Theory Laurent_Convergence
theory Laurent_Convergence
imports "HOL-Computational_Algebra.Formal_Laurent_Series" "HOL-Library.Landau_Symbols"
Residue_Theorem
begin
instance fps :: (semiring_char_0) semiring_char_0
proof
show "inj (of_nat :: nat ⇒ 'a fps)"
proof
fix m n :: nat
assume "of_nat m = (of_nat n :: 'a fps)"
hence "fps_nth (of_nat m) 0 = (fps_nth (of_nat n) 0 :: 'a)"
by (simp only: )
thus "m = n"
by simp
qed
qed
instance fls :: (semiring_char_0) semiring_char_0
proof
show "inj (of_nat :: nat ⇒ 'a fls)"
by (metis fls_regpart_of_nat injI of_nat_eq_iff)
qed
lemma fls_const_eq_0_iff [simp]: "fls_const c = 0 ⟷ c = 0"
using fls_const_0 fls_const_nonzero by blast
lemma fls_subdegree_add_eq1:
assumes "f ≠ 0" "fls_subdegree f < fls_subdegree g"
shows "fls_subdegree (f + g) = fls_subdegree f"
proof (intro antisym)
from assms have *: "fls_nth (f + g) (fls_subdegree f) ≠ 0"
by auto
from * show "fls_subdegree (f + g) ≤ fls_subdegree f"
by (rule fls_subdegree_leI)
from * have "f + g ≠ 0"
using fls_nonzeroI by blast
thus "fls_subdegree f ≤ fls_subdegree (f + g)"
using assms(2) fls_plus_subdegree by force
qed
lemma fls_subdegree_add_eq2:
assumes "g ≠ 0" "fls_subdegree g < fls_subdegree f"
shows "fls_subdegree (f + g) = fls_subdegree g"
proof (intro antisym)
from assms have *: "fls_nth (f + g) (fls_subdegree g) ≠ 0"
by auto
from * show "fls_subdegree (f + g) ≤ fls_subdegree g"
by (rule fls_subdegree_leI)
from * have "f + g ≠ 0"
using fls_nonzeroI by blast
thus "fls_subdegree g ≤ fls_subdegree (f + g)"
using assms(2) fls_plus_subdegree by force
qed
lemma fls_subdegree_diff_eq1:
assumes "f ≠ 0" "fls_subdegree f < fls_subdegree g"
shows "fls_subdegree (f - g) = fls_subdegree f"
using fls_subdegree_add_eq1[of f "-g"] assms by simp
lemma fls_subdegree_diff_eq2:
assumes "g ≠ 0" "fls_subdegree g < fls_subdegree f"
shows "fls_subdegree (f - g) = fls_subdegree g"
using fls_subdegree_add_eq2[of "-g" f] assms by simp
lemma nat_minus_fls_subdegree_plus_const_eq:
"nat (-fls_subdegree (F + fls_const c)) = nat (-fls_subdegree F)"
proof (cases "fls_subdegree F < 0")
case True
hence "fls_subdegree (F + fls_const c) = fls_subdegree F"
by (intro fls_subdegree_add_eq1) auto
thus ?thesis
by simp
next
case False
thus ?thesis
by (auto simp: fls_subdegree_ge0I)
qed
lemma at_to_0': "NO_MATCH 0 z ⟹ at z = filtermap (λx. x + z) (at 0)"
for z :: "'a::real_normed_vector"
by (rule at_to_0)
lemma nhds_to_0: "nhds (x :: 'a :: real_normed_vector) = filtermap ((+) x) (nhds 0)"
proof -
have "(λxa. xa - - x) = (+) x"
by auto
thus ?thesis
using filtermap_nhds_shift[of "-x" 0] by simp
qed
lemma nhds_to_0': "NO_MATCH 0 x ⟹ nhds (x :: 'a :: real_normed_vector) = filtermap ((+) x) (nhds 0)"
by (rule nhds_to_0)
definition%important fls_conv_radius :: "complex fls ⇒ ereal" where
"fls_conv_radius f = fps_conv_radius (fls_regpart f)"
definition%important eval_fls :: "complex fls ⇒ complex ⇒ complex" where
"eval_fls F z = eval_fps (fls_base_factor_to_fps F) z * z powi fls_subdegree F"
definition
has_laurent_expansion :: "(complex ⇒ complex) ⇒ complex fls ⇒ bool"
(infixl ‹has'_laurent'_expansion› 60)
where "(f has_laurent_expansion F) ⟷
fls_conv_radius F > 0 ∧ eventually (λz. eval_fls F z = f z) (at 0)"
lemma has_laurent_expansion_schematicI:
"f has_laurent_expansion F ⟹ F = G ⟹ f has_laurent_expansion G"
by simp
lemma has_laurent_expansion_cong:
assumes "eventually (λx. f x = g x) (at 0)" "F = G"
shows "(f has_laurent_expansion F) ⟷ (g has_laurent_expansion G)"
proof -
have "eventually (λz. eval_fls F z = g z) (at 0)"
if "eventually (λz. eval_fls F z = f z) (at 0)" "eventually (λx. f x = g x) (at 0)" for f g
using that by eventually_elim auto
from this[of f g] this[of g f] show ?thesis
using assms by (auto simp: eq_commute has_laurent_expansion_def)
qed
lemma has_laurent_expansion_cong':
assumes "eventually (λx. f x = g x) (at z)" "F = G" "z = z'"
shows "((λx. f (z + x)) has_laurent_expansion F) ⟷ ((λx. g (z' + x)) has_laurent_expansion G)"
by (intro has_laurent_expansion_cong)
(use assms in ‹auto simp: at_to_0' eventually_filtermap add_ac›)
lemma fls_conv_radius_altdef:
"fls_conv_radius F = fps_conv_radius (fls_base_factor_to_fps F)"
proof -
have "conv_radius (λn. fls_nth F (int n)) = conv_radius (λn. fls_nth F (int n + fls_subdegree F))"
proof (cases "fls_subdegree F ≥ 0")
case True
hence "conv_radius (λn. fls_nth F (int n + fls_subdegree F)) =
conv_radius (λn. fls_nth F (int (n + nat (fls_subdegree F))))"
by auto
thus ?thesis
by (subst (asm) conv_radius_shift) auto
next
case False
hence "conv_radius (λn. fls_nth F (int n)) =
conv_radius (λn. fls_nth F (fls_subdegree F + int (n + nat (-fls_subdegree F))))"
by auto
thus ?thesis
by (subst (asm) conv_radius_shift) (auto simp: add_ac)
qed
thus ?thesis
by (simp add: fls_conv_radius_def fps_conv_radius_def)
qed
lemma eval_fps_of_nat [simp]: "eval_fps (of_nat n) z = of_nat n"
and eval_fps_of_int [simp]: "eval_fps (of_int m) z = of_int m"
by (simp_all flip: fps_of_nat fps_of_int)
lemma fls_subdegree_numeral [simp]: "fls_subdegree (numeral n) = 0"
by (metis fls_subdegree_of_nat of_nat_numeral)
lemma fls_regpart_numeral [simp]: "fls_regpart (numeral n) = numeral n"
by (metis fls_regpart_of_nat of_nat_numeral)
lemma fps_conv_radius_of_nat [simp]: "fps_conv_radius (of_nat n) = ∞"
and fps_conv_radius_of_int [simp]: "fps_conv_radius (of_int m) = ∞"
by (simp_all flip: fps_of_nat fps_of_int)
lemma fps_conv_radius_fls_regpart: "fps_conv_radius (fls_regpart F) = fls_conv_radius F"
by (simp add: fls_conv_radius_def)
lemma fls_conv_radius_0 [simp]: "fls_conv_radius 0 = ∞"
and fls_conv_radius_1 [simp]: "fls_conv_radius 1 = ∞"
and fls_conv_radius_const [simp]: "fls_conv_radius (fls_const c) = ∞"
and fls_conv_radius_numeral [simp]: "fls_conv_radius (numeral num) = ∞"
and fls_conv_radius_of_nat [simp]: "fls_conv_radius (of_nat n) = ∞"
and fls_conv_radius_of_int [simp]: "fls_conv_radius (of_int m) = ∞"
and fls_conv_radius_X [simp]: "fls_conv_radius fls_X = ∞"
and fls_conv_radius_X_inv [simp]: "fls_conv_radius fls_X_inv = ∞"
and fls_conv_radius_X_intpow [simp]: "fls_conv_radius (fls_X_intpow m) = ∞"
by (simp_all add: fls_conv_radius_def fls_X_intpow_regpart)
lemma fls_conv_radius_shift [simp]: "fls_conv_radius (fls_shift n F) = fls_conv_radius F"
unfolding fls_conv_radius_altdef by (subst fls_base_factor_to_fps_shift) (rule refl)
lemma fls_conv_radius_fps_to_fls [simp]: "fls_conv_radius (fps_to_fls F) = fps_conv_radius F"
by (simp add: fls_conv_radius_def)
lemma fls_conv_radius_deriv [simp]: "fls_conv_radius (fls_deriv F) ≥ fls_conv_radius F"
proof -
have "fls_conv_radius (fls_deriv F) = fps_conv_radius (fls_regpart (fls_deriv F))"
by (simp add: fls_conv_radius_def)
also have "fls_regpart (fls_deriv F) = fps_deriv (fls_regpart F)"
by (intro fps_ext) (auto simp: add_ac)
also have "fps_conv_radius … ≥ fls_conv_radius F"
by (simp add: fls_conv_radius_def fps_conv_radius_deriv)
finally show ?thesis .
qed
lemma fls_conv_radius_uminus [simp]: "fls_conv_radius (-F) = fls_conv_radius F"
by (simp add: fls_conv_radius_def)
lemma fls_conv_radius_add: "fls_conv_radius (F + G) ≥ min (fls_conv_radius F) (fls_conv_radius G)"
by (simp add: fls_conv_radius_def fps_conv_radius_add)
lemma fls_conv_radius_diff: "fls_conv_radius (F - G) ≥ min (fls_conv_radius F) (fls_conv_radius G)"
by (simp add: fls_conv_radius_def fps_conv_radius_diff)
lemma fls_conv_radius_mult: "fls_conv_radius (F * G) ≥ min (fls_conv_radius F) (fls_conv_radius G)"
proof (cases "F = 0 ∨ G = 0")
case False
hence [simp]: "F ≠ 0" "G ≠ 0"
by auto
have "fls_conv_radius (F * G) = fps_conv_radius (fls_regpart (fls_shift (fls_subdegree F + fls_subdegree G) (F * G)))"
by (simp add: fls_conv_radius_altdef)
also have "fls_regpart (fls_shift (fls_subdegree F + fls_subdegree G) (F * G)) =
fls_base_factor_to_fps F * fls_base_factor_to_fps G"
by (simp add: fls_times_def)
also have "fps_conv_radius … ≥ min (fls_conv_radius F) (fls_conv_radius G)"
unfolding fls_conv_radius_altdef by (rule fps_conv_radius_mult)
finally show ?thesis .
qed auto
lemma fps_conv_radius_add_ge:
"fps_conv_radius F ≥ r ⟹ fps_conv_radius G ≥ r ⟹ fps_conv_radius (F + G) ≥ r"
using fps_conv_radius_add[of F G] by (simp add: min_def split: if_splits)
lemma fps_conv_radius_diff_ge:
"fps_conv_radius F ≥ r ⟹ fps_conv_radius G ≥ r ⟹ fps_conv_radius (F - G) ≥ r"
using fps_conv_radius_diff[of F G] by (simp add: min_def split: if_splits)
lemma fps_conv_radius_mult_ge:
"fps_conv_radius F ≥ r ⟹ fps_conv_radius G ≥ r ⟹ fps_conv_radius (F * G) ≥ r"
using fps_conv_radius_mult[of F G] by (simp add: min_def split: if_splits)
lemma fls_conv_radius_add_ge:
"fls_conv_radius F ≥ r ⟹ fls_conv_radius G ≥ r ⟹ fls_conv_radius (F + G) ≥ r"
using fls_conv_radius_add[of F G] by (simp add: min_def split: if_splits)
lemma fls_conv_radius_diff_ge:
"fls_conv_radius F ≥ r ⟹ fls_conv_radius G ≥ r ⟹ fls_conv_radius (F - G) ≥ r"
using fls_conv_radius_diff[of F G] by (simp add: min_def split: if_splits)
lemma fls_conv_radius_mult_ge:
"fls_conv_radius F ≥ r ⟹ fls_conv_radius G ≥ r ⟹ fls_conv_radius (F * G) ≥ r"
using fls_conv_radius_mult[of F G] by (simp add: min_def split: if_splits)
lemma fls_conv_radius_power: "fls_conv_radius (F ^ n) ≥ fls_conv_radius F"
by (induction n) (auto intro!: fls_conv_radius_mult_ge)
lemma eval_fls_0 [simp]: "eval_fls 0 z = 0"
and eval_fls_1 [simp]: "eval_fls 1 z = 1"
and eval_fls_const [simp]: "eval_fls (fls_const c) z = c"
and eval_fls_numeral [simp]: "eval_fls (numeral num) z = numeral num"
and eval_fls_of_nat [simp]: "eval_fls (of_nat n) z = of_nat n"
and eval_fls_of_int [simp]: "eval_fls (of_int m) z = of_int m"
and eval_fls_X [simp]: "eval_fls fls_X z = z"
and eval_fls_X_intpow [simp]: "eval_fls (fls_X_intpow m) z = z powi m"
by (simp_all add: eval_fls_def)
lemma eval_fls_at_0: "eval_fls F 0 = (if fls_subdegree F ≥ 0 then fls_nth F 0 else 0)"
by (cases "fls_subdegree F = 0")
(simp_all add: eval_fls_def fls_regpart_def eval_fps_at_0)
lemma eval_fps_to_fls:
assumes "norm z < fps_conv_radius F"
shows "eval_fls (fps_to_fls F) z = eval_fps F z"
proof (cases "F = 0")
case [simp]: False
have "eval_fps F z = eval_fps (unit_factor F * normalize F) z"
by (metis unit_factor_mult_normalize)
also have "… = eval_fps (unit_factor F * fps_X ^ subdegree F) z"
by simp
also have "… = eval_fps (unit_factor F) z * z ^ subdegree F"
using assms by (subst eval_fps_mult) auto
also have "… = eval_fls (fps_to_fls F) z"
unfolding eval_fls_def fls_base_factor_to_fps_to_fls fls_subdegree_fls_to_fps
power_int_of_nat ..
finally show ?thesis ..
qed auto
lemma eval_fls_shift:
assumes [simp]: "z ≠ 0"
shows "eval_fls (fls_shift n F) z = eval_fls F z * z powi -n"
proof (cases "F = 0")
case [simp]: False
show ?thesis
unfolding eval_fls_def
by (subst fls_base_factor_to_fps_shift, subst fls_shift_subdegree[OF ‹F ≠ 0›], subst power_int_diff)
(auto simp: power_int_minus divide_simps)
qed auto
lemma eval_fls_add:
assumes "ereal (norm z) < fls_conv_radius F" "ereal (norm z) < fls_conv_radius G" "z ≠ 0"
shows "eval_fls (F + G) z = eval_fls F z + eval_fls G z"
using assms
proof (induction "fls_subdegree F" "fls_subdegree G" arbitrary: F G rule: linorder_wlog)
case (sym F G)
show ?case
using sym(1)[of G F] sym(2-) by (simp add: add_ac)
next
case (le F G)
show ?case
proof (cases "F = 0 ∨ G = 0")
case False
hence [simp]: "F ≠ 0" "G ≠ 0"
by auto
note [simp] = ‹z ≠ 0›
define F' G' where "F' = fls_base_factor_to_fps F" "G' = fls_base_factor_to_fps G"
define m n where "m = fls_subdegree F" "n = fls_subdegree G"
have "m ≤ n"
using le by (auto simp: m_n_def)
have conv1: "ereal (cmod z) < fps_conv_radius F'" "ereal (cmod z) < fps_conv_radius G'"
using assms le by (simp_all add: F'_G'_def fls_conv_radius_altdef)
have conv2: "ereal (cmod z) < fps_conv_radius (G' * fps_X ^ nat (n - m))"
using conv1 by (intro less_le_trans[OF _ fps_conv_radius_mult]) auto
have conv3: "ereal (cmod z) < fps_conv_radius (F' + G' * fps_X ^ nat (n - m))"
using conv1 conv2 by (intro less_le_trans[OF _ fps_conv_radius_add]) auto
have "eval_fls F z + eval_fls G z = eval_fps F' z * z powi m + eval_fps G' z * z powi n"
unfolding eval_fls_def m_n_def[symmetric] F'_G'_def[symmetric]
by (simp add: power_int_add algebra_simps)
also have "… = (eval_fps F' z + eval_fps G' z * z powi (n - m)) * z powi m"
by (simp add: algebra_simps power_int_diff)
also have "eval_fps G' z * z powi (n - m) = eval_fps (G' * fps_X ^ nat (n - m)) z"
using assms ‹m ≤ n› conv1 by (subst eval_fps_mult) (auto simp: power_int_def)
also have "eval_fps F' z + … = eval_fps (F' + G' * fps_X ^ nat (n - m)) z"
using conv1 conv2 by (subst eval_fps_add) auto
also have "… = eval_fls (fps_to_fls (F' + G' * fps_X ^ nat (n - m))) z"
using conv3 by (subst eval_fps_to_fls) auto
also have "… * z powi m = eval_fls (fls_shift (-m) (fps_to_fls (F' + G' * fps_X ^ nat (n - m)))) z"
by (subst eval_fls_shift) auto
also have "fls_shift (-m) (fps_to_fls (F' + G' * fps_X ^ nat (n - m))) = F + G"
using ‹m ≤ n›
by (simp add: fls_times_fps_to_fls fps_to_fls_power fls_X_power_conv_shift_1
fls_shifted_times_simps F'_G'_def m_n_def)
finally show ?thesis ..
