈ {w ∈ ball z r - {z}. f w = 0}" for w
using that by (auto simp: g_def)
qed (use r that g_holo in auto)
have "eventually (λw. w ∈ ball z r - {z}) (at z)"
using r by (intro eventually_at_in_open) auto
thus "eventually (λw. f w = 0) (at z)"
by (metis freq non_zero_neighbour not_eventually not_frequently sing)
qed
lemma pole_imp_not_constant:
fixes f :: "'a :: {perfect_space} ⇒ _"
assumes "is_pole f x" "open A" "x ∈ A" "A ⊆ insert x B"
shows "¬f constant_on B"
proof
assume *: "f constant_on B"
then obtain c where c: "∀x∈B. f x = c"
by (auto simp: constant_on_def)
have "eventually (λy. y ∈ A - {x}) (at x)"
using assms by (intro eventually_at_in_open) auto
hence "eventually (λy. f y = c) (at x)"
by eventually_elim (use c assms in auto)
hence **: "f ─x→ c"
by (simp add: tendsto_eventually)
show False
using not_tendsto_and_filterlim_at_infinity[OF _ ** assms(1)[unfolded is_pole_def]] by simp
qed
lemma neg_zorder_imp_is_pole:
assumes iso:"isolated_singularity_at f z" and f_ness:"not_essential f z"
and "zorder f z < 0" and fre_nz:"∃⇩F w in at z. f w ≠ 0 "
shows "is_pole f z"
proof -
define P where "P = zor_poly f z"
define n where "n = zorder f z"
have "n<0" unfolding n_def by (simp add: assms(3))
define nn where "nn = nat (-n)"
obtain r where "P z ≠ 0" "r>0" and r_holo:"P holomorphic_on cball z r" and
w_Pn:"(∀w∈cball z r - {z}. f w = P w * (w - z) powi n ∧ P w ≠ 0)"
using zorder_exist[OF iso f_ness fre_nz,folded P_def n_def] by auto
have "is_pole (λw. P w * (w - z) powi n) z"
unfolding is_pole_def
proof (rule tendsto_mult_filterlim_at_infinity)
show "P ─z→ P z"
by (meson open_ball ‹0 < r› ball_subset_cball centre_in_ball
continuous_on_eq_continuous_at continuous_on_subset
holomorphic_on_imp_continuous_on isContD r_holo)
show "P z≠0" by (simp add: ‹P z ≠ 0›)
have "LIM x at z. inverse ((x - z) ^ nat (-n)) :> at_infinity"
apply (subst filterlim_inverse_at_iff[symmetric])
using ‹n<0›
by (auto intro!:tendsto_eq_intros filterlim_atI
simp add:eventually_at_filter)
then show "LIM x at z. (x - z) powi n :> at_infinity"
proof (elim filterlim_mono_eventually)
have "inverse ((x - z) ^ nat (-n)) = (x - z) powi n"
if "x≠z" for x
by (metis ‹n < 0› linorder_not_le power_int_def power_inverse)
then show "∀⇩F x in at z. inverse ((x - z) ^ nat (-n))
= (x - z) powi n"
by (simp add: eventually_at_filter)
qed auto
qed
moreover have "∀⇩F w in at z. f w = P w * (w - z) powi n"
unfolding eventually_at_le
using w_Pn ‹r>0› by (force simp add: dist_commute)
ultimately show ?thesis using is_pole_cong by fast
qed
lemma is_pole_divide_zorder:
fixes f g::"complex ⇒ complex" and z::complex
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
and f_ness:"not_essential f z" and g_ness:"not_essential g z"
and fg_nconst: "∃⇩Fw in (at z). f w * g w≠ 0"
and z_less:"zorder f z < zorder g z"
shows "is_pole (λz. f z / g z) z"
proof -
define fn gn fg where "fn=zorder f z" and "gn=zorder g z"
and "fg=(λw. f w / g w)"
have "isolated_singularity_at fg z"
unfolding fg_def using f_iso g_iso g_ness
by (auto intro:singularity_intros)
moreover have "not_essential fg z"
unfolding fg_def using f_iso g_iso g_ness f_ness
by (auto intro:singularity_intros)
moreover have "zorder fg z < 0"
proof -
have "zorder fg z = fn - gn"
using zorder_divide[OF f_iso g_iso f_ness g_ness
fg_nconst,folded fn_def gn_def fg_def] .
