Theory Complex_Singularities
theory Complex_Singularities
imports Conformal_Mappings
begin
subsection ‹Non-essential singular points›
definition is_pole ::
"('a::topological_space ⇒ 'b::real_normed_vector) ⇒ 'a ⇒ bool" where
"is_pole f a = (LIM x (at a). f x :> at_infinity)"
lemma is_pole_cong:
assumes "eventually (λx. f x = g x) (at a)" "a=b"
shows "is_pole f a ⟷ is_pole g b"
unfolding is_pole_def using assms by (intro filterlim_cong,auto)
lemma is_pole_transform:
assumes "is_pole f a" "eventually (λx. f x = g x) (at a)" "a=b"
shows "is_pole g b"
using is_pole_cong assms by auto
lemma is_pole_shift_iff:
fixes f :: "('a::real_normed_vector ⇒ 'b::real_normed_vector)"
shows "is_pole (f ∘ (+) d) a = is_pole f (a + d)"
by (metis add_diff_cancel_right' filterlim_shift_iff is_pole_def)
lemma is_pole_tendsto:
fixes f::"('a::topological_space ⇒ 'b::real_normed_div_algebra)"
shows "is_pole f x ⟹ ((inverse o f) ⤏ 0) (at x)"
unfolding is_pole_def
by (auto simp add:filterlim_inverse_at_iff[symmetric] comp_def filterlim_at)
lemma is_pole_inverse_holomorphic:
assumes "open s"
and f_holo:"f holomorphic_on (s-{z})"
and pole:"is_pole f z"
and non_z:"∀x∈s-{z}. f x≠0"
shows "(λx. if x=z then 0 else inverse (f x)) holomorphic_on s"
proof -
define g where "g ≡ λx. if x=z then 0 else inverse (f x)"
have "isCont g z" unfolding isCont_def using is_pole_tendsto[OF pole]
by (simp add: g_def cong: LIM_cong)
moreover have "continuous_on (s-{z}) f" using f_holo holomorphic_on_imp_continuous_on by auto
hence "continuous_on (s-{z}) (inverse o f)" unfolding comp_def
by (auto elim!:continuous_on_inverse simp add:non_z)
hence "continuous_on (s-{z}) g" unfolding g_def
apply (subst continuous_on_cong[where t="s-{z}" and g="inverse o f"])
by auto
ultimately have "continuous_on s g" using open_delete[OF ‹open s›] ‹open s›
by (auto simp add:continuous_on_eq_continuous_at)
moreover have "(inverse o f) holomorphic_on (s-{z})"
unfolding comp_def using f_holo
by (auto elim!:holomorphic_on_inverse simp add:non_z)
hence "g holomorphic_on (s-{z})"
apply (subst holomorphic_cong[where t="s-{z}" and g="inverse o f"])
by (auto simp add:g_def)
ultimately show ?thesis unfolding g_def using ‹open s›
by (auto elim!: no_isolated_singularity)
qed
lemma not_is_pole_holomorphic:
assumes "open A" "x ∈ A" "f holomorphic_on A"
shows "¬is_pole f x"
proof -
have "continuous_on A f" by (intro holomorphic_on_imp_continuous_on) fact
with assms have "isCont f x" by (simp add: continuous_on_eq_continuous_at)
hence "f ─x→ f x" by (simp add: isCont_def)
thus "¬is_pole f x" unfolding is_pole_def
using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto
qed
lemma is_pole_inverse_power: "n > 0 ⟹ is_pole (λz::complex. 1 / (z - a) ^ n) a"
unfolding is_pole_def inverse_eq_divide [symmetric]
by (intro filterlim_compose[OF filterlim_inverse_at_infinity] tendsto_intros)
(auto simp: filterlim_at eventually_at intro!: exI[of _ 1] tendsto_eq_intros)
lemma is_pole_inverse: "is_pole (λz::complex. 1 / (z - a)) a"
using is_pole_inverse_power[of 1 a] by simp
lemma is_pole_divide:
fixes f :: "'a :: t2_space ⇒ 'b :: real_normed_field"
assumes "isCont f z" "filterlim g (at 0) (at z)" "f z ≠ 0"
shows "is_pole (λz. f z / g z) z"
proof -
have "filterlim (λz. f z * inverse (g z)) at_infinity (at z)"
by (intro tendsto_mult_filterlim_at_infinity[of _ "f z"]
filterlim_compose[OF filterlim_inverse_at_infinity])+
(insert assms, auto simp: isCont_def)
thus ?thesis by (simp add: field_split_simps is_pole_def)
qed
lemma is_pole_basic:
assumes "f holomorphic_on A" "open A" "z ∈ A" "f z ≠ 0" "n > 0"
shows "is_pole (λw. f w / (w - z) ^ n) z"
proof (rule is_pole_divide)
have "continuous_on A f" by (rule holomorphic_on_imp_continuous_on) fact
with assms show "isCont f z" by (auto simp: continuous_on_eq_continuous_at)
have "filterlim (λw. (w - z) ^ n) (nhds 0) (at z)"
using assms by (auto intro!: tendsto_eq_intros)
thus "filterlim (λw. (w - z) ^ n) (at 0) (at z)"
by (intro filterlim_atI tendsto_eq_intros)
(insert assms, auto simp: eventually_at_filter)
qed fact+
lemma is_pole_basic':
assumes "f holomorphic_on A" "open A" "0 ∈ A" "f 0 ≠ 0" "n > 0"
shows "is_pole (λw. f w / w ^ n) 0"
using is_pole_basic[of f A 0] assms by simp
text ‹The proposition
\<^term>‹∃x. ((f::complex⇒complex) ⤏ x) (at z) ∨ is_pole f z›
can be interpreted as the complex function \<^term>‹f› has a non-essential singularity at \<^term>‹z›
(i.e. the singularity is either removable or a pole).›
definition not_essential::"[complex ⇒ complex, complex] ⇒ bool" where
"not_essential f z = (∃x. f─z→x ∨ is_pole f z)"
definition isolated_singularity_at::"[complex ⇒ complex, complex] ⇒ bool" where
"isolated_singularity_at f z = (∃r>0. f analytic_on ball z r-{z})"
named_theorems singularity_intros "introduction rules for singularities"
lemma holomorphic_factor_unique:
fixes f::"complex ⇒ complex" and z::complex and r::real and m n::int
assumes "r>0" "g z≠0" "h z≠0"
and asm:"∀w∈ball z r-{z}. f w = g