qed auto
qed
lemma eval_fls_minus:
assumes "ereal (norm z) < fls_conv_radius F"
shows "eval_fls (-F) z = -eval_fls F z"
using assms by (simp add: eval_fls_def eval_fps_minus fls_conv_radius_altdef)
lemma eval_fls_diff:
assumes "ereal (norm z) < fls_conv_radius F" "ereal (norm z) < fls_conv_radius G"
and [simp]: "z ≠ 0"
shows "eval_fls (F - G) z = eval_fls F z - eval_fls G z"
proof -
have "eval_fls (F + (-G)) z = eval_fls F z - eval_fls G z"
using assms by (subst eval_fls_add) (auto simp: eval_fls_minus)
thus ?thesis
by simp
qed
lemma eval_fls_mult:
assumes "ereal (norm z) < fls_conv_radius F" "ereal (norm z) < fls_conv_radius G" "z ≠ 0"
shows "eval_fls (F * G) z = eval_fls F z * eval_fls G z"
proof (cases "F = 0 ∨ G = 0")
case False
hence [simp]: "F ≠ 0" "G ≠ 0"
by auto
note [simp] = ‹z ≠ 0›
define F' G' where "F' = fls_base_factor_to_fps F" "G' = fls_base_factor_to_fps G"
define m n where "m = fls_subdegree F" "n = fls_subdegree G"
have "eval_fls F z * eval_fls G z = (eval_fps F' z * eval_fps G' z) * z powi (m + n)"
unfolding eval_fls_def m_n_def[symmetric] F'_G'_def[symmetric]
by (simp add: power_int_add algebra_simps)
also have "… = eval_fps (F' * G') z * z powi (m + n)"
using assms by (subst eval_fps_mult) (auto simp: F'_G'_def fls_conv_radius_altdef)
also have "… = eval_fls (F * G) z"
by (simp add: eval_fls_def F'_G'_def m_n_def) (simp add: fls_times_def)
finally show ?thesis ..
qed auto
lemma eval_fls_power:
assumes "ereal (norm z) < fls_conv_radius F" "z ≠ 0"
shows "eval_fls (F ^ n) z = eval_fls F z ^ n"
proof (induction n)
case (Suc n)
have "eval_fls (F ^ Suc n) z = eval_fls (F * F ^ n) z"
by simp
also have "… = eval_fls F z * eval_fls (F ^ n) z"
using assms by (subst eval_fls_mult) (auto intro!: less_le_trans[OF _ fls_conv_radius_power])
finally show ?case
using Suc by simp
qed auto
lemma norm_summable_fls:
"norm z < fls_conv_radius f ⟹ summable (λn. norm (fls_nth f n * z ^ n))"
using norm_summable_fps[of z "fls_regpart f"] by (simp add: fls_conv_radius_def)
lemma norm_summable_fls':
"norm z < fls_conv_radius f ⟹ summable (λn. norm (fls_nth f (n + fls_subdegree f) * z ^ n))"
using norm_summable_fps[of z "fls_base_factor_to_fps f"] by (simp add: fls_conv_radius_altdef)
lemma summable_fls:
"norm z < fls_conv_radius f ⟹ summable (λn. fls_nth f n * z ^ n)"
by (rule summable_norm_cancel[OF norm_summable_fls])
theorem sums_eval_fls:
fixes f
defines "n ≡ fls_subdegree f"
assumes "norm z < fls_conv_radius f" and "z ≠ 0 ∨ n ≥ 0"
shows "(λk. fls_nth f (int k + n) * z powi (int k + n)) sums eval_fls f z"
proof (cases "z = 0")
case [simp]: False
have "(λk. fps_nth (fls_base_factor_to_fps f) k * z ^ k * z powi n) sums
(eval_fps (fls_base_factor_to_fps f) z * z powi n)"
using assms(2) by (intro sums_eval_fps sums_mult2) (auto simp: fls_conv_radius_altdef)
thus ?thesis
by (simp add: power_int_add n_def eval_fls_def mult_ac)
next
case [simp]: True
with assms have "n ≥ 0"
by auto
have "(λk. fls_nth f (int k + n) * z powi (int k + n)) sums
(∑k∈(if n ≤ 0 then {nat (-n)} else {}). fls_nth f (int k + n) * z powi (int k + n))"
by (intro sums_finite) (auto split: if_splits)
also have "… = eval_fls f z"
using ‹n ≥ 0› by (auto simp: eval_fls_at_0 n_def not_le)
finally show ?thesis .
qed
lemma holomorphic_on_eval_fls:
fixes f
defines "n ≡ fls_subdegree f"
assumes "A ⊆ eball 0 (fls_conv_radius f) - (if n ≥ 0 then {} else {0})"
shows "eval_fls f holomorphic_on A"
proof (cases "n ≥ 0")
case True
have "eval_fls f = (λz. eval_fps (fls_base_factor_to_fps f) z * z ^ nat n)"
using True by (simp add: fun_eq_iff eval_fls_def power_int_def n_def)
moreover have "… holomorphic_on A"
using True assms(2) by (intro holomorphic_intros) (auto simp: fls_conv_radius_altdef)
ultimately show ?thesis
by simp
next
case False
show ?thesis using assms
unfolding eval_fls_def by (intro holomorphic_intros) (auto simp: fls_conv_radius_altdef)
qed
lemma holomorphic_on_eval_fls' [holomorphic_intros]:
assumes "g holomorphic_on A"
assumes "g ` A ⊆ eball 0 (fls_conv_radius f) - (if fls_subdegree f ≥ 0 then {} else {0})"
shows "(λx. eval_fls f (g x)) holomorphic_on A"
by (meson assms holomorphic_on_compose holomorphic_on_eval_fls holomorphic_transform o_def)
lemma continuous_on_eval_fls:
fixes f
defines "n ≡ fls_subdegree f"
assumes "A ⊆ eball 0 (fls_conv_radius f) - (if n ≥ 0 then {} else {0})"
shows "continuous_on A (eval_fls f)"
using assms holomorphic_on_eval_fls holomorphic_on_imp_continuous_on by blast
lemma continuous_on_eval_fls' [continuous_intros]:
fixes f
defines "n ≡ fls_subdegree f"
assumes "g ` A ⊆ eball 0 (fls_conv_radius f) - (if n ≥ 0 then {} else {0})"
assumes "continuous_on A g"
shows "continuous_on A (λx. eval_fls f (g x))"
by (metis assms continuous_on_compose2 continuous_on_eval_fls order.refl)
lemmas has_field_derivative_eval_fps' [derivative_intros] =
DERIV_chain2[OF has_field_derivative_eval_fps]
lemma fps_deriv_fls_regpart: "fps_deriv (fls_regpart F) = fls_regpart (fls_deriv F)"
by (intro fps_ext) (auto simp: add_ac)
lemma has_field_derivative_eval_fls:
assumes "z ∈ eball 0 (fls_conv_radius f) - {0}"
shows "(eval_fls f has_field_derivative eval_fls (fls_deriv f) z) (at z within A)"
proof -
define g where "g = fls_base_factor_to_fps f"
define n where "n = fls_subdegree f"
have [simp]: "fps_conv_radius g = fls_conv_radius f"
by (simp add: fls_conv_radius_altdef g_def)
have conv1: "fps_conv_radius (fps_deriv g * fps_X) ≥ fls_conv_radius f"
by (intro fps_conv_radius_mult_ge order.trans[OF _ fps_conv_radius_deriv]) auto
have conv2: "fps_conv_radius (of_int n * g) ≥ fls_conv_radius f"
by (intro fps_conv_radius_mult_ge) auto
have conv3: "fps_conv_radius (fps_deriv g * fps_X + of_int n * g) ≥ fls_conv_radius f"
by (intro fps_conv_radius_add_ge conv1 conv2)
have [simp]: "fps_conv_radius g = fls_conv_radius f"
by (simp add: g_def fls_conv_radius_altdef)
have "((λz. eval_fps g z * z powi fls_subdegree f) has_field_derivative
(eval_fps (fps_deriv g) z * z powi n + of_int n * z powi (n - 1) * eval_fps g z))
(at z within A)"
using assms by (auto intro!: derivative_eq_intros simp: n_def)
also have "(λz. eval_fps g z * z powi fls_subdegree f) = eval_fls f"
by (simp add: eval_fls_def g_def fun_eq_iff)
also have "eval_fps (fps_deriv g) z * z powi n + of_int n * z powi (n - 1) * eval_fps g z =
(z * eval_fps (fps_deriv g) z + of_int n * eval_fps g z) * z powi (n - 1)"
using assms by (auto simp: power_int_diff field_simps)
also have "(z * eval_fps (fps_deriv g) z + of_int n * eval_fps g z) =
eval_fps (fps_deriv g * fps_X + of_int n * g) z"
using conv1 conv2 assms fps_conv_radius_deriv[of g]
by (subst eval_fps_add) (auto simp: eval_fps_mult)
also have "… = eval_fls (fps_to_fls (fps_deriv g * fps_X + of_int n * g)) z"
using conv3 assms by (subst eval_fps_to_fls) auto
also have "… * z powi (n - 1) = eval_fls (fls_shift (1 - n) (fps_to_fls (fps_deriv g * fps_X + of_int n * g))) z"
using assms by (subst eval_fls_shift) auto
also have "fls_shift (1 - n) (fps_to_fls (fps_deriv g * fps_X + of_int n * g)) = fls_deriv f"
by (intro fls_eqI) (auto simp: g_def n_def algebra_simps eq_commute[of _ "fls_subdegree f"])
finally show ?thesis .
qed
lemma eval_fls_deriv:
assumes "z ∈ eball 0 (fls_conv_radius F) - {0}"
shows "eval_fls (fls_deriv F) z = deriv (eval_fls F) z"
by (metis DERIV_imp_deriv assms has_field_derivative_eval_fls)
lemma analytic_on_eval_fls:
assumes "A ⊆ eball 0 (fls_conv_radius f) - (if fls_subdegree f ≥ 0 then {} else {0})"
shows "eval_fls f analytic_on A"
proof (rule analytic_on_subset [OF _ assms])
show "eval_fls f analytic_on eball 0 (fls_conv_radius f) - (if fls_subdegree f ≥ 0 then {} else {0})"
using holomorphic_on_eval_fls[OF order.refl]
by (subst analytic_on_open) auto
qed
lemma analytic_on_eval_fls' [analytic_intros]:
assumes "g analytic_on A"
assumes "g ` A ⊆ eball 0 (fls_conv_radius f) - (if fls_subdegree f ≥ 0 then {} else {0})"
shows "(λx. eval_fls f (g x)) analytic_on A"
proof -
have "eval_fls f ∘ g analytic_on A"
by (intro analytic_on_compose[OF assms(1) analytic_on_eval_fls]) (use assms in auto)
thus ?thesis
by (simp add: o_def)
qed
lemma continuous_eval_fls [continuous_intros]:
assumes "z ∈ eball 0 (fls_conv_radius F) - (if fls_subdegree F ≥ 0 then {} else {0})"
shows "continuous (at z within A) (eval_fls F)"
proof -
have "isCont (eval_fls F) z"
using continuous_on_eval_fls[OF order.refl] assms
by (subst (asm) continuous_on_eq_continuous_at) auto
thus ?thesis
using continuous_at_imp_continuous_at_within by blast
qed
named_theorems laurent_expansion_intros
lemma has_laurent_expansion_imp_asymp_equiv_0:
assumes F: "f has_laurent_expansion F"
defines "n ≡ fls_subdegree F"
shows "f ∼[at 0] (λz. fls_nth F n * z powi n)"
proof (cases "F = 0")
case True
thus ?thesis using assms
by (auto simp: has_laurent_expansion_def)
next
case [simp]: False
define G where "G = fls_base_factor_to_fps F"
have "fls_conv_radius F > 0"
using F by (auto simp: has_laurent_expansion_def)
hence "isCont (eval_fps G) 0"
by (intro continuous_intros) (auto simp: G_def fps_conv_radius_fls_regpart zero_ereal_def)
hence lim: "eval_fps G ─0→ eval_fps G 0"
by (meson isContD)
have [simp]: "fps_nth G 0 ≠ 0"
by (auto simp: G_def)
have "f ∼[at 0] eval_fls F"
using F by (intro asymp_equiv_refl_ev) (auto simp: has_laurent_expansion_def eq_commute)
also have "… = (λz. eval_fps G z * z powi n)"
by (intro ext) (simp_all add: eval_fls_def G_def n_def)
also have "… ∼[at 0] (λz. fps_nth G 0 * z powi n)" using lim
by (intro asymp_equiv_intros tendsto_imp_asymp_equiv_const) (auto simp: eval_fps_at_0)
also have "fps_nth G 0 = fls_nth F n"
by (simp add: G_def n_def)
finally show ?thesis
by simp
qed
lemma has_laurent_expansion_imp_asymp_equiv:
assumes F: "(λw. f (z + w)) has_laurent_expansion F"
defines "n ≡ fls_subdegree F"
shows "f ∼[at z] (λw. fls_nth F n * (w - z) powi n)"
using has_laurent_expansion_imp_asymp_equiv_0[OF assms(1)] unfolding n_def
by (simp add: at_to_0[of z] asymp_equiv_filtermap_iff add_ac)
lemmas [tendsto_intros del] = tendsto_power_int
lemma has_laurent_expansion_imp_tendsto_0:
assumes F: "f has_laurent_expansion F" and "fls_subdegree F ≥ 0"
shows "f ─0→ fls_nth F 0"
proof (rule asymp_equiv_tendsto_transfer)
show "(λz. fls_nth F (fls_subdegree F) * z powi fls_subdegree F) ∼[at 0] f"
by (rule asymp_equiv_symI, rule has_laurent_expansion_imp_asymp_equiv_0) fact
show "(λz. fls_nth F (fls_subdegree F) * z powi fls_subdegree F) ─0→ fls_nth F 0"
by (rule tendsto_eq_intros refl | use assms(2) in simp)+
(use assms(2) in ‹auto simp: power_int_0_left_if›)
qed
lemma has_laurent_expansion_imp_tendsto:
assumes F: "(λw. f (z + w)) has_laurent_expansion F" and "fls_subdegree F ≥ 0"
shows "f ─z→ fls_nth F 0"
using has_laurent_expansion_imp_tendsto_0[OF assms]
by (simp add: at_to_0[of z] filterlim_filtermap add_ac)
lemma has_laurent_expansion_imp_filterlim_infinity_0:
assumes F: "f has_laurent_expansion F" and "fls_subdegree F < 0"
shows "filterlim f at_infinity (at 0)"
proof (rule asymp_equiv_at_infinity_transfer)
have [simp]: "F ≠ 0"
using assms(2) by auto
show "(λz. fls_nth F (fls_subdegree F) * z powi fls_subdegree F) ∼[at 0] f"
by (rule asymp_equiv_symI, rule has_laurent_expansion_imp_asymp_equiv_0) fact
show "filterlim (λz. fls_nth F (fls_subdegree F) * z powi fls_subdegree F) at_infinity (at 0)"
by (rule tendsto_mult_filterlim_at_infinity tendsto_const
filterlim_power_int_neg_at_infinity | use assms(2) in simp)+
(auto simp: eventually_at_filter)
qed
lemma has_laurent_expansion_imp_neg_fls_subdegree:
assumes F: "f has_laurent_expansion F"
and infy:"filterlim f at_infinity (at 0)"
shows "fls_subdegree F < 0"
proof (rule ccontr)
assume asm:"¬ fls_subdegree F < 0"
define ff where "ff=(λz. fls_nth F (fls_subdegree F)
* z powi fls_subdegree F)"
have "(ff ⤏ (if fls_subdegree F =0 then fls_nth F 0 else 0)) (at 0)"
using asm unfolding ff_def
by (auto intro!: tendsto_eq_intros)
moreover have "filterlim ff at_infinity (at 0)"
proof (rule asymp_equiv_at_infinity_transfer)
show "f ∼[at 0] ff" unfolding ff_def
using has_laurent_expansion_imp_asymp_equiv_0[OF F] unfolding ff_def .