then show ?thesis
using z_less by (simp add: fn_def gn_def)
qed
moreover have "∃⇩F w in at z. fg w ≠ 0"
using fg_nconst unfolding fg_def by force
ultimately show "is_pole fg z"
using neg_zorder_imp_is_pole by auto
qed
lemma isolated_pole_imp_nzero_times:
assumes f_iso:"isolated_singularity_at f z"
and "is_pole f z"
shows "∃⇩Fw in (at z). deriv f w * f w ≠ 0"
proof (rule ccontr)
assume "¬ (∃⇩F w in at z. deriv f w * f w ≠ 0)"
then have "∀⇩F x in at z. deriv f x * f x = 0"
unfolding not_frequently by simp
moreover have "∀⇩F w in at z. f w ≠ 0"
using non_zero_neighbour_pole[OF ‹is_pole f z›] .
moreover have "∀⇩F w in at z. deriv f w ≠ 0"
using is_pole_deriv[OF ‹is_pole f z› f_iso,THEN non_zero_neighbour_pole]
.
ultimately have "∀⇩F w in at z. False"
by eventually_elim auto
then show False by auto
qed
lemma isolated_pole_imp_neg_zorder:
assumes "isolated_singularity_at f z" and "is_pole f z"
shows "zorder f z < 0"
using analytic_imp_holomorphic assms centre_in_ball isolated_singularity_at_def zorder_exist_pole by blast
lemma isolated_singularity_at_deriv[singularity_intros]:
assumes "isolated_singularity_at f x"
shows "isolated_singularity_at (deriv f) x"
by (meson analytic_deriv assms isolated_singularity_at_def)
lemma zorder_deriv_minus_1:
fixes f g::"complex ⇒ complex" and z::complex
assumes f_iso:"isolated_singularity_at f z"
and f_ness:"not_essential f z"
and f_nconst:"∃⇩F w in at z. f w ≠ 0"
and f_ord:"zorder f z ≠0"
shows "zorder (deriv f) z = zorder f z - 1"
proof -
define P where "P = zor_poly f z"
define n where "n = zorder f z"
have "n≠0" unfolding n_def using f_ord by auto
obtain r where "P z ≠ 0" "r>0" and P_holo:"P holomorphic_on cball z r"
and "(∀w∈cball z r - {z}. f w
= P w * (w - z) powi n ∧ P w ≠ 0)"
using zorder_exist[OF f_iso f_ness f_nconst,folded P_def n_def] by auto
from this(4)
have f_eq:"(∀w∈cball z r - {z}. f w
= P w * (w - z) powi n ∧ P w ≠ 0)"
using complex_powr_of_int f_ord n_def by presburger
define D where "D = (λw. (deriv P w * (w - z) + of_int n * P w)
* (w - z) powi (n - 1))"
have deriv_f_eq:"deriv f w = D w" if "w ∈ ball z r - {z}" for w
proof -
have ev': "eventually (λw. w ∈ ball z r - {z}) (nhds w)"
using that by (intro eventually_nhds_in_open) auto
define wz where "wz = w - z"
have "wz ≠0" unfolding wz_def using that by auto
moreover have "(P has_field_derivative deriv P w) (at w)"
by (meson DiffD1 Elementary_Metric_Spaces.open_ball P_holo
ball_subset_cball holomorphic_derivI holomorphic_on_subset that)
ultimately have "((λw. P w * (w - z) powi n) has_field_derivative D w) (at w)"
unfolding D_def using that
apply (auto intro!: derivative_eq_intros)
apply (fold wz_def)
by (auto simp:algebra_simps simp flip:power_int_add_1')
also have "?this ⟷ (f has_field_derivative D w) (at w)"
using f_eq
by (intro has_field_derivative_cong_ev refl eventually_mono[OF ev']) auto
ultimately have "(f has_field_derivative D w) (at w)" by simp
moreover have "(f has_field_derivative deriv f w) (at w)"
by (metis DERIV_imp_deriv calculation)
ultimately show ?thesis using DERIV_imp_deriv by blast
qed
show "zorder (deriv f) z = n - 1"
proof (rule zorder_eqI)
show "open (ball z r)" "z ∈ ball z r"
using ‹r > 0› by auto
define g where "g=(λw. (deriv P w * (w - z) + of_int n * P w))"
show "g holomorphic_on ball z r"
unfolding g_def using P_holo
by (auto intro!:holomorphic_intros)
show "g z ≠ 0"
unfolding g_def using ‹P z ≠ 0› ‹n≠0› by auto
show "deriv f w =
(deriv P w * (w - z) + of_int n * P w) * (w - z) powi (n - 1)"
if "w ∈ ball z r" "w ≠ z" for w
using D_def deriv_f_eq that by blast
qed
qed
lemma deriv_divide_is_pole:
fixes f g::"complex ⇒ complex" and z::complex
assumes f_iso:"isolated_singularity_at f z"
and f_ness:"not_essential f z"
and fg_nconst: "∃⇩Fw in (at z). deriv f w * f w ≠ 0"
and f_ord:"zorder f z ≠0"
shows "is_pole (λz. deriv f z / f z) z"
proof (rule neg_zorder_imp_is_pole)
define ff where "ff=(λw. deriv f w / f w)"
show "isolated_singularity_at ff z"
using f_iso f_ness unfolding ff_def
by (auto intro:singularity_intros)
show "not_essential ff z"
unfolding ff_def using f_ness f_iso
by (auto intro:singularity_intros)
have "zorder ff z = zorder (deriv f) z - zorder f z"
unfolding ff_def using f_iso f_ness fg_nconst
using isolated_singularity_at_deriv not_essential_deriv zorder_divide by blast
moreover have "zorder (deriv f) z = zorder f z - 1"
using f_iso f_ness f_ord fg_nconst frequently_elim1 zorder_deriv_minus_1 by fastforce
ultimately show "zorder ff z < 0" by auto
show "∃⇩F w in at z. ff w ≠ 0"
unfolding ff_def using fg_nconst by auto
qed
lemma is_pole_deriv_divide_is_pole:
fixes f g::"complex ⇒ complex" and z::complex
assumes f_iso:"isolated_singularity_at f z"
and "is_pole f z"
shows "is_pole (λz. deriv f z / f z) z"
proof (rule deriv_divide_is_pole[OF f_iso])
show "not_essential f z"
using ‹is_pole f z› unfolding not_essential_def by auto
show "∃⇩F w in at z. deriv f w * f w ≠ 0"
using assms f_iso isolated_pole_imp_nzero_times by blast
show "zorder f z ≠ 0"
using isolated_pole_imp_neg_zorder assms by fastforce
qed
subsection ‹Isolated zeroes›
definition isolated_zero :: "(complex ⇒ complex) ⇒ complex ⇒ bool" where
"isolated_zero f z ⟷ f z = 0 ∧ eventually (λz. f z ≠ 0) (at z)"
lemma isolated_zero_altdef: "isolated_zero f z ⟷ f z = 0 ∧ ¬z islimpt {z. f z = 0}"
unfolding isolated_zero_def eventually_at_filter eventually_nhds islimpt_def by blast
lemma isolated_zero_mult1:
assumes "isolated_zero f x" "isolated_zero g x"
shows "isolated_zero (λx. f x * g x) x"
proof -
have "∀⇩F x in at x. f x ≠ 0" "∀⇩F x in at x. g x ≠ 0"
using assms unfolding isolated_zero_def by auto
hence "eventually (λx. f x * g x ≠ 0) (at x)"
by eventually_elim auto
with assms show ?thesis
by (auto simp: isolated_zero_def)
qed
lemma isolated_zero_mult2:
assumes "isolated_zero f x" "g x ≠ 0" "g analytic_on {x}"
shows "isolated_zero (λx. f x * g x) x"
proof -
have "eventually (λx. f x ≠ 0) (at x)"
using assms unfolding isolated_zero_def by auto
moreover have "eventually (λx. g x ≠ 0) (at x)"
using analytic_at_neq_imp_eventually_neq[of g x 0] assms by auto
ultimately have "eventually (λx. f x * g x ≠ 0) (at x)"
by eventually_elim auto
thus ?thesis
using assms(1) by (auto simp: isolated_zero_def)
qed
lemma isolated_zero_mult3:
assumes "isolated_zero f x" "g x ≠ 0" "g analytic_on {x}"