show "filterlim f at_infinity (at 0)" by fact
qed
ultimately show False
using not_tendsto_and_filterlim_at_infinity[of "at (0::complex)"] by auto
qed
lemma has_laurent_expansion_imp_filterlim_infinity:
assumes F: "(λw. f (z + w)) has_laurent_expansion F" and "fls_subdegree F < 0"
shows "filterlim f at_infinity (at z)"
using has_laurent_expansion_imp_filterlim_infinity_0[OF assms]
by (simp add: at_to_0[of z] filterlim_filtermap add_ac)
lemma has_laurent_expansion_imp_is_pole_0:
assumes F: "f has_laurent_expansion F" and "fls_subdegree F < 0"
shows "is_pole f 0"
using has_laurent_expansion_imp_filterlim_infinity_0[OF assms]
by (simp add: is_pole_def)
lemma is_pole_0_imp_neg_fls_subdegree:
assumes F: "f has_laurent_expansion F" and "is_pole f 0"
shows "fls_subdegree F < 0"
using F assms(2) has_laurent_expansion_imp_neg_fls_subdegree is_pole_def
by blast
lemma has_laurent_expansion_imp_is_pole:
assumes F: "(λx. f (z + x)) has_laurent_expansion F" and "fls_subdegree F < 0"
shows "is_pole f z"
using has_laurent_expansion_imp_is_pole_0[OF assms]
by (simp add: is_pole_shift_0')
lemma is_pole_imp_neg_fls_subdegree:
assumes F: "(λx. f (z + x)) has_laurent_expansion F" and "is_pole f z"
shows "fls_subdegree F < 0"
proof -
have "is_pole (λx. f (z + x)) 0"
using assms(2) is_pole_shift_0 by blast
then show ?thesis
using F is_pole_0_imp_neg_fls_subdegree by blast
qed
lemma is_pole_fls_subdegree_iff:
assumes "(λx. f (z + x)) has_laurent_expansion F"
shows "is_pole f z ⟷ fls_subdegree F < 0"
using assms is_pole_imp_neg_fls_subdegree has_laurent_expansion_imp_is_pole
by auto
lemma
assumes "f has_laurent_expansion F"
shows has_laurent_expansion_isolated_0: "isolated_singularity_at f 0"
and has_laurent_expansion_not_essential_0: "not_essential f 0"
proof -
from assms have "eventually (λz. eval_fls F z = f z) (at 0)"
by (auto simp: has_laurent_expansion_def)
then obtain r where r: "r > 0" "⋀z. z ∈ ball 0 r - {0} ⟹ eval_fls F z = f z"
by (auto simp: eventually_at_filter ball_def eventually_nhds_metric)
have "fls_conv_radius F > 0"
using assms by (auto simp: has_laurent_expansion_def)
then obtain R :: real where R: "R > 0" "R ≤ min r (fls_conv_radius F)"
using ‹r > 0› by (metis dual_order.strict_implies_order ereal_dense2 ereal_less(2) min_def)
have "eval_fls F holomorphic_on ball 0 R - {0}"
using r R by (intro holomorphic_intros ball_eball_mono Diff_mono) (auto simp: ereal_le_less)
also have "?this ⟷ f holomorphic_on ball 0 R - {0}"
using r R by (intro holomorphic_cong) auto
also have "… ⟷ f analytic_on ball 0 R - {0}"
by (subst analytic_on_open) auto
finally show "isolated_singularity_at f 0"
unfolding isolated_singularity_at_def using ‹R > 0› by blast
show "not_essential f 0"
proof (cases "fls_subdegree F ≥ 0")
case True
hence "f ─0→ fls_nth F 0"
by (intro has_laurent_expansion_imp_tendsto_0[OF assms])
thus ?thesis
by (auto simp: not_essential_def)
next
case False
hence "is_pole f 0"
by (intro has_laurent_expansion_imp_is_pole_0[OF assms]) auto
thus ?thesis
by (auto simp: not_essential_def)
qed
qed
lemma
assumes "(λw. f (z + w)) has_laurent_expansion F"
shows has_laurent_expansion_isolated: "isolated_singularity_at f z"
and has_laurent_expansion_not_essential: "not_essential f z"
using has_laurent_expansion_isolated_0[OF assms] has_laurent_expansion_not_essential_0[OF assms]
by (simp_all add: isolated_singularity_at_shift_0 not_essential_shift_0)
lemma has_laurent_expansion_fps:
assumes "f has_fps_expansion F"
shows "f has_laurent_expansion fps_to_fls F"
proof -
from assms have radius: "0 < fps_conv_radius F" and eval: "∀⇩F z in nhds 0. eval_fps F z = f z"
by (auto simp: has_fps_expansion_def)
from eval have eval': "∀⇩F z in at 0. eval_fps F z = f z"
using eventually_at_filter eventually_mono by fastforce
moreover have "eventually (λz. z ∈ eball 0 (fps_conv_radius F) - {0}) (at 0)"
using radius by (intro eventually_at_in_open) (auto simp: zero_ereal_def)
ultimately have "eventually (λz. eval_fls (fps_to_fls F) z = f z) (at 0)"
by eventually_elim (auto simp: eval_fps_to_fls)
thus ?thesis using radius
by (auto simp: has_laurent_expansion_def)
qed
lemma has_laurent_expansion_const [simp, intro, laurent_expansion_intros]:
"(λ_. c) has_laurent_expansion fls_const c"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_0 [simp, intro, laurent_expansion_intros]:
"(λ_. 0) has_laurent_expansion 0"
by (auto simp: has_laurent_expansion_def)
lemma has_fps_expansion_0_iff: "f has_fps_expansion 0 ⟷ eventually (λz. f z = 0) (nhds 0)"
by (auto simp: has_fps_expansion_def)
lemma has_laurent_expansion_1 [simp, intro, laurent_expansion_intros]:
"(λ_. 1) has_laurent_expansion 1"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_numeral [simp, intro, laurent_expansion_intros]:
"(λ_. numeral n) has_laurent_expansion numeral n"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_fps_X_power [laurent_expansion_intros]:
"(λx. x ^ n) has_laurent_expansion (fls_X_intpow n)"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_fps_X_power_int [laurent_expansion_intros]:
"(λx. x powi n) has_laurent_expansion (fls_X_intpow n)"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_fps_X [laurent_expansion_intros]:
"(λx. x) has_laurent_expansion fls_X"
by (auto simp: has_laurent_expansion_def)
lemma has_laurent_expansion_cmult_left [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(λx. c * f x) has_laurent_expansion fls_const c * F"
proof -
from assms have "eventually (λz. z ∈ eball 0 (fls_conv_radius F) - {0}) (at 0)"
by (intro eventually_at_in_open) (auto simp: has_laurent_expansion_def zero_ereal_def)
moreover from assms have "eventually (λz. eval_fls F z = f z) (at 0)"
by (auto simp: has_laurent_expansion_def)
ultimately have "eventually (λz. eval_fls (fls_const c * F) z = c * f z) (at 0)"
by eventually_elim (simp_all add: eval_fls_mult)
with assms show ?thesis
by (auto simp: has_laurent_expansion_def intro!: less_le_trans[OF _ fls_conv_radius_mult])
qed
lemma has_laurent_expansion_cmult_right [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(λx. f x * c) has_laurent_expansion F * fls_const c"
proof -
have "F * fls_const c = fls_const c * F"
by (intro fls_eqI) (auto simp: mult.commute)
with has_laurent_expansion_cmult_left [OF assms] show ?thesis
by (simp add: mult.commute)
qed
lemma has_fps_expansion_scaleR [fps_expansion_intros]:
fixes F :: "'a :: {banach, real_normed_div_algebra, comm_ring_1} fps"
shows "f has_fps_expansion F ⟹ (λx. c *⇩R f x) has_fps_expansion fps_const (of_real c) * F"
unfolding scaleR_conv_of_real by (intro fps_expansion_intros)
lemma has_laurent_expansion_scaleR [laurent_expansion_intros]:
"f has_laurent_expansion F ⟹ (λx. c *⇩R f x) has_laurent_expansion fls_const (of_real c) * F"
unfolding scaleR_conv_of_real by (intro laurent_expansion_intros)
lemma has_laurent_expansion_minus [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(λx. - f x) has_laurent_expansion -F"
proof -
from assms have "eventually (λx. x ∈ eball 0 (fls_conv_radius F) - {0}) (at 0)"
by (intro eventually_at_in_open) (auto simp: has_laurent_expansion_def zero_ereal_def)
moreover from assms have "eventually (λx. eval_fls F x = f x) (at 0)"
by (auto simp: has_laurent_expansion_def)
ultimately have "eventually (λx. eval_fls (-F) x = -f x) (at 0)"
by eventually_elim (auto simp: eval_fls_minus)
thus ?thesis using assms by (auto simp: has_laurent_expansion_def)
qed
lemma has_laurent_expansion_add [laurent_expansion_intros]:
assumes "f has_laurent_expansion F" "g has_laurent_expansion G"
shows "(λx. f x + g x) has_laurent_expansion F + G"
proof -
from assms have "0 < min (fls_conv_radius F) (fls_conv_radius G)"
by (auto simp: has_laurent_expansion_def)
also have "… ≤ fls_conv_radius (F + G)"
by (rule fls_conv_radius_add)
finally have radius: "… > 0" .
from assms have "eventually (λx. x ∈ eball 0 (fls_conv_radius F) - {0}) (at 0)"
"eventually (λx. x ∈ eball 0 (fls_conv_radius G) - {0}) (at 0)"
by (intro eventually_at_in_open; force simp: has_laurent_expansion_def zero_ereal_def)+
moreover have "eventually (λx. eval_fls F x = f x) (at 0)"
and "eventually (λx. eval_fls G x = g x) (at 0)"
using assms by (auto simp: has_laurent_expansion_def)
ultimately have "eventually (λx. eval_fls (F + G) x = f x + g x) (at 0)"
by eventually_elim (auto simp: eval_fls_add)
with radius show ?thesis by (auto simp: has_laurent_expansion_def)
qed
lemma has_laurent_expansion_diff [laurent_expansion_intros]:
assumes "f has_laurent_expansion F" "g has_laurent_expansion G"
shows "(λx. f x - g x) has_laurent_expansion F - G"
using has_laurent_expansion_add[of f F "λx. - g x" "-G"] assms
by (simp add: has_laurent_expansion_minus)
lemma has_laurent_expansion_mult [laurent_expansion_intros]:
assumes "f has_laurent_expansion F" "g has_laurent_expansion G"
shows "(λx. f x * g x) has_laurent_expansion F * G"
proof -
from assms have "0 < min (fls_conv_radius F) (fls_conv_radius G)"
by (auto simp: has_laurent_expansion_def)
also have "… ≤ fls_conv_radius (F * G)"
by (rule fls_conv_radius_mult)
finally have radius: "… > 0" .
from assms have "eventually (λx. x ∈ eball 0 (fls_conv_radius F) - {0}) (at 0)"
"eventually (λx. x ∈ eball 0 (fls_conv_radius G) - {0}) (at 0)"
by (intro eventually_at_in_open; force simp: has_laurent_expansion_def zero_ereal_def)+
moreover have "eventually (λx. eval_fls F x = f x) (at 0)"
and "eventually (λx. eval_fls G x = g x) (at 0)"
using assms by (auto simp: has_laurent_expansion_def)
ultimately have "eventually (λx. eval_fls (F * G) x = f x * g x) (at 0)"
by eventually_elim (auto simp: eval_fls_mult)
with radius show ?thesis by (auto simp: has_laurent_expansion_def)
qed
lemma has_fps_expansion_power [fps_expansion_intros]:
fixes F :: "'a :: {banach, real_normed_div_algebra, comm_ring_1} fps"
shows "f has_fps_expansion F ⟹ (λx. f x ^ m) has_fps_expansion F ^ m"
by (induction m) (auto intro!: fps_expansion_intros)
lemma has_laurent_expansion_power [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(λx. f x ^ n) has_laurent_expansion F ^ n"
by (induction n) (auto intro!: laurent_expansion_intros assms)
lemma has_laurent_expansion_sum [laurent_expansion_intros]:
assumes "⋀x. x ∈ I ⟹ f x has_laurent_expansion F x"
shows "(λy. ∑x∈I. f x y) has_laurent_expansion (∑x∈I. F x)"
using assms by (induction I rule: infinite_finite_induct) (auto intro!: laurent_expansion_intros)
lemma has_laurent_expansion_prod [laurent_expansion_intros]:
assumes "⋀x. x ∈ I ⟹ f x has_laurent_expansion F x"
shows "(λy. ∏x∈I. f x y) has_laurent_expansion (∏x∈I. F x)"
using assms by (induction I rule: infinite_finite_induct) (auto intro!: laurent_expansion_intros)
lemma has_laurent_expansion_deriv [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "deriv f has_laurent_expansion fls_deriv F"
proof -
have "eventually (λz. z ∈ eball 0 (fls_conv_radius F) - {0}) (at 0)"
using assms by (intro eventually_at_in_open)
(auto simp: has_laurent_expansion_def zero_ereal_def)
moreover from assms have "eventually (λz. eval_fls F z = f z) (at 0)"
by (auto simp: has_laurent_expansion_def)
then obtain s where "open s" "0 ∈ s" and s: "⋀w. w ∈ s - {0} ⟹ eval_fls F w = f w"
by (auto simp: eventually_nhds eventually_at_filter)
hence "eventually (λw. w ∈ s - {0}) (at 0)"
by (intro eventually_at_in_open) auto
ultimately have "eventually (λz. eval_fls (fls_deriv F) z = deriv f z) (at 0)"
proof eventually_elim
case (elim z)
hence "eval_fls (fls_deriv F) z = deriv (eval_fls F) z"
by (simp add: eval_fls_deriv)
also have "eventually (λw. w ∈ s - {0}) (nhds z)"
using elim and ‹open s› by (intro eventually_nhds_in_open) auto
hence "eventually (λw. eval_fls F w = f w) (nhds z)"
by eventually_elim (use s in auto)
hence "deriv (eval_fls F) z = deriv f z"
by (intro deriv_cong_ev refl)
finally show ?case .
qed
with assms show ?thesis
by (auto simp: has_laurent_expansion_def intro!: less_le_trans[OF _ fls_conv_radius_deriv])
qed
lemma has_laurent_expansion_shift [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(λx. f x * x powi n) has_laurent_expansion (fls_shift (-n) F)"
proof -
have "eventually (λx. x ∈ eball 0 (fls_conv_radius F) - {0}) (at 0)"
using assms by (intro eventually_at_in_open) (auto simp: has_laurent_expansion_def zero_ereal_def)
moreover have "eventually (λx. eval_fls F x = f x) (at 0)"
using assms by (auto simp: has_laurent_expansion_def)
ultimately have "eventually (λx. eval_fls (fls_shift (-n) F) x = f x * x powi n) (at 0)"
by eventually_elim (auto simp: eval_fls_shift assms)
with assms show ?thesis by (auto simp: has_laurent_expansion_def)
qed
lemma has_laurent_expansion_shift' [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(λx. f x * x powi (-n)) has_laurent_expansion (fls_shift n F)"
using has_laurent_expansion_shift[OF assms, of "-n"] by simp
lemma has_laurent_expansion_deriv':
assumes "f has_laurent_expansion F"
assumes "open A" "0 ∈ A" "⋀x. x ∈ A - {0} ⟹ (f has_field_derivative f' x) (at x)"
shows "f' has_laurent_expansion fls_deriv F"
proof -
have "deriv f has_laurent_expansion fls_deriv F"
by (intro laurent_expansion_intros assms)
also have "?this ⟷ ?thesis"
proof (intro has_laurent_expansion_cong refl)
have "eventually (λz. z ∈ A - {0}) (at 0)"
by (intro eventually_at_in_open assms)
thus "eventually (λz. deriv f z = f' z) (at 0)"
by eventually_elim (auto intro!: DERIV_imp_deriv assms)
qed
finally show ?thesis .
qed
definition laurent_expansion :: "(complex ⇒ complex) ⇒ complex ⇒ complex fls" where
"laurent_expansion f z =
(if eventually (λz. f z = 0) (at z) then 0
else fls_shift (-zorder f z) (fps_to_fls (fps_expansion (zor_poly f z) z)))"
lemma laurent_expansion_cong:
assumes "eventually (λw. f w = g w) (at z)" "z = z'"
shows "laurent_expansion f z = laurent_expansion g z'"
unfolding laurent_expansion_def
using zor_poly_cong[OF assms(1,2)] zorder_cong[OF assms] assms
by (intro if_cong refl) (auto elim: eventually_elim2)
theorem not_essential_has_laurent_expansion_0:
assumes "isolated_singularity_at f 0" "not_essential f 0"
shows "f has_laurent_expansion laurent_expansion f 0"
proof (cases "∃⇩F w in at 0. f w ≠ 0")
case False
have "(λ_. 0) has_laurent_expansion 0"
by simp
also have "?this ⟷ f has_laurent_expansion 0"
using False by (intro has_laurent_expansion_cong) (auto simp: frequently_def)
finally show ?thesis
using False by (simp add: laurent_expansion_def frequently_def)
next
case True
define n where "n = zorder f 0"
obtain r where r: "zor_poly f 0 0 ≠ 0" "zor_poly f 0 holomorphic_on cball 0 r" "r > 0"
"∀w∈cball 0 r - {0}. f w = zor_poly f 0 w * w powi n ∧
zor_poly f 0 w ≠ 0"
using zorder_exist[OF assms True] unfolding n_def by auto
have holo: "zor_poly f 0 holomorphic_on ball 0 r"
by (rule holomorphic_on_subset[OF r(2)]) auto
define F where "F = fps_expansion (zor_poly f 0) 0"
have F: "zor_poly f 0 has_fps_expansion F"
unfolding F_def by (rule has_fps_expansion_fps_expansion[OF _ _ holo]) (use ‹r > 0› in auto)
have "(λz. zor_poly f 0 z * z powi n) has_laurent_expansion fls_shift (-n) (fps_to_fls F)"
by (intro laurent_expansion_intros has_laurent_expansion_fps[OF F])
also have "?this ⟷ f has_laurent_expansion fls_shift (-n) (fps_to_fls F)"
by (intro has_laurent_expansion_cong refl eventually_mono[OF eventually_at_in_open[of "ball 0 r"]])
(use r in ‹auto simp: complex_powr_of_int›)
finally show ?thesis using True
by (simp add: laurent_expansion_def F_def n_def frequently_def)
qed
lemma not_essential_has_laurent_expansion:
assumes "isolated_singularity_at f z" "not_essential f z"
shows "(λx. f (z + x)) has_laurent_expansion laurent_expansion f z"
proof -
from assms(1) have iso:"isolated_singularity_at (λx. f (z + x)) 0"
by (simp add: isolated_singularity_at_shift_0)
moreover from assms(2) have ness:"not_essential (λx. f (z + x)) 0"
by (simp add: not_essential_shift_0)
ultimately have "(λx. f (z + x)) has_laurent_expansion laurent_expansion (λx. f (z + x)) 0"
by (rule not_essential_has_laurent_expansion_0)
also have "… = laurent_expansion f z"
proof (cases "∃⇩F w in at z. f w ≠ 0")
case False
then have "∀⇩F w in at z. f w = 0" using not_frequently by force
then have "laurent_expansion (λx. f (z + x)) 0 = 0"
by (smt (verit, best) add.commute eventually_at_to_0 eventually_mono
laurent_expansion_def)
moreover have "laurent_expansion f z = 0"
using ‹∀⇩F w in at z. f w = 0› unfolding laurent_expansion_def by auto
ultimately show ?thesis by auto
next
case True
define df where "df=zor_poly (λx. f (z + x)) 0"
define g where "g=(λu. u-z)"
have "fps_expansion df 0
= fps_expansion (df o g) z"
proof -
have "∃⇩F w in at 0. f (z + w) ≠ 0" using True
by (smt (verit, best) add.commute eventually_at_to_0
eventually_mono not_frequently)
from zorder_exist[OF iso ness this,folded df_def]
obtain r where "r>0" and df_holo:"df holomorphic_on cball 0 r" and "df 0 ≠ 0"
"∀w∈cball 0 r - {0}.
f (z + w) = df w * w powi (zorder (λw. f (z + w)) 0) ∧
df w ≠ 0"
by auto
then have df_nz:"∀w∈ball 0 r. df w≠0" by auto
have "(deriv ^^ n) df 0 = (deriv ^^ n) (df ∘ g) z" for n
unfolding comp_def g_def
proof (subst higher_deriv_compose_linear'[where u=1 and c="-z",simplified])
show "df holomorphic_on ball 0 r"
using df_holo by auto
show "open (ball z r)" "open (ball 0 r)" "z ∈ ball z r"
using ‹r>0› by auto
show " ⋀w. w ∈ ball z r ⟹ w - z ∈ ball 0 r"
by (simp add: dist_norm)
qed auto
then show ?thesis
unfolding fps_expansion_def by auto
qed
also have "... = fps_expansion (zor_poly f z) z"
proof (rule fps_expansion_cong)
have "∀⇩F w in nhds z. zor_poly f z w
= zor_poly (λu. f (z + u)) 0 (w - z)"
apply (rule zor_poly_shift)
using True assms by auto
then show "∀⇩F w in nhds z. (df ∘ g) w = zor_poly f z w"
unfolding df_def g_def comp_def
by (auto elim:eventually_mono)
qed
finally show ?thesis unfolding df_def
by (auto simp: laurent_expansion_def at_to_0[of z]
eventually_filtermap add_ac zorder_shift')
qed
finally show ?thesis .
qed
lemma has_fps_expansion_to_laurent:
"f has_fps_expansion F ⟷ f has_laurent_expansion fps_to_fls F ∧ f 0 = fps_nth F 0"
proof safe
assume *: "f has_laurent_expansion fps_to_fls F" "f 0 = fps_nth F 0"
have "eventually (λz. z ∈ eball 0 (fps_conv_radius F)) (nhds 0)"
using * by (intro eventually_nhds_in_open) (auto simp: has_laurent_expansion_def zero_ereal_def)
moreover have "eventually (λz. z ≠ 0 ⟶ eval_fls (fps_to_fls F) z = f z) (nhds 0)"
using * by (auto simp: has_laurent_expansion_def eventually_at_filter)
ultimately have "eventually (λz. f z = eval_fps F z) (nhds 0)"
by eventually_elim
(auto simp: has_laurent_expansion_def eventually_at_filter eval_fps_at_0 eval_fps_to_fls *(2))
thus "f has_fps_expansion F"
using * by (auto simp: has_fps_expansion_def has_laurent_expansion_def eq_commute)
next
assume "f has_fps_expansion F"
thus "f 0 = fps_nth F 0"
by (metis eval_fps_at_0 has_fps_expansion_imp_holomorphic)
qed (auto intro: has_laurent_expansion_fps)
lemma eval_fps_fls_base_factor [simp]:
assumes "z ≠ 0"
shows "eval_fps (fls_base_factor_to_fps F) z = eval_fls F z * z powi -fls_subdegree F"
using assms unfolding eval_fls_def by (simp add: power_int_minus field_simps)
lemma has_fps_expansion_imp_analytic_0:
assumes "f has_fps_expansion F"
shows "f analytic_on {0}"
by (meson analytic_at_two assms has_fps_expansion_imp_holomorphic)
lemma has_fps_expansion_imp_analytic:
assumes "(λx. f (z + x)) has_fps_expansion F"
shows "f analytic_on {z}"
proof -
have "(λx. f (z + x)) analytic_on {0}"
by (rule has_fps_expansion_imp_analytic_0) fact
hence "(λx. f (z + x)) ∘ (λx. x - z) analytic_on {z}"
by (intro analytic_on_compose_gen analytic_intros) auto
thus ?thesis
by (simp add: o_def)
qed
lemma is_pole_cong_asymp_equiv:
assumes "f ∼[at z] g" "z = z'"
shows "is_pole f z = is_pole g z'"
using asymp_equiv_at_infinity_transfer[OF assms(1)]
asymp_equiv_at_infinity_transfer[OF asymp_equiv_symI[OF assms(1)]] assms(2)
unfolding is_pole_def by auto
lemma not_is_pole_const [simp]: "¬is_pole (λ_::'a::perfect_space. c :: complex) z"
using not_tendsto_and_filterlim_at_infinity[of "at z" "λ_::'a. c" c] by (auto simp: is_pole_def)
lemma has_laurent_expansion_imp_is_pole_iff:
assumes F: "(λx. f (z + x)) has_laurent_expansion F"
shows "is_pole f z ⟷ fls_subdegree F < 0"
proof
assume pole: "is_pole f z"
have [simp]: "F ≠ 0"
proof
assume "F = 0"
hence "is_pole f z ⟷ is_pole (λ_. 0 :: complex) z" using assms
by (intro is_pole_cong)
(auto simp: has_laurent_expansion_def at_to_0[of z] eventually_filtermap add_ac)
with pole show False
by simp
qed
note pole
also have "is_pole f z ⟷
is_pole (λw. fls_nth F (fls_subdegree F) * (w - z) powi fls_subdegree F) z"
using has_laurent_expansion_imp_asymp_equiv[OF F] by (intro is_pole_cong_asymp_equiv refl)
also have "… ⟷ is_pole (λw. (w - z) powi fls_subdegree F) z"
by simp
finally have pole': … .
have False if "fls_subdegree F ≥ 0"
proof -
have "(λw. (w - z) powi fls_subdegree F) holomorphic_on UNIV"
using that by (intro holomorphic_intros) auto
hence "¬is_pole (λw. (w - z) powi fls_subdegree F) z"
by (meson UNIV_I not_is_pole_holomorphic open_UNIV)
with pole' show False
by simp
qed
thus "fls_subdegree F < 0"
by force
qed (use has_laurent_expansion_imp_is_pole[OF assms] in auto)
lemma analytic_at_imp_has_fps_expansion_0:
assumes "f analytic_on {0}"
shows "f has_fps_expansion fps_expansion f 0"
using assms has_fps_expansion_fps_expansion analytic_at by fast
lemma deriv_shift_0: "deriv f z = deriv (f ∘ (λx. z + x)) 0"
proof -
have *: "(f ∘ (+) z has_field_derivative D) (at z')"
if "(f has_field_derivative D) (at (z + z'))" for D z z' and f :: "'a ⇒ 'a"
proof -
have "(f ∘ (+) z has_field_derivative D * 1) (at z')"
by (rule DERIV_chain that derivative_eq_intros refl)+ auto
thus ?thesis by simp
qed
have "(λD. (f has_field_derivative D) (at z)) = (λ D. (f ∘ (+) z has_field_derivative D) (at 0))"
using *[of f _ z 0] *[of "f ∘ (+) z" _ "-z" z] by (intro ext iffI) (auto simp: o_def)
thus ?thesis
by (simp add: deriv_def)
qed
lemma deriv_shift_0': "NO_MATCH 0 z ⟹ deriv f z = deriv (f ∘ (λx. z + x)) 0"
by (rule deriv_shift_0)
lemma higher_deriv_shift_0: "(deriv ^^ n) f z = (deriv ^^ n) (f ∘ (λx. z + x)) 0"
proof (induction n arbitrary: f)
case (Suc n)
have "(deriv ^^ Suc n) f z = (deriv ^^ n) (deriv f) z"
by (subst funpow_Suc_right) auto
also have "… = (deriv ^^ n) (λx. deriv f (z + x)) 0"
by (subst Suc) (auto simp: o_def)
also have "… = (deriv ^^ n) (λx. deriv (λxa. f (z + x + xa)) 0) 0"
by (subst deriv_shift_0) (auto simp: o_def)
also have "(λx. deriv (λxa. f (z + x + xa)) 0) = deriv (λx. f (z + x))"
by (rule ext) (simp add: deriv_shift_0' o_def add_ac)
also have "(deriv ^^ n) … 0 = (deriv ^^ Suc n) (f ∘ (λx. z + x)) 0"
by (subst funpow_Suc_right) (auto simp: o_def)
finally show ?case .
qed auto
lemma higher_deriv_shift_0': "NO_MATCH 0 z ⟹ (deriv ^^ n) f z = (deriv ^^ n) (f ∘ (λx. z + x)) 0"
by (rule higher_deriv_shift_0)
lemma analytic_at_imp_has_fps_expansion:
assumes "f analytic_on {z}"
shows "(λx. f (z + x)) has_fps_expansion fps_expansion f z"
proof -
have "f ∘ (λx. z + x) analytic_on {0}"
by (intro analytic_on_compose_gen[OF _ assms] analytic_intros) auto
hence "(f ∘ (λx. z + x)) has_fps_expansion fps_expansion (f ∘ (λx. z + x)) 0"
unfolding o_def by (intro analytic_at_imp_has_fps_expansion_0) auto
also have "… = fps_expansion f z"
by (simp add: fps_expansion_def higher_deriv_shift_0')
finally show ?thesis by (simp add: add_ac)
qed
lemma has_laurent_expansion_zorder_0:
assumes "f has_laurent_expansion F" "F ≠ 0"
shows "zorder f 0 = fls_subdegree F"
proof -
define G where "G = fls_base_factor_to_fps F"
from assms obtain A where A: "0 ∈ A" "open A" "⋀x. x ∈ A - {0} ⟹ eval_fls F x = f x"
unfolding has_laurent_expansion_def eventually_at_filter eventually_nhds
by blast
show ?thesis
proof (rule zorder_eqI)
show "open (A ∩ eball 0 (fls_conv_radius F))" "0 ∈ A ∩ eball 0 (fls_conv_radius F)"
using assms A by (auto simp: has_laurent_expansion_def zero_ereal_def)
show "eval_fps G holomorphic_on A ∩ eball 0 (fls_conv_radius F)"
by (intro holomorphic_intros) (auto simp: fls_conv_radius_altdef G_def)
show "eval_fps G 0 ≠ 0" using ‹F ≠ 0›
by (auto simp: eval_fps_at_0 G_def)
next
fix w :: complex assume "w ∈ A ∩ eball 0 (fls_conv_radius F)" "w ≠ 0"
thus "f w = eval_fps G w * (w - 0) powi (fls_subdegree F)"
using A unfolding G_def
by (subst eval_fps_fls_base_factor)
(auto simp: complex_powr_of_int power_int_minus field_simps)
qed
qed
lemma has_laurent_expansion_zorder:
assumes "(λw. f (z + w)) has_laurent_expansion F" "F ≠ 0"
shows "zorder f z = fls_subdegree F"
using has_laurent_expansion_zorder_0[OF assms] by (simp add: zorder_shift' add_ac)
lemma has_fps_expansion_zorder_0:
assumes "f has_fps_expansion F" "F ≠ 0"
shows "zorder f 0 = int (subdegree F)"
using assms has_laurent_expansion_zorder_0[of f "fps_to_fls F"]
by (auto simp: has_fps_expansion_to_laurent fls_subdegree_fls_to_fps)
lemma has_fps_expansion_zorder:
assumes "(λw. f (z + w)) has_fps_expansion F" "F ≠ 0"
shows "zorder f z = int (subdegree F)"
using has_fps_expansion_zorder_0[OF assms]
by (simp add: zorder_shift' add_ac)
lemma has_fps_expansion_fls_base_factor_to_fps:
assumes "f has_laurent_expansion F"
defines "n ≡ fls_subdegree F"
defines "c ≡ fps_nth (fls_base_factor_to_fps F) 0"
shows "(λz. if z = 0 then c else f z * z powi -n) has_fps_expansion fls_base_factor_to_fps F"
proof -
have "(λz. f z * z powi -n) has_laurent_expansion fls_shift (-(-n)) F"
by (intro laurent_expansion_intros assms)
also have "fls_shift (-(-n)) F = fps_to_fls (fls_base_factor_to_fps F)"
by (simp add: n_def fls_shift_nonneg_subdegree)
also have "(λz. f z * z powi - n) has_laurent_expansion fps_to_fls (fls_base_factor_to_fps F) ⟷
(λz. if z = 0 then c else f z * z powi -n) has_laurent_expansion fps_to_fls (fls_base_factor_to_fps F)"
by (intro has_laurent_expansion_cong) (auto simp: eventually_at_filter)
also have "… ⟷ (λz. if z = 0 then c else f z * z powi -n) has_fps_expansion fls_base_factor_to_fps F"
by (subst has_fps_expansion_to_laurent) (auto simp: c_def)
finally show ?thesis .