shows "isolated_zero (λx. g x * f x) x"
using isolated_zero_mult2[OF assms] by (simp add: mult_ac)
lemma isolated_zero_prod:
assumes "⋀x. x ∈ I ⟹ isolated_zero (f x) z" "I ≠ {}" "finite I"
shows "isolated_zero (λy. ∏x∈I. f x y) z"
using assms(3,2,1) by (induction rule: finite_ne_induct) (auto intro: isolated_zero_mult1)
lemma non_isolated_zero':
assumes "isolated_singularity_at f z" "not_essential f z" "f z = 0" "¬isolated_zero f z"
shows "eventually (λz. f z = 0) (at z)"
by (metis assms isolated_zero_def non_zero_neighbour not_eventually)
lemma non_isolated_zero:
assumes "¬isolated_zero f z" "f analytic_on {z}" "f z = 0"
shows "eventually (λz. f z = 0) (nhds z)"
proof -
have "eventually (λz. f z = 0) (at z)"
by (rule non_isolated_zero')
(use assms in ‹auto intro: not_essential_analytic isolated_singularity_at_analytic›)
with ‹f z = 0› show ?thesis
unfolding eventually_at_filter by (auto elim!: eventually_mono)
qed
lemma not_essential_compose:
assumes "not_essential f (g z)" "g analytic_on {z}"
shows "not_essential (λx. f (g x)) z"
proof (cases "isolated_zero (λw. g w - g z) z")
case False
hence "eventually (λw. g w - g z = 0) (nhds z)"
by (rule non_isolated_zero) (use assms in ‹auto intro!: analytic_intros›)
hence "not_essential (λx. f (g x)) z ⟷ not_essential (λ_. f (g z)) z"
by (intro not_essential_cong refl)
(auto elim!: eventually_mono simp: eventually_at_filter)
thus ?thesis
by (simp add: not_essential_const)
next
case True
hence ev: "eventually (λw. g w ≠ g z) (at z)"
by (auto simp: isolated_zero_def)
from assms consider c where "f ─g z→ c" | "is_pole f (g z)"
by (auto simp: not_essential_def)
have "isCont g z"
by (rule analytic_at_imp_isCont) fact
hence lim: "g ─z→ g z"
using isContD by blast
from assms(1) consider c where "f ─g z→ c" | "is_pole f (g z)"
unfolding not_essential_def by blast
thus ?thesis
proof cases
fix c assume "f ─g z→ c"
hence "(λx. f (g x)) ─z→ c"
by (rule filterlim_compose) (use lim ev in ‹auto simp: filterlim_at›)
thus ?thesis
by (auto simp: not_essential_def)
next
assume "is_pole f (g z)"
hence "is_pole (λx. f (g x)) z"
by (rule is_pole_compose) fact+
thus ?thesis
by (auto simp: not_essential_def)
qed
qed
subsection ‹Isolated points›
definition isolated_points_of :: "complex set ⇒ complex set" where
"isolated_points_of A = {z∈A. eventually (λw. w ∉ A) (at z)}"
lemma isolated_points_of_altdef: "isolated_points_of A = {z∈A. ¬z islimpt A}"
unfolding isolated_points_of_def islimpt_def eventually_at_filter eventually_nhds by blast
lemma isolated_points_of_empty [simp]: "isolated_points_of {} = {}"
and isolated_points_of_UNIV [simp]: "isolated_points_of UNIV = {}"
by (auto simp: isolated_points_of_def)
lemma isolated_points_of_open_is_empty [simp]: "open A ⟹ isolated_points_of A = {}"
unfolding isolated_points_of_altdef
by (simp add: interior_limit_point interior_open)
lemma isolated_points_of_subset: "isolated_points_of A ⊆ A"
by (auto simp: isolated_points_of_def)
lemma isolated_points_of_discrete:
assumes "discrete A"
shows "isolated_points_of A = A"
using assms by (auto simp: isolated_points_of_def discrete_altdef)