qed
lemma zero_has_laurent_expansion_imp_eq_0:
assumes "(λ_. 0) has_laurent_expansion F"
shows "F = 0"
proof -
have "at (0 :: complex) ≠ bot"
by auto
moreover have "(λz. if z = 0 then fls_nth F (fls_subdegree F) else 0) has_fps_expansion
fls_base_factor_to_fps F" (is "?f has_fps_expansion _")
using has_fps_expansion_fls_base_factor_to_fps[OF assms] by (simp cong: if_cong)
hence "isCont ?f 0"
using has_fps_expansion_imp_continuous by blast
hence "?f ─0→ fls_nth F (fls_subdegree F)"
by (auto simp: isCont_def)
moreover have "?f ─0→ 0 ⟷ (λ_::complex. 0 :: complex) ─0→ 0"
by (intro filterlim_cong) (auto simp: eventually_at_filter)
hence "?f ─0→ 0"
by simp
ultimately have "fls_nth F (fls_subdegree F) = 0"
by (rule tendsto_unique)
thus ?thesis
by (meson nth_fls_subdegree_nonzero)
qed
lemma has_laurent_expansion_unique:
assumes "f has_laurent_expansion F" "f has_laurent_expansion G"
shows "F = G"
proof -
from assms have "(λx. f x - f x) has_laurent_expansion F - G"
by (intro laurent_expansion_intros)
hence "(λ_. 0) has_laurent_expansion F - G"
by simp
hence "F - G = 0"
by (rule zero_has_laurent_expansion_imp_eq_0)
thus ?thesis
by simp
qed
lemma laurent_expansion_eqI:
assumes "(λx. f (z + x)) has_laurent_expansion F"
shows "laurent_expansion f z = F"
using assms has_laurent_expansion_isolated has_laurent_expansion_not_essential
has_laurent_expansion_unique not_essential_has_laurent_expansion by blast
lemma laurent_expansion_0_eqI:
assumes "f has_laurent_expansion F"
shows "laurent_expansion f 0 = F"
using assms laurent_expansion_eqI[of f 0] by simp
lemma has_laurent_expansion_nonzero_imp_eventually_nonzero:
assumes "f has_laurent_expansion F" "F ≠ 0"
shows "eventually (λx. f x ≠ 0) (at 0)"
proof (rule ccontr)
assume "¬eventually (λx. f x ≠ 0) (at 0)"
with assms have "eventually (λx. f x = 0) (at 0)"
by (intro not_essential_frequently_0_imp_eventually_0 has_laurent_expansion_isolated
has_laurent_expansion_not_essential)
(auto simp: frequently_def)
hence "(f has_laurent_expansion 0) ⟷ ((λ_. 0) has_laurent_expansion 0)"
by (intro has_laurent_expansion_cong) auto
hence "f has_laurent_expansion 0"
by simp
with assms(1) have "F = 0"
using has_laurent_expansion_unique by blast
with ‹F ≠ 0› show False
by contradiction
qed
lemma has_laurent_expansion_eventually_nonzero_iff':
assumes "f has_laurent_expansion F"
shows "eventually (λx. f x ≠ 0) (at 0) ⟷ F ≠ 0 "
proof
assume "∀⇩F x in at 0. f x ≠ 0"
moreover have "¬ (∀⇩F x in at 0. f x ≠ 0)" if "F=0"
proof -
have "∀⇩F x in at 0. f x = 0"
using assms that unfolding has_laurent_expansion_def by simp
then show ?thesis unfolding not_eventually
by (auto elim:eventually_frequentlyE)
qed
ultimately show "F ≠ 0" by auto
qed (simp add:has_laurent_expansion_nonzero_imp_eventually_nonzero[OF assms])
lemma has_laurent_expansion_eventually_nonzero_iff:
assumes "(λw. f (z+w)) has_laurent_expansion F"
shows "eventually (λx. f x ≠ 0) (at z) ⟷ F ≠ 0"
apply (subst eventually_at_to_0)
apply (rule has_laurent_expansion_eventually_nonzero_iff')
using assms by (simp add:add.commute)
lemma has_laurent_expansion_inverse [laurent_expansion_intros]:
assumes "f has_laurent_expansion F"
shows "(λx. inverse (f x)) has_laurent_expansion inverse F"
proof (cases "F = 0")
case True
thus ?thesis using assms
by (auto simp: has_laurent_expansion_def)
next
case False
define G where "G = laurent_expansion (λx. inverse (f x)) 0"
from False have ev: "eventually (λz. f z ≠ 0) (at 0)"
by (intro has_laurent_expansion_nonzero_imp_eventually_nonzero[OF assms])
have *: "(λx. inverse (f x)) has_laurent_expansion G" unfolding G_def
by (intro not_essential_has_laurent_expansion_0 isolated_singularity_at_inverse not_essential_inverse
has_laurent_expansion_isolated_0[OF assms] has_laurent_expansion_not_essential_0[OF assms])
have "(λx. f x * inverse (f x)) has_laurent_expansion F * G"
by (intro laurent_expansion_intros assms *)
also have "?this ⟷ (λx. 1) has_laurent_expansion F * G"
by (intro has_laurent_expansion_cong refl eventually_mono[OF ev]) auto
finally have "(λ_. 1) has_laurent_expansion F * G" .
moreover have "(λ_. 1) has_laurent_expansion 1"
by simp
ultimately have "F * G = 1"
using has_laurent_expansion_unique by blast
hence "G = inverse F"
using inverse_unique by blast
with * show ?thesis
by simp
qed
lemma has_laurent_expansion_power_int [laurent_expansion_intros]:
"f has_laurent_expansion F ⟹ (λx. f x powi n) has_laurent_expansion (F powi n)"
by (auto simp: power_int_def intro!: laurent_expansion_intros)
lemma has_fps_expansion_0_analytic_continuation:
assumes "f has_fps_expansion 0" "f holomorphic_on A"
assumes "open A" "connected A" "0 ∈ A" "x ∈ A"
shows "f x = 0"
proof -
have "eventually (λz. z ∈ A ∧ f z = 0) (nhds 0)" using assms
by (intro eventually_conj eventually_nhds_in_open) (auto simp: has_fps_expansion_def)
then obtain B where B: "open B" "0 ∈ B" "∀z∈B. z ∈ A ∧ f z = 0"
unfolding eventually_nhds by blast
show ?thesis
proof (rule analytic_continuation_open[where f = f and g = "λ_. 0"])
show "B ≠ {}"
using ‹open B› B by auto
show "connected A"
using assms by auto
qed (use assms B in auto)
qed
lemma has_laurent_expansion_0_analytic_continuation:
assumes "f has_laurent_expansion 0" "f holomorphic_on A - {0}"
assumes "open A" "connected A" "0 ∈ A" "x ∈ A - {0}"
shows "f x = 0"
proof -
have "eventually (λz. z ∈ A - {0} ∧ f z = 0) (at 0)" using assms
by (intro eventually_conj eventually_at_in_open) (auto simp: has_laurent_expansion_def)
then obtain B where B: "open B" "0 ∈ B" "∀z∈B - {0}. z ∈ A - {0} ∧ f z = 0"
unfolding eventually_at_filter eventually_nhds by blast
show ?thesis
proof (rule analytic_continuation_open[where f = f and g = "λ_. 0"])
show "B - {0} ≠ {}"
using ‹open B› ‹0 ∈ B› by (metis insert_Diff not_open_singleton)
show "connected (A - {0})"
using assms by (intro connected_open_delete) auto
qed (use assms B in auto)
qed
lemma has_fps_expansion_cong:
assumes "eventually (λx. f x = g x) (nhds 0)" "F = G"
shows "f has_fps_expansion F ⟷ g has_fps_expansion G"
using assms(2) by (auto simp: has_fps_expansion_def elim!: eventually_elim2[OF assms(1)])
lemma zor_poly_has_fps_expansion:
assumes "f has_laurent_expansion F" "F ≠ 0"
shows "zor_poly f 0 has_fps_expansion fls_base_factor_to_fps F"
proof -
note [simp] = ‹F ≠ 0›
have "eventually (λz. f z ≠ 0) (at 0)"
by (rule has_laurent_expansion_nonzero_imp_eventually_nonzero[OF assms])
hence freq: "frequently (λz. f z ≠ 0) (at 0)"
by (rule eventually_frequently[rotated]) auto
have *: "isolated_singularity_at f 0" "not_essential f 0"
using has_laurent_expansion_isolated_0[OF assms(1)] has_laurent_expansion_not_essential_0[OF assms(1)]
by auto
define G where "G = fls_base_factor_to_fps F"
define n where "n = zorder f 0"
have n_altdef: "n = fls_subdegree F"
using has_laurent_expansion_zorder_0 [OF assms(1)] by (simp add: n_def)
obtain r where r: "zor_poly f 0 0 ≠ 0" "zor_poly f 0 holomorphic_on cball 0 r" "r > 0"
"∀w∈cball 0 r - {0}. f w = zor_poly f 0 w * w powi n ∧
zor_poly f 0 w ≠ 0"
using zorder_exist[OF * freq] unfolding n_def by auto
obtain r' where r': "r' > 0" "∀x∈ball 0 r'-{0}. eval_fls F x = f x"
using assms(1) unfolding has_laurent_expansion_def eventually_at_filter eventually_nhds_metric ball_def
by (auto simp: dist_commute)
have holo: "zor_poly f 0 holomorphic_on ball 0 r"
by (rule holomorphic_on_subset[OF r(2)]) auto
have "(λz. if z = 0 then fps_nth G 0 else f z * z powi -n) has_fps_expansion G"
unfolding G_def n_altdef by (intro has_fps_expansion_fls_base_factor_to_fps assms)
also have "?this ⟷ zor_poly f 0 has_fps_expansion G"
proof (intro has_fps_expansion_cong)
have "eventually (λz. z ∈ ball 0 (min r r')) (nhds 0)"
using ‹r > 0› ‹r' > 0› by (intro eventually_nhds_in_open) auto
thus "∀⇩F x in nhds 0. (if x = 0 then G $ 0 else f x * x powi - n) = zor_poly f 0 x"
proof eventually_elim
case (elim w)
have w: "w ∈ ball 0 r" "w ∈ ball 0 r'"
using elim by auto
show ?case
proof (cases "w = 0")
case False
hence "f w = zor_poly f 0 w * w powi n"
using r w by auto
thus ?thesis using False
by (simp add: powr_minus complex_powr_of_int power_int_minus)
next
case [simp]: True
obtain R where R: "R > 0" "R ≤ r" "R ≤ r'" "R ≤ fls_conv_radius F"
using ‹r > 0› ‹r' > 0› assms(1) unfolding has_laurent_expansion_def
by (smt (verit, ccfv_SIG) ereal_dense2 ereal_less(2) less_ereal.simps(1) order.strict_implies_order order_trans)
have "eval_fps G 0 = zor_poly f 0 0"
proof (rule analytic_continuation_open[where f = "eval_fps G" and g = "zor_poly f 0"])
show "connected (ball 0 R :: complex set)"
by auto
have "of_real R / 2 ∈ ball 0 R - {0 :: complex}"
using R by auto
thus "ball 0 R - {0 :: complex} ≠ {}"
by blast
show "eval_fps G holomorphic_on ball 0 R"
using R less_le_trans[OF _ R(4)] unfolding G_def
by (intro holomorphic_intros) (auto simp: fls_conv_radius_altdef)
show "zor_poly f 0 holomorphic_on ball 0 R"
by (rule holomorphic_on_subset[OF holo]) (use R in auto)
show "eval_fps G z = zor_poly f 0 z" if "z ∈ ball 0 R - {0}" for z
using that r r' R n_altdef unfolding G_def
by (subst eval_fps_fls_base_factor)
(auto simp: complex_powr_of_int field_simps power_int_minus n_def)
qed (use R in auto)
hence "zor_poly f 0 0 = fps_nth G 0"
by (simp add: eval_fps_at_0)
thus ?thesis by simp
qed
qed
qed (use r' in auto)
finally show ?thesis
by (simp add: G_def)
qed
lemma zorder_geI_0:
assumes "f analytic_on {0}" "f holomorphic_on A" "open A" "connected A" "0 ∈ A" "z ∈ A" "f z ≠ 0"
assumes "⋀k. k < n ⟹ (deriv ^^ k) f 0 = 0"
shows "zorder f 0 ≥ n"
proof -
define F where "F = fps_expansion f 0"
from assms have "f has_fps_expansion F"
unfolding F_def using analytic_at_imp_has_fps_expansion_0 by blast
hence laurent: "f has_laurent_expansion fps_to_fls F" and [simp]: "f 0 = fps_nth F 0"
by (simp_all add: has_fps_expansion_to_laurent)
have [simp]: "F ≠ 0"
proof
assume [simp]: "F = 0"
hence "f z = 0"
proof (cases "z = 0")
case False
have "f has_laurent_expansion 0"
using laurent by simp
thus ?thesis
proof (rule has_laurent_expansion_0_analytic_continuation)
show "f holomorphic_on A - {0}"
using assms(2) by (rule holomorphic_on_subset) auto
qed (use assms False in auto)
qed auto
with ‹f z ≠ 0› show False by contradiction
qed
have "zorder f 0 = int (subdegree F)"
using has_laurent_expansion_zorder_0[OF laurent] by (simp add: fls_subdegree_fls_to_fps)
also have "subdegree F ≥ n"
using assms by (intro subdegree_geI ‹F ≠ 0›) (auto simp: F_def fps_expansion_def)
hence "int (subdegree F) ≥ int n"
by simp
finally show ?thesis .
qed
lemma zorder_geI:
assumes "f analytic_on {x}" "f holomorphic_on A" "open A" "connected A" "x ∈ A" "z ∈ A" "f z ≠ 0"
assumes "⋀k. k < n ⟹ (deriv ^^ k) f x = 0"
shows "zorder f x ≥ n"
proof -
have "zorder f x = zorder (f ∘ (λu. u + x)) 0"
by (subst zorder_shift) (auto simp: o_def)
also have "… ≥ n"
proof (rule zorder_geI_0)
show "(f ∘ (λu. u + x)) analytic_on {0}"
by (intro analytic_on_compose_gen[OF _ assms(1)] analytic_intros) auto
show "f ∘ (λu. u + x) holomorphic_on ((+) (-x)) ` A"
by (intro holomorphic_on_compose_gen[OF _ assms(2)] holomorphic_intros) auto
show "connected ((+) (- x) ` A)"
by (intro connected_continuous_image continuous_intros assms)
show "open ((+) (- x) ` A)"
by (intro open_translation assms)
show "z - x ∈ (+) (- x) ` A"
using ‹z ∈ A› by auto
show "0 ∈ (+) (- x) ` A"
using ‹x ∈ A› by auto
show "(f ∘ (λu. u + x)) (z - x) ≠ 0"
using ‹f z ≠ 0› by auto
next
fix k :: nat assume "k < n"
hence "(deriv ^^ k) f x = 0"
using assms by simp
also have "(deriv ^^ k) f x = (deriv ^^ k) (f ∘ (+) x) 0"
by (subst higher_deriv_shift_0) auto
finally show "(deriv ^^ k) (f ∘ (λu. u + x)) 0 = 0"
by (subst add.commute) auto
qed
finally show ?thesis .
qed
lemma has_laurent_expansion_divide [laurent_expansion_intros]:
assumes "f has_laurent_expansion F" and "g has_laurent_expansion G"
shows "(λx. f x / g x) has_laurent_expansion (F / G)"
proof -
have "(λx. f x * inverse (g x)) has_laurent_expansion (F * inverse G)"
by (intro laurent_expansion_intros assms)
thus ?thesis
by (simp add: field_simps)
qed
lemma vector_derivative_translate [simp]:
"vector_derivative ((+) z ∘ g) (at x within A) = vector_derivative g (at x within A)"
proof -
have "(((+) z ∘ g) has_vector_derivative g') (at x within A)"
if "(g has_vector_derivative g') (at x within A)" for g :: "real ⇒ 'a" and z g'
unfolding o_def using that by (auto intro!: derivative_eq_intros)
from this[of g _ z] this[of "λx. z + g x" _ "-z"] show ?thesis
unfolding vector_derivative_def
by (intro arg_cong[where f = Eps] ext) (auto simp: o_def algebra_simps)
qed
lemma has_contour_integral_translate:
"(f has_contour_integral I) ((+) z ∘ g) ⟷ ((λx. f (x + z)) has_contour_integral I) g"
by (simp add: has_contour_integral_def add_ac)
lemma contour_integrable_translate:
"f contour_integrable_on ((+) z ∘ g) ⟷ (λx. f (x + z)) contour_integrable_on g"
by (simp add: contour_integrable_on_def has_contour_integral_translate)
lemma contour_integral_translate:
"contour_integral ((+) z ∘ g) f = contour_integral g (λx. f (x + z))"
by (simp add: contour_integral_def contour_integrable_translate has_contour_integral_translate)
lemma residue_shift_0: "residue f z = residue (λx. f (z + x)) 0"
proof -
define Q where
"Q = (λr f z ε. (f has_contour_integral complex_of_real (2 * pi) * 𝗂 * r) (circlepath z ε))"
define P where
"P = (λr f z. ∃e>0. ∀ε>0. ε < e ⟶ Q r f z ε)"
have path_eq: "circlepath (z - w) ε = (+) (-w) ∘ circlepath z ε" for z w ε
by (simp add: circlepath_def o_def part_circlepath_def algebra_simps)
have *: "P r f z" if "P r (λx. f (x + w)) (z - w)" for r w f z
using that by (auto simp: P_def Q_def path_eq has_contour_integral_translate)
have "(SOME r. P r f z) = (SOME r. P r (λx. f (z + x)) 0)"
using *[of _ f z z] *[of _ "λx. f (z + x)" "-z"]
by (intro arg_cong[where f = Eps] ext iffI) (simp_all add: add_ac)
thus ?thesis
by (simp add: residue_def P_def Q_def)
qed
lemma residue_shift_0': "NO_MATCH 0 z ⟹ residue f z = residue (λx. f (z + x)) 0"
by (rule residue_shift_0)
lemma has_laurent_expansion_residue_0:
assumes "f has_laurent_expansion F"
shows "residue f 0 = fls_residue F"
proof (cases "fls_subdegree F ≥ 0")
case True
have "residue f 0 = residue (eval_fls F) 0"
using assms by (intro residue_cong) (auto simp: has_laurent_expansion_def eq_commute)
also have "… = 0"
by (rule residue_holo[OF _ _ holomorphic_on_eval_fls[OF order.refl]])
(use True assms in ‹auto simp: has_laurent_expansion_def zero_ereal_def›)
also have "… = fls_residue F"
using True by simp
finally show ?thesis .
next
case False
hence "F ≠ 0"
by auto
have *: "zor_poly f 0 has_fps_expansion fls_base_factor_to_fps F"
by (intro zor_poly_has_fps_expansion False assms ‹F ≠ 0›)
have "residue f 0 = (deriv ^^ (nat (-zorder f 0) - 1)) (zor_poly f 0) 0 / fact (nat (- zorder f 0) - 1)"
by (intro residue_pole_order has_laurent_expansion_isolated_0[OF assms]
has_laurent_expansion_imp_is_pole_0[OF assms]) (use False in auto)
also have "… = fls_residue F"
using has_laurent_expansion_zorder_0[OF assms ‹F ≠ 0›] False
by (subst fps_nth_fps_expansion [OF *, symmetric]) (auto simp: of_nat_diff)
finally show ?thesis .
qed
lemma has_laurent_expansion_residue:
assumes "(λx. f (z + x)) has_laurent_expansion F"
shows "residue f z = fls_residue F"
using has_laurent_expansion_residue_0[OF assms] by (simp add: residue_shift_0')
lemma eval_fls_has_laurent_expansion [laurent_expansion_intros]:
assumes "fls_conv_radius F > 0"
shows "eval_fls F has_laurent_expansion F"
using assms by (auto simp: has_laurent_expansion_def)
lemma fps_expansion_unique_complex:
fixes F G :: "complex fps"
assumes "f has_fps_expansion F" "f has_fps_expansion G"
shows "F = G"
using assms unfolding fps_eq_iff by (auto simp: fps_eq_iff fps_nth_fps_expansion)
lemma fps_expansion_eqI:
assumes "f has_fps_expansion F"
shows "fps_expansion f 0 = F"
using assms unfolding fps_eq_iff
by (auto simp: fps_eq_iff fps_nth_fps_expansion fps_expansion_def)
lemma has_fps_expansion_imp_eval_fps_eq:
assumes "f has_fps_expansion F" "norm z < r"
assumes "f holomorphic_on ball 0 r"
shows "eval_fps F z = f z"
proof -
have [simp]: "fps_expansion f 0 = F"
by (rule fps_expansion_eqI) fact
have *: "f holomorphic_on eball 0 (ereal r)"
using assms by simp
from conv_radius_fps_expansion[OF *] have "fps_conv_radius F ≥ ereal r"
by simp
have "eval_fps (fps_expansion f 0) z = f (0 + z)"
by (rule eval_fps_expansion'[OF *]) (use assms in auto)
thus ?thesis
by simp
qed
lemma fls_conv_radius_ge:
assumes "f has_laurent_expansion F"
assumes "f holomorphic_on eball 0 r - {0}"
shows "fls_conv_radius F ≥ r"
proof -
define n where "n = fls_subdegree F"
define G where "G = fls_base_factor_to_fps F"
define g where "g = (λz. if z = 0 then fps_nth G 0 else f z * z powi -n)"
have G: "g has_fps_expansion G"
unfolding G_def g_def n_def
by (intro has_fps_expansion_fls_base_factor_to_fps assms)
have "(λz. f z * z powi -n) holomorphic_on eball 0 r - {0}"
by (intro holomorphic_intros assms) auto
also have "?this ⟷ g holomorphic_on eball 0 r - {0}"
by (intro holomorphic_cong) (auto simp: g_def)
finally have "g analytic_on eball 0 r - {0}"
by (subst analytic_on_open) auto
moreover have "g analytic_on {0}"
using G has_fps_expansion_imp_analytic_0 by auto
ultimately have "g analytic_on (eball 0 r - {0} ∪ {0})"
by (subst analytic_on_Un) auto
hence "g analytic_on eball 0 r"
by (rule analytic_on_subset) auto
hence "g holomorphic_on eball 0 r"
by (subst (asm) analytic_on_open) auto
hence "fps_conv_radius (fps_expansion g 0) ≥ r"
by (intro conv_radius_fps_expansion)
also have "fps_expansion g 0 = G"
using G by (intro fps_expansion_eqI)
finally show ?thesis
by (simp add: fls_conv_radius_altdef G_def)
qed
lemma connected_eball [intro]: "connected (eball (z :: 'a :: real_normed_vector) r)"
by (cases r) auto
lemma eval_fls_eqI:
assumes "f has_laurent_expansion F" "f holomorphic_on eball 0 r - {0}"
assumes "z ∈ eball 0 r - {0}"
shows "eval_fls F z = f z"
proof -
have conv: "fls_conv_radius F ≥ r"
by (intro fls_conv_radius_ge[OF assms(1,2)])
have "(λz. eval_fls F z - f z) has_laurent_expansion F - F"
using assms by (intro laurent_expansion_intros assms) (auto simp: has_laurent_expansion_def)
hence "(λz. eval_fls F z - f z) has_laurent_expansion 0"
by simp
hence "eval_fls F z - f z = 0"
proof (rule has_laurent_expansion_0_analytic_continuation)
have "ereal 0 ≤ ereal (norm z)"
by simp
also have "norm z < r"
using assms by auto
finally have "r > 0"
by (simp add: zero_ereal_def)
thus "open (eball 0 r :: complex set)" "connected (eball 0 r :: complex set)"
"0 ∈ eball 0 r" "z ∈ eball 0 r - {0}"
using assms by (auto simp: zero_ereal_def)
qed (auto intro!: holomorphic_intros assms less_le_trans[OF _ conv] split: if_splits)
thus ?thesis by simp
qed
lemma fls_nth_as_contour_integral:
assumes F: "f has_laurent_expansion F"
assumes holo: "f holomorphic_on ball 0 r - {0}"
assumes R: "0 < R" "R < r"
shows "((λz. f z * z powi (-(n+1))) has_contour_integral
complex_of_real (2 * pi) * 𝗂 * fls_nth F n) (circlepath 0 R)"
proof -
define I where "I = (λz. f z * z powi (-(n+1)))"
have "(I has_contour_integral complex_of_real (2 * pi) * 𝗂 * residue I 0) (circlepath 0 R)"
proof (rule base_residue)
show "open (ball (0::complex) r)" "0 ∈ ball (0::complex) r"
using R F by (auto simp: has_laurent_expansion_def zero_ereal_def)
qed (use R in ‹auto intro!: holomorphic_intros holomorphic_on_subset[OF holo]
simp: I_def split: if_splits›)
also have "residue I 0 = fls_residue (fls_shift (n + 1) F)"
unfolding I_def by (intro has_laurent_expansion_residue_0 laurent_expansion_intros F)
also have "… = fls_nth F n"
by simp
finally show ?thesis
by (simp add: I_def)
qed
lemma tendsto_0_subdegree_iff_0:
assumes F:"f has_laurent_expansion F" and "F≠0"
shows "(f ─0→0) ⟷ fls_subdegree F > 0"
proof -
have ?thesis if "is_pole f 0"
proof -
have "fls_subdegree F <0"
using is_pole_0_imp_neg_fls_subdegree[OF F that] .
moreover then have "¬ f ─0→0"
using ‹is_pole f 0› F at_neq_bot
has_laurent_expansion_imp_filterlim_infinity_0
not_tendsto_and_filterlim_at_infinity that
by blast
ultimately show ?thesis by auto
qed
moreover have ?thesis if "¬is_pole f 0" "∃x. f ─0→x"
proof -
have "fls_subdegree F ≥0"
using has_laurent_expansion_imp_is_pole_0[OF F] that(1)
by linarith
have "f ─0→0" if "fls_subdegree F > 0"
using fls_eq0_below_subdegree[OF that]
by (metis F ‹0 ≤ fls_subdegree F› has_laurent_expansion_imp_tendsto_0)
moreover have "fls_subdegree F > 0" if "f ─0→0"
proof -
have False if "fls_subdegree F = 0"
proof -
have "f ─0→ fls_nth F 0"
using has_laurent_expansion_imp_tendsto_0
[OF F ‹fls_subdegree F ≥0›] .
then have "fls_nth F 0 = 0" using ‹f ─0→0›
using LIM_unique by blast
then have "F = 0"
using nth_fls_subdegree_zero_iff ‹fls_subdegree F = 0›
by metis
with ‹F≠0› show False by auto
qed
with ‹fls_subdegree F ≥0›
show ?thesis by fastforce
qed
ultimately show ?thesis by auto
qed
moreover have "is_pole f 0 ∨ (∃x. f ─0→x)"
proof -
have "not_essential f 0"
using F has_laurent_expansion_not_essential_0 by auto
then show ?thesis unfolding not_essential_def
by auto
qed
ultimately show ?thesis by auto
qed
lemma tendsto_0_subdegree_iff:
assumes F:"(λw. f (z+w)) has_laurent_expansion F" and "F≠0"
shows "(f ─z→0) ⟷ fls_subdegree F > 0"
apply (subst Lim_at_zero)
apply (rule tendsto_0_subdegree_iff_0)
using assms by auto
lemma is_pole_0_deriv_divide_iff:
assumes F:"f has_laurent_expansion F" and "F≠0"
shows "is_pole (λx. deriv f x / f x) 0 ⟷ is_pole f 0 ∨ (f ─0→0)"
proof -
have "(λx. deriv f x / f x) has_laurent_expansion fls_deriv F / F"
using F by (auto intro:laurent_expansion_intros)
have "is_pole (λx. deriv f x / f x) 0 ⟷
fls_subdegree (fls_deriv F / F) < 0"
apply (rule is_pole_fls_subdegree_iff)
using F by (auto intro:laurent_expansion_intros)
also have "... ⟷ is_pole f 0 ∨ (f ─0→0)"
proof (cases "fls_subdegree F = 0")
case True
then have "fls_subdegree (fls_deriv F / F) ≥ 0"
by (metis diff_zero div_0 ‹F≠0› fls_deriv_subdegree0
fls_divide_subdegree)
moreover then have "¬ is_pole f 0"
by (metis F True is_pole_0_imp_neg_fls_subdegree less_le)
moreover have "¬ (f ─0→0)"
using tendsto_0_subdegree_iff_0[OF F ‹F≠0›] True by auto
ultimately show ?thesis by auto
next
case False
then have "fls_deriv F ≠ 0"
by (metis fls_const_subdegree fls_deriv_eq_0_iff)
then have "fls_subdegree (fls_deriv F / F) =
fls_subdegree (fls_deriv F) - fls_subdegree F"
by (rule fls_divide_subdegree[OF _ ‹F≠0›])
moreover have "fls_subdegree (fls_deriv F) = fls_subdegree F - 1"
using fls_subdegree_deriv[OF False] .
ultimately have "fls_subdegree (fls_deriv F / F) < 0" by auto
moreover have "f ─0→ 0 = (0 < fls_subdegree F)"
using tendsto_0_subdegree_iff_0[OF F ‹F ≠ 0›] .
moreover have "is_pole f 0 = (fls_subdegree F < 0)"
using is_pole_fls_subdegree_iff F by auto
ultimately show ?thesis using False by auto
qed
finally show ?thesis .
qed
lemma is_pole_deriv_divide_iff:
assumes F:"(λw. f (z+w)) has_laurent_expansion F" and "F≠0"
shows "is_pole (λx. deriv f x / f x) z ⟷ is_pole f z ∨ (f ─z→0)"
proof -
define ff df where "ff=(λw. f (z+w))" and "df=(λw. deriv f (z + w))"
have "is_pole (λx. deriv f x / f x) z
⟷ is_pole (λx. deriv ff x / ff x) 0"
unfolding ff_def df_def
by (simp add:deriv_shift_0' is_pole_shift_0' comp_def algebra_simps)
moreover have "is_pole f z ⟷ is_pole ff 0"
unfolding ff_def by (auto simp:is_pole_shift_0')
moreover have "(f ─z→0) ⟷ (ff ─0→0)"
unfolding ff_def by (simp add: LIM_offset_zero_iff)
moreover have "is_pole (λx. deriv ff x / ff x) 0 = (is_pole ff 0 ∨ ff ─0→ 0)"
apply (rule is_pole_0_deriv_divide_iff)
using F ff_def ‹F≠0› by blast+
ultimately show ?thesis by auto
qed
lemma subdegree_imp_eventually_deriv_nzero_0:
assumes F:"f has_laurent_expansion F" and "fls_subdegree F≠0"
shows "eventually (λz. deriv f z ≠ 0) (at 0)"
proof -
have "deriv f has_laurent_expansion fls_deriv F"
using has_laurent_expansion_deriv[OF F] .
moreover have "fls_deriv F≠0"
using ‹fls_subdegree F≠0›
by (metis fls_const_subdegree fls_deriv_eq_0_iff)
ultimately show ?thesis
using has_laurent_expansion_eventually_nonzero_iff' by blast
qed
lemma subdegree_imp_eventually_deriv_nzero:
assumes F:"(λw. f (z+w)) has_laurent_expansion F"
and "fls_subdegree F≠0"
shows "eventually (λw. deriv f w ≠ 0) (at z)"
proof -
have "∀⇩F x in at 0. deriv (λw. f (z + w)) x ≠ 0"
using subdegree_imp_eventually_deriv_nzero_0 assms by auto
then show ?thesis
apply (subst eventually_at_to_0)
by (simp add:deriv_shift_0' comp_def algebra_simps)
qed
lemma has_fps_expansion_imp_asymp_equiv_0:
fixes f :: "complex ⇒ complex"
assumes F: "f has_fps_expansion F"
defines "n ≡ subdegree F"
shows "f ∼[nhds 0] (λz. fps_nth F n * z ^ n)"
proof -
have F': "f has_laurent_expansion fps_to_fls F"
using F has_laurent_expansion_fps by blast
have "f ∼[at 0] (λz. fps_nth F n * z ^ n)"
using has_laurent_expansion_imp_asymp_equiv_0[OF F']
by (simp add: fls_subdegree_fls_to_fps n_def)
moreover have "f 0 = fps_nth F n * 0 ^ n"
using F by (auto simp: n_def has_fps_expansion_to_laurent power_0_left)
ultimately show ?thesis
by (auto simp: asymp_equiv_nhds_iff)
qed
lemma has_fps_expansion_imp_tendsto_0:
fixes f :: "complex ⇒ complex"
assumes "f has_fps_expansion F"
shows "(f ⤏ fps_nth F 0) (nhds 0)"
proof (rule asymp_equiv_tendsto_transfer)
show "(λz. fps_nth F (subdegree F) * z ^ subdegree F) ∼[nhds 0] f"
by (rule asymp_equiv_symI, rule has_fps_expansion_imp_asymp_equiv_0) fact
have "((λz. F $ subdegree F * z ^ subdegree F) ⤏ F $ 0) (at 0)"
by (rule tendsto_eq_intros refl | simp)+ (auto simp: power_0_left)
thus "((λz. F $ subdegree F * z ^ subdegree F) ⤏ F $ 0) (nhds 0)"
by (auto simp: tendsto_nhds_iff power_0_left)
qed
lemma has_fps_expansion_imp_0_eq_fps_nth_0:
assumes "f has_fps_expansion F"
shows "f 0 = fps_nth F 0"
proof -
have "eventually (λx. f x = eval_fps F x) (nhds 0)"
using assms by (auto simp: has_fps_expansion_def eq_commute)
then obtain A where "open A" "0 ∈ A" "∀x∈A. f x = eval_fps F x"
unfolding eventually_nhds by blast
hence "f 0 = eval_fps F 0"
by blast
thus ?thesis
by (simp add: eval_fps_at_0)
qed
lemma fls_nth_compose_aux:
assumes "f has_fps_expansion F"
assumes G: "g has_fps_expansion G" "fps_nth G 0 = 0" "fps_deriv G ≠ 0"
assumes "(f ∘ g) has_laurent_expansion H"
shows "fls_nth H (int n) = fps_nth (fps_compose F G) n"
using assms(1,5)
proof (induction n arbitrary: f F H rule: less_induct)
case (less n f F H)
have [simp]: "g 0 = 0"
using has_fps_expansion_imp_0_eq_fps_nth_0[OF G(1)] G(2) by simp
have ana_f: "f analytic_on {0}"
using less.prems by (meson has_fps_expansion_imp_analytic_0)
have ana_g: "g analytic_on {0}"
using G by (meson has_fps_expansion_imp_analytic_0)
have "(f ∘ g) has_laurent_expansion fps_to_fls (fps_expansion (f ∘ g) 0)"
by (rule analytic_at_imp_has_fps_expansion_0 analytic_intros has_laurent_expansion_fps
analytic_on_compose_gen ana_f ana_g)+ auto
with less.prems have "H = fps_to_fls (fps_expansion (f ∘ g) 0)"
using has_laurent_expansion_unique by blast
also have "fls_subdegree … ≥ 0"
by (simp add: fls_subdegree_fls_to_fps)
finally have subdeg: "fls_subdegree H ≥ 0" .