lemmas uniform_discreteI1 = uniformI1
lemmas uniform_discreteI2 = uniformI2
lemma isolated_singularity_at_compose:
assumes "isolated_singularity_at f (g z)" "g analytic_on {z}"
shows "isolated_singularity_at (λx. f (g x)) z"
proof (cases "isolated_zero (λw. g w - g z) z")
case False
hence "eventually (λw. g w - g z = 0) (nhds z)"
by (rule non_isolated_zero) (use assms in ‹auto intro!: analytic_intros›)
hence "isolated_singularity_at (λx. f (g x)) z ⟷ isolated_singularity_at (λ_. f (g z)) z"
by (intro isolated_singularity_at_cong refl)
(auto elim!: eventually_mono simp: eventually_at_filter)
thus ?thesis
by (simp add: isolated_singularity_at_const)
next
case True
from assms(1) obtain r where r: "r > 0" "f analytic_on ball (g z) r - {g z}"
by (auto simp: isolated_singularity_at_def)
hence holo_f: "f holomorphic_on ball (g z) r - {g z}"
by (subst (asm) analytic_on_open) auto
from assms(2) obtain r' where r': "r' > 0" "g holomorphic_on ball z r'"
by (auto simp: analytic_on_def)
have "continuous_on (ball z r') g"
using holomorphic_on_imp_continuous_on r' by blast
hence "isCont g z"
using r' by (subst (asm) continuous_on_eq_continuous_at) auto
hence "g ─z→ g z"
using isContD by blast
hence "eventually (λw. g w ∈ ball (g z) r) (at z)"
using ‹r > 0› unfolding tendsto_def by force
moreover have "eventually (λw. g w ≠ g z) (at z)" using True
by (auto simp: isolated_zero_def elim!: eventually_mono)
ultimately have "eventually (λw. g w ∈ ball (g z) r - {g z}) (at z)"
by eventually_elim auto
then obtain r'' where r'': "r'' > 0" "∀w∈ball z r''-{z}. g w ∈ ball (g z) r - {g z}"
unfolding eventually_at_filter eventually_nhds_metric ball_def
by (auto simp: dist_commute)
have "f ∘ g holomorphic_on ball z (min r' r'') - {z}"
proof (rule holomorphic_on_compose_gen)
show "g holomorphic_on ball z (min r' r'') - {z}"
by (rule holomorphic_on_subset[OF r'(2)]) auto
show "f holomorphic_on ball (g z) r - {g z}"
by fact
show "g ` (ball z (min r' r'') - {z}) ⊆ ball (g z) r - {g z}"
using r'' by force
qed
hence "f ∘ g analytic_on ball z (min r' r'') - {z}"
by (subst analytic_on_open) auto
thus ?thesis using ‹r' > 0› ‹r'' > 0›
by (auto simp: isolated_singularity_at_def o_def intro!: exI[of _ "min r' r''"])
qed
lemma is_pole_power_int_0:
assumes "f analytic_on {x}" "isolated_zero f x" "n < 0"
shows "is_pole (λx. f x powi n) x"
proof -
have "f ─x→ f x"
using assms(1) by (simp add: analytic_at_imp_isCont isContD)
with assms show ?thesis
unfolding is_pole_def
by (intro filterlim_power_int_neg_at_infinity) (auto simp: isolated_zero_def)
qed
lemma isolated_zero_imp_not_constant_on:
assumes "isolated_zero f x" "x ∈ A" "open A"
shows "¬f constant_on A"
proof
assume "f constant_on A"
then obtain c where c: "⋀x. x ∈ A ⟹ f x = c"
by (auto simp: constant_on_def)
from assms and c[of x] have [simp]: "c = 0"
by (auto simp: isolated_zero_def)
have "eventually (λx. f x ≠ 0) (at x)"
using assms by (auto simp: isolated_zero_def)
moreover have "eventually (λx. x ∈ A) (at x)"
using assms by (intro eventually_at_in_open') auto
ultimately have "eventually (λx. False) (at x)"
by eventually_elim (use c in auto)
thus False
by simp
qed
end