show ?case
proof (cases "n = 0")
case [simp]: True
have lim_g: "g ─0→ 0"
using has_laurent_expansion_imp_tendsto_0[of g "fps_to_fls G"] G
by (auto simp: fls_subdegree_fls_to_fps_gt0 has_fps_expansion_to_laurent)
have lim_f: "(f ⤏ fps_nth F 0) (nhds 0)"
by (intro has_fps_expansion_imp_tendsto_0 less.prems)
have "(λx. f (g x)) ─0→ fps_nth F 0"
by (rule filterlim_compose[OF lim_f lim_g])
moreover have "(f ∘ g) ─0→ fls_nth H 0"
by (intro has_laurent_expansion_imp_tendsto_0 less.prems subdeg)
ultimately have "fps_nth F 0 = fls_nth H 0"
using tendsto_unique by (force simp: o_def)
thus ?thesis
by simp
next
case n: False
define GH where "GH = (fls_deriv H / fls_deriv (fps_to_fls G))"
define GH' where "GH' = fls_regpart GH"
have "(λx. deriv (f ∘ g) x / deriv g x) has_laurent_expansion
fls_deriv H / fls_deriv (fps_to_fls G)"
by (intro laurent_expansion_intros less.prems has_laurent_expansion_fps[of _ G] G)
also have "?this ⟷ (deriv f ∘ g) has_laurent_expansion fls_deriv H / fls_deriv (fps_to_fls G)"
proof (rule has_laurent_expansion_cong)
from ana_f obtain r1 where r1: "r1 > 0" "f holomorphic_on ball 0 r1"
unfolding analytic_on_def by blast
from ana_g obtain r2 where r2: "r2 > 0" "g holomorphic_on ball 0 r2"
unfolding analytic_on_def by blast
have lim_g: "g ─0→ 0"
using has_laurent_expansion_imp_tendsto_0[of g "fps_to_fls G"] G
by (auto simp: fls_subdegree_fls_to_fps_gt0 has_fps_expansion_to_laurent)
moreover have "open (ball 0 r1)" "0 ∈ ball 0 r1"
using r1 by auto
ultimately have "eventually (λx. g x ∈ ball 0 r1) (at 0)"
unfolding tendsto_def by blast
moreover have "eventually (λx. deriv g x ≠ 0) (at 0)"
using G fps_to_fls_eq_0_iff has_fps_expansion_deriv has_fps_expansion_to_laurent
has_laurent_expansion_nonzero_imp_eventually_nonzero by blast
moreover have "eventually (λx. x ∈ ball 0 (min r1 r2) - {0}) (at 0)"
by (intro eventually_at_in_open) (use r1 r2 in auto)
ultimately show "eventually (λx. deriv (f ∘ g) x / deriv g x = (deriv f ∘ g) x) (at 0)"
proof eventually_elim
case (elim x)
thus ?case using r1 r2
by (subst deriv_chain)
(auto simp: field_simps holomorphic_on_def at_within_open[of _ "ball _ _"])
qed
qed auto
finally have GH: "(deriv f ∘ g) has_laurent_expansion GH"
unfolding GH_def .
have "(deriv f ∘ g) has_laurent_expansion fps_to_fls (fps_expansion (deriv f ∘ g) 0)"
by (rule analytic_at_imp_has_fps_expansion_0 analytic_intros has_laurent_expansion_fps
analytic_on_compose_gen ana_f ana_g)+ auto
with GH have "GH = fps_to_fls (fps_expansion (deriv f ∘ g) 0)"
using has_laurent_expansion_unique by blast
also have "fls_subdegree … ≥ 0"
by (simp add: fls_subdegree_fls_to_fps)
finally have subdeg': "fls_subdegree GH ≥ 0" .
have "deriv f has_fps_expansion fps_deriv F"
by (intro fps_expansion_intros less.prems)
from this and GH have IH: "fls_nth GH (int k) = fps_nth (fps_compose (fps_deriv F) G) k"
if "k < n" for k
by (intro less.IH that)
have "fps_nth (fps_compose (fps_deriv F) G) n = (∑i=0..n. of_nat (Suc i) * F $ Suc i * G ^ i $ n)"
by (simp add: fps_compose_nth)
have "fps_nth (fps_compose F G) n =
fps_nth (fps_deriv (fps_compose F G)) (n - 1) / of_nat n"
using n by (cases n) (auto simp del: of_nat_Suc)
also have "fps_deriv (fps_compose F G) = fps_compose (fps_deriv F) G * fps_deriv G "
using G by (subst fps_compose_deriv) auto
also have "fps_nth … (n - 1) = (∑i=0..n-1. (fps_deriv F oo G) $ i * fps_deriv G $ (n - 1 - i))"
unfolding fps_mult_nth ..
also have "… = (∑i=0..n-1. fps_nth GH' i * of_nat (n - i) * G $ (n - i))"
using n by (intro sum.cong) (auto simp: IH Suc_diff_Suc GH'_def)
also have "… = (∑i=0..n. fps_nth GH' i * of_nat (n - i) * G $ (n - i))"
by (intro sum.mono_neutral_left) auto
also have "… = fps_nth (GH' * Abs_fps (λi. of_nat i * fps_nth G i)) n"
by (simp add: fps_mult_nth mult_ac)
also have "Abs_fps (λi. of_nat i * fps_nth G i) = fps_X * fps_deriv G"
by (simp add: fps_mult_fps_X_deriv_shift)
also have "fps_nth (GH' * (fps_X * fps_deriv G)) n =
fls_nth (fps_to_fls (GH' * (fps_X * fps_deriv G))) (int n)"
by simp
also have "fps_to_fls (GH' * (fps_X * fps_deriv G)) =
GH * fps_to_fls (fps_deriv G) * fls_X"
using subdeg' by (simp add: mult_ac fls_times_fps_to_fls GH'_def)
also have "GH * fps_to_fls (fps_deriv G) = fls_deriv H"
unfolding GH_def using G by (simp add: fls_deriv_fps_to_fls)
also have "fls_deriv H * fls_X = fls_shift (-1) (fls_deriv H)"
using fls_X_times_conv_shift(2) by blast
finally show ?thesis
using n by simp
qed
qed
lemma has_fps_expansion_compose [fps_expansion_intros]:
fixes f g :: "complex ⇒ complex"
assumes F: "f has_fps_expansion F"
assumes G: "g has_fps_expansion G" "fps_nth G 0 = 0"
shows "(f ∘ g) has_fps_expansion fps_compose F G"
proof (cases "fps_deriv G = 0")
case False
have [simp]: "g 0 = 0"
using has_fps_expansion_imp_0_eq_fps_nth_0[OF G(1)] G(2) False by simp
have ana_f: "f analytic_on {0}"
using F by (meson has_fps_expansion_imp_analytic_0)
have ana_g: "g analytic_on {0}"
using G by (meson has_fps_expansion_imp_analytic_0)
have fg: "(f ∘ g) has_fps_expansion fps_expansion (f ∘ g) 0"
by (rule analytic_at_imp_has_fps_expansion_0 analytic_intros
analytic_on_compose_gen ana_f ana_g)+ auto
have "fls_nth (fps_to_fls (fps_expansion (f ∘ g) 0)) (int n) = fps_nth (fps_compose F G) n" for n
by (rule fls_nth_compose_aux has_laurent_expansion_fps F G False fg)+
hence "fps_expansion (f ∘ g) 0 = fps_compose F G"
by (simp add: fps_eq_iff)
thus ?thesis using fg
by simp
next
case True
have [simp]: "f 0 = fps_nth F 0"
using F by (auto dest: has_fps_expansion_imp_0_eq_fps_nth_0)
from True have "fps_nth G n = 0" for n
using G(2) by (cases n) (auto simp del: of_nat_Suc)
hence [simp]: "G = 0"
by (auto simp: fps_eq_iff)
have "(λ_. f 0) has_fps_expansion fps_const (f 0)"
by (intro fps_expansion_intros)
also have "eventually (λx. g x = 0) (nhds 0)"
using G by (auto simp: has_fps_expansion_def)
hence "(λ_. f 0) has_fps_expansion fps_const (f 0) ⟷ (f ∘ g) has_fps_expansion fps_const (f 0)"
by (intro has_fps_expansion_cong) (auto elim!: eventually_mono)
thus ?thesis
by simp
qed
lemma has_fps_expansion_fps_to_fls:
assumes "f has_laurent_expansion fps_to_fls F"
shows "(λz. if z = 0 then fps_nth F 0 else f z) has_fps_expansion F"
(is "?f' has_fps_expansion _")
proof -
have "f has_laurent_expansion fps_to_fls F ⟷ ?f' has_laurent_expansion fps_to_fls F"
by (intro has_laurent_expansion_cong) (auto simp: eventually_at_filter)
with assms show ?thesis
by (auto simp: has_fps_expansion_to_laurent)
qed
lemma has_laurent_expansion_compose [laurent_expansion_intros]:
fixes f g :: "complex ⇒ complex"
assumes F: "f has_laurent_expansion F"
assumes G: "g has_laurent_expansion fps_to_fls G" "fps_nth G 0 = 0" "G ≠ 0"
shows "(f ∘ g) has_laurent_expansion fls_compose_fps F G"
proof -
from assms have lim_g: "g ─0→ 0"
by (subst tendsto_0_subdegree_iff_0[OF G(1)])
(auto simp: fls_subdegree_fls_to_fps subdegree_pos_iff)
have ev1: "eventually (λz. g z ≠ 0) (at 0)"
using ‹G ≠ 0› G(1) fps_to_fls_eq_0_iff has_laurent_expansion_fps
has_laurent_expansion_nonzero_imp_eventually_nonzero by blast
moreover have "eventually (λz. z ≠ 0) (at (0 :: complex))"
by (auto simp: eventually_at_filter)
ultimately have ev: "eventually (λz. z ≠ 0 ∧ g z ≠ 0) (at 0)"
by eventually_elim blast
from ev1 and lim_g have lim_g': "filterlim g (at 0) (at 0)"
by (auto simp: filterlim_at)
define g' where "g' = (λz. if z = 0 then fps_nth G 0 else g z)"
show ?thesis
proof (cases "F = 0")
assume [simp]: "F = 0"
have "eventually (λz. f z = 0) (at 0)"
using F by (auto simp: has_laurent_expansion_def)
hence "eventually (λz. f (g z) = 0) (at 0)"
using lim_g' by (rule eventually_compose_filterlim)
thus ?thesis
by (auto simp: has_laurent_expansion_def)
next
assume [simp]: "F ≠ 0"
define n where "n = fls_subdegree F"
define f' where
"f' = (λz. if z = 0 then fps_nth (fls_base_factor_to_fps F) 0 else f z * z powi -n)"
have "((λz. (f' ∘ g') z * g z powi n)) has_laurent_expansion fls_compose_fps F G"
unfolding f'_def n_def fls_compose_fps_def g'_def
by (intro fps_expansion_intros laurent_expansion_intros has_fps_expansion_fps_to_fls
has_fps_expansion_fls_base_factor_to_fps assms has_laurent_expansion_fps)
also have "?this ⟷ ?thesis"
by (intro has_laurent_expansion_cong eventually_mono[OF ev])
(auto simp: f'_def power_int_minus g'_def)
finally show ?thesis .
qed
qed
lemma has_laurent_expansion_fls_X_inv [laurent_expansion_intros]:
"inverse has_laurent_expansion fls_X_inv"
using has_laurent_expansion_inverse[OF has_laurent_expansion_fps_X]
by (simp add: fls_inverse_X)
lemma zorder_times_analytic:
assumes "f analytic_on {z}" "g analytic_on {z}"
assumes "eventually (λz. f z * g z ≠ 0) (at z)"
shows "zorder (λz. f z * g z) z = zorder f z + zorder g z"
proof -
have *: "(λw. f (z + w)) has_fps_expansion fps_expansion f z"
"(λw. g (z + w)) has_fps_expansion fps_expansion g z"
"(λw. f (z + w) * g (z + w)) has_fps_expansion fps_expansion f z * fps_expansion g z"
by (intro fps_expansion_intros analytic_at_imp_has_fps_expansion assms)+
have [simp]: "fps_expansion f z ≠ 0"
proof
assume "fps_expansion f z = 0"
hence "eventually (λz. f z * g z = 0) (at z)" using *(1)
by (auto simp: has_fps_expansion_0_iff nhds_to_0' eventually_filtermap eventually_at_filter
elim: eventually_mono)
with assms(3) have "eventually (λz. False) (at z)"
by eventually_elim auto
thus False by simp
qed
have [simp]: "fps_expansion g z ≠ 0"
proof
assume "fps_expansion g z = 0"
hence "eventually (λz. f z * g z = 0) (at z)" using *(2)
by (auto simp: has_fps_expansion_0_iff nhds_to_0' eventually_filtermap eventually_at_filter
elim: eventually_mono)
with assms(3) have "eventually (λz. False) (at z)"
by eventually_elim auto
thus False by simp
qed
from *[THEN has_fps_expansion_zorder] show ?thesis
by auto
qed
lemma zorder_const [simp]: "c ≠ 0 ⟹ zorder (λ_. c) z = 0"
by (intro zorder_eqI[where S = UNIV]) auto
lemma zorder_prod_analytic:
assumes "⋀x. x ∈ A ⟹ f x analytic_on {z}"
assumes "eventually (λz. (∏x∈A. f x z) ≠ 0) (at z)"
shows "zorder (λz. ∏x∈A. f x z) z = (∑x∈A. zorder (f x) z)"
using assms
proof (induction A rule: infinite_finite_induct)
case (insert x A)
have "zorder (λz. f x z * (∏x∈A. f x z)) z = zorder (f x) z + zorder (λz. ∏x∈A. f x z) z"
using insert.prems insert.hyps by (intro zorder_times_analytic analytic_intros) auto
also have "zorder (λz. ∏x∈A. f x z) z = (∑x∈A. zorder (f x) z)"
using insert.prems insert.hyps by (intro insert.IH) (auto elim!: eventually_mono)
finally show ?case using insert
by simp
qed auto
lemma zorder_eq_0I:
assumes "g analytic_on {z}" "g z ≠ 0"
shows "zorder g z = 0"
using analytic_at assms zorder_eqI by fastforce
lemma zorder_pos_iff:
assumes "f holomorphic_on A" "open A" "z ∈ A" "frequently (λz. f z ≠ 0) (at z)"
shows "zorder f z > 0 ⟷ f z = 0"
proof -
have "f analytic_on {z}"
using assms analytic_at by blast
hence *: "(λw. f (z + w)) has_fps_expansion fps_expansion f z"
using analytic_at_imp_has_fps_expansion by blast
have nz: "fps_expansion f z ≠ 0"
proof
assume "fps_expansion f z = 0"
hence "eventually (λz. f z = 0) (nhds z)"
using * by (auto simp: has_fps_expansion_def nhds_to_0' eventually_filtermap add_ac)
hence "eventually (λz. f z = 0) (at z)"
by (auto simp: eventually_at_filter elim: eventually_mono)
with assms show False
by (auto simp: frequently_def)
qed
from has_fps_expansion_zorder[OF * this] have eq: "zorder f z = int (subdegree (fps_expansion f z))"
by auto
moreover have "subdegree (fps_expansion f z) = 0 ⟷ fps_nth (fps_expansion f z) 0 ≠ 0"
using nz by (auto simp: subdegree_eq_0_iff)
moreover have "fps_nth (fps_expansion f z) 0 = f z"
by (auto simp: fps_expansion_def)
ultimately show ?thesis
by auto
qed
lemma zorder_pos_iff':
assumes "f analytic_on {z}" "frequently (λz. f z ≠ 0) (at z)"
shows "zorder f z > 0 ⟷ f z = 0"
using analytic_at assms zorder_pos_iff by blast
lemma zorder_ge_0:
assumes "f analytic_on {z}" "frequently (λz. f z ≠ 0) (at z)"
shows "zorder f z ≥ 0"
proof -
have *: "(λw. f (z + w)) has_laurent_expansion fps_to_fls (fps_expansion f z)"
using assms by (simp add: analytic_at_imp_has_fps_expansion has_laurent_expansion_fps)
from * assms(2) have "fps_to_fls (fps_expansion f z) ≠ 0"
by (auto simp: has_laurent_expansion_def frequently_def at_to_0' eventually_filtermap add_ac)
with has_laurent_expansion_zorder[OF *] show ?thesis
by (simp add: fls_subdegree_fls_to_fps)
qed
lemma zorder_eq_0_iff:
assumes "f analytic_on {z}" "frequently (λw. f w ≠ 0) (at z)"
shows "zorder f z = 0 ⟷ f z ≠ 0"
using assms zorder_eq_0I zorder_pos_iff' by fastforce
lemma dist_mult_left:
"dist (a * b) (a * c :: 'a :: real_normed_field) = norm a * dist b c"
unfolding dist_norm right_diff_distrib [symmetric] norm_mult by simp
lemma dist_mult_right:
"dist (b * a) (c * a :: 'a :: real_normed_field) = norm a * dist b c"
using dist_mult_left[of a b c] by (simp add: mult_ac)
lemma zorder_scale:
assumes "f analytic_on {a * z}" "eventually (λw. f w ≠ 0) (at (a * z))" "a ≠ 0"
shows "zorder (λw. f (a * w)) z = zorder f (a * z)"
proof -
from assms(1) obtain r where r: "r > 0" "f holomorphic_on ball (a * z) r"
by (auto simp: analytic_on_def)
have *: "open (ball (a * z) r)" "connected (ball (a * z) r)" "a * z ∈ ball (a * z) r"
using r ‹a ≠ 0› by (auto simp: dist_norm)
from assms(2) have "eventually (λw. f w ≠ 0 ∧ w ∈ ball (a * z) r - {a * z}) (at (a * z))"
using ‹r > 0› by (intro eventually_conj eventually_at_in_open) auto
then obtain z0 where "f z0 ≠ 0 ∧ z0 ∈ ball (a * z) r - {a * z}"
using eventually_happens[of _ "at (a * z)"] by force
hence **: "∃w∈ball (a * z) r. f w ≠ 0"
by blast
define n where "n = nat (zorder f (a * z))"
obtain r' where r':
"(if f (a * z) = 0 then 0 < zorder f (a * z) else zorder f (a * z) = 0)"
"r' > 0" "cball (a * z) r' ⊆ ball (a * z) r" "zor_poly f (a * z) holomorphic_on cball (a * z) r'"
"⋀w. w ∈ cball (a * z) r' ⟹
f w = zor_poly f (a * z) w * (w - a * z) ^ n ∧ zor_poly f (a * z) w ≠ 0"
unfolding n_def using zorder_exist_zero[OF r(2) * **] by blast
show ?thesis
proof (rule zorder_eqI)
show "open (ball z (r' / norm a))" "z ∈ ball z (r' / norm a)"
using r ‹r' > 0› ‹a ≠ 0› by auto
have "(*) a ` ball z (r' / cmod a) ⊆ cball (a * z) r'"
proof safe
fix w assume "w ∈ ball z (r' / cmod a)"
thus "a * w ∈ cball (a * z) r'"
using dist_mult_left[of a z w] ‹a ≠ 0› by (auto simp: divide_simps mult_ac)
qed
thus "(λw. a ^ n * (zor_poly f (a * z) ∘ (λw. a * w)) w) holomorphic_on ball z (r' / norm a)"
using ‹a ≠ 0› by (intro holomorphic_on_compose_gen[OF _ r'(4)] holomorphic_intros) auto
show "a ^ n * (zor_poly f (a * z) ∘ (λw. a * w)) z ≠ 0"
using r' ‹a ≠ 0› by auto
show "f (a * w) = a ^ n * (zor_poly f (a * z) ∘ (*) a) w * (w - z) powi (zorder f (a * z))"
if "w ∈ ball z (r' / norm a)" "w ≠ z" for w
proof -
have "f (a * w) = zor_poly f (a * z) (a * w) * (a * (w - z)) ^ n"
using that r'(5)[of "a * w"] dist_mult_left[of a z w] ‹a ≠ 0› unfolding ring_distribs
by (auto simp: divide_simps mult_ac)
also have "… = a ^ n * zor_poly f (a * z) (a * w) * (w - z) ^ n"
by (subst power_mult_distrib) (auto simp: mult_ac)
also have "(w - z) ^ n = (w - z) powi of_nat n"
by simp
also have "of_nat n = zorder f (a * z)"
using r'(1) by (auto simp: n_def split: if_splits)
finally show ?thesis
unfolding o_def n_def .
qed
qed
qed
lemma subdegree_fps_compose [simp]:
fixes F G :: "'a :: idom fps"
assumes [simp]: "fps_nth G 0 = 0"
shows "subdegree (fps_compose F G) = subdegree F * subdegree G"
proof (cases "G = 0"; cases "F = 0")
assume [simp]: "G ≠ 0" "F ≠ 0"
define m where "m = subdegree F"
define F' where "F' = fps_shift m F"
have F_eq: "F = F' * fps_X ^ m"
unfolding F'_def by (simp add: fps_shift_times_fps_X_power m_def)
have [simp]: "F' ≠ 0"
using ‹F ≠ 0› unfolding F_eq by auto
have "subdegree (fps_compose F G) = subdegree (fps_compose F' G) + m * subdegree G"
by (simp add: F_eq fps_compose_mult_distrib fps_compose_eq_0_iff flip: fps_compose_power)
also have "subdegree (fps_compose F' G) = 0"
by (intro subdegree_eq_0) (auto simp: F'_def m_def)
finally show ?thesis by (simp add: m_def)
qed auto
lemma fls_subdegree_power_int [simp]:
fixes F :: "'a :: field fls"
shows "fls_subdegree (F powi n) = n * fls_subdegree F"
by (auto simp: power_int_def fls_subdegree_pow)
lemma subdegree_fls_compose_fps [simp]:
fixes G :: "'a :: field fps"
assumes [simp]: "fps_nth G 0 = 0"
shows "fls_subdegree (fls_compose_fps F G) = fls_subdegree F * subdegree G"
proof (cases "F = 0"; cases "G = 0")
assume [simp]: "G ≠ 0" "F ≠ 0"
have nz1: "fls_base_factor_to_fps F ≠ 0"
using ‹F ≠ 0› fls_base_factor_to_fps_nonzero by blast
show ?thesis
unfolding fls_compose_fps_def using nz1
by (subst fls_subdegree_mult) (simp_all add: fps_compose_eq_0_iff fls_subdegree_fls_to_fps)
qed (auto simp: fls_compose_fps_0_right)
lemma zorder_compose_aux:
assumes "isolated_singularity_at f 0" "not_essential f 0"
assumes G: "g has_fps_expansion G" "G ≠ 0" "g 0 = 0"
assumes "eventually (λw. f w ≠ 0) (at 0)"
shows "zorder (f ∘ g) 0 = zorder f 0 * subdegree G"
proof -
obtain F where F: "f has_laurent_expansion F"
using not_essential_has_laurent_expansion_0[OF assms(1,2)] by blast
have [simp]: "fps_nth G 0 = 0"
using G ‹g 0 = 0› by (simp add: has_fps_expansion_imp_0_eq_fps_nth_0)
note [simp] = ‹G ≠ 0› ‹g 0 = 0›
have [simp]: "F ≠ 0"
using has_laurent_expansion_eventually_nonzero_iff[of f 0 F] F assms by simp
have FG: "(f ∘ g) has_laurent_expansion fls_compose_fps F G"
by (intro has_laurent_expansion_compose has_laurent_expansion_fps F G) auto
have "zorder (f ∘ g) 0 = fls_subdegree (fls_compose_fps F G)"
using has_laurent_expansion_zorder_0 [OF FG] by (auto simp: fls_compose_fps_eq_0_iff)
also have "… = fls_subdegree F * int (subdegree G)"
by simp
also have "fls_subdegree F = zorder f 0"
using has_laurent_expansion_zorder_0 [OF F] by auto
finally show ?thesis .
qed
lemma zorder_compose:
assumes "isolated_singularity_at f (g z)" "not_essential f (g z)"
assumes G: "(λx. g (z + x) - g z) has_fps_expansion G" "G ≠ 0"
assumes "eventually (λw. f w ≠ 0) (at (g z))"
shows "zorder (f ∘ g) z = zorder f (g z) * subdegree G"
proof -
define f' where "f' = (λw. f (g z + w))"
define g' where "g' = (λw. g (z + w) - g z)"
have "zorder f (g z) = zorder f' 0"
by (simp add: f'_def zorder_shift' add_ac)
have "zorder (λx. g x - g z) z = zorder g' 0"
by (simp add: g'_def zorder_shift' add_ac)
have "zorder (f ∘ g) z = zorder (f' ∘ g') 0"
by (simp add: zorder_shift' f'_def g'_def add_ac o_def)
also have "… = zorder f' 0 * int (subdegree G)"
proof (rule zorder_compose_aux)
show "isolated_singularity_at f' 0" unfolding f'_def
using assms has_laurent_expansion_isolated_0 not_essential_has_laurent_expansion by blast
show "not_essential f' 0" unfolding f'_def
using assms has_laurent_expansion_not_essential_0 not_essential_has_laurent_expansion by blast
qed (use assms in ‹auto simp: f'_def g'_def at_to_0' eventually_filtermap add_ac›)
also have "zorder f' 0 = zorder f (g z)"
by (simp add: f'_def zorder_shift' add_ac)
finally show ?thesis .
qed
lemma fps_to_fls_eq_fls_const_iff [simp]: "fps_to_fls F = fls_const c ⟷ F = fps_const c"
using fps_to_fls_eq_iff by fastforce
lemma zorder_compose':
assumes "isolated_singularity_at f (g z)" "not_essential f (g z)"
assumes "g analytic_on {z}"
assumes "eventually (λw. f w ≠ 0) (at (g z))"
assumes "eventually (λw. g w ≠ g z) (at z)"
shows "zorder (f ∘ g) z = zorder f (g z) * zorder (λx. g x - g z) z"
proof -
obtain G where G [fps_expansion_intros]: "(λx. g (z + x)) has_fps_expansion G"
using assms analytic_at_imp_has_fps_expansion by blast
have G': "(λx. g (z + x) - g z) has_fps_expansion G - fps_const (g z)"
by (intro fps_expansion_intros)
hence G'': "(λx. g (z + x) - g z) has_laurent_expansion fps_to_fls (G - fps_const (g z))"
using has_laurent_expansion_fps by blast
have nz: "G - fps_const (g z) ≠ 0"
using has_laurent_expansion_eventually_nonzero_iff[OF G''] assms by auto
have "zorder (f ∘ g) z = zorder f (g z) * subdegree (G - fps_const (g z))"
proof (rule zorder_compose)
show "(λx. g (z + x) - g z) has_fps_expansion G - fps_const (g z)"
by (intro fps_expansion_intros)
qed (use assms nz in auto)
also have "int (subdegree (G - fps_const (g z))) = fls_subdegree (fps_to_fls G - fls_const (g z))"
by (simp flip: fls_subdegree_fls_to_fps)
also have "… = zorder (λx. g x - g z) z"
using has_laurent_expansion_zorder [OF G''] nz by auto
finally show ?thesis .
qed
lemma analytic_at_cong:
assumes "eventually (λx. f x = g x) (nhds x)" "x = y"
shows "f analytic_on {x} ⟷ g analytic_on {y}"
proof -
have "g analytic_on {x}" if "f analytic_on {x}" "eventually (λx. f x = g x) (nhds x)" for f g
proof -
have "(λy. f (x + y)) has_fps_expansion fps_expansion f x"
by (rule analytic_at_imp_has_fps_expansion) fact
also have "?this ⟷ (λy. g (x + y)) has_fps_expansion fps_expansion f x"
using that by (intro has_fps_expansion_cong refl) (auto simp: nhds_to_0' eventually_filtermap)
finally show ?thesis
by (rule has_fps_expansion_imp_analytic)
qed
from this[of f g] this[of g f] show ?thesis using assms
by (auto simp: eq_commute)
qed
lemma has_laurent_expansion_sin' [laurent_expansion_intros]:
"sin has_laurent_expansion fps_to_fls (fps_sin 1)"
using has_fps_expansion_sin' has_fps_expansion_to_laurent by blast
lemma has_laurent_expansion_cos' [laurent_expansion_intros]:
"cos has_laurent_expansion fps_to_fls (fps_cos 1)"
using has_fps_expansion_cos' has_fps_expansion_to_laurent by blast
lemma has_laurent_expansion_sin [laurent_expansion_intros]:
"(λz. sin (c * z)) has_laurent_expansion fps_to_fls (fps_sin c)"
by (intro has_laurent_expansion_fps has_fps_expansion_sin)
lemma has_laurent_expansion_cos [laurent_expansion_intros]:
"(λz. cos (c * z)) has_laurent_expansion fps_to_fls (fps_cos c)"
by (intro has_laurent_expansion_fps has_fps_expansion_cos)
lemma has_laurent_expansion_tan' [laurent_expansion_intros]:
"tan has_laurent_expansion fps_to_fls (fps_tan 1)"
using has_fps_expansion_tan' has_fps_expansion_to_laurent by blast
lemma has_laurent_expansion_tan [laurent_expansion_intros]:
"(λz. tan (c * z)) has_laurent_expansion fps_to_fls (fps_tan c)"
by (intro has_laurent_expansion_fps has_fps_expansion_tan)
subsection ‹More Laurent expansions›
lemma has_laurent_expansion_frequently_zero_iff:
assumes "(λw. f (z + w)) has_laurent_expansion F"
shows "frequently (λz. f z = 0) (at z) ⟷ F = 0"
using assms by (simp add: frequently_def has_laurent_expansion_eventually_nonzero_iff)
lemma has_laurent_expansion_eventually_zero_iff:
assumes "(λw. f (z + w)) has_laurent_expansion F"
shows "eventually (λz. f z = 0) (at z) ⟷ F = 0"
using assms
by (metis has_laurent_expansion_frequently_zero_iff has_laurent_expansion_isolated
has_laurent_expansion_not_essential laurent_expansion_def
not_essential_frequently_0_imp_eventually_0 not_essential_has_laurent_expansion)
lemma has_laurent_expansion_frequently_nonzero_iff:
assumes "(λw. f (z + w)) has_laurent_expansion F"
shows "frequently (λz. f z ≠ 0) (at z) ⟷ F ≠ 0"
using assms by (metis has_laurent_expansion_eventually_zero_iff not_eventually)
lemma has_laurent_expansion_sum_list [laurent_expansion_intros]:
assumes "⋀x. x ∈ set xs ⟹ f x has_laurent_expansion F x"
shows "(λy. ∑x←xs. f x y) has_laurent_expansion (∑x←xs. F x)"
using assms by (induction xs) (auto intro!: laurent_expansion_intros)
lemma has_laurent_expansion_prod_list [laurent_expansion_intros]:
assumes "⋀x. x ∈ set xs ⟹ f x has_laurent_expansion F x"
shows "(λy. ∏x←xs. f x y) has_laurent_expansion (∏x←xs. F x)"
using assms by (induction xs) (auto intro!: laurent_expansion_intros)
lemma has_laurent_expansion_sum_mset [laurent_expansion_intros]:
assumes "⋀x. x ∈# I ⟹ f x has_laurent_expansion F x"
shows "(λy. ∑x∈#I. f x y) has_laurent_expansion (∑x∈#I. F x)"
using assms by (induction I) (auto intro!: laurent_expansion_intros)
lemma has_laurent_expansion_prod_mset [laurent_expansion_intros]:
assumes "⋀x. x ∈# I ⟹ f x has_laurent_expansion F x"
shows "(λy. ∏x∈#I. f x y) has_laurent_expansion (∏x∈#I. F x)"
using assms by (induction I) (auto intro!: laurent_expansion_intros)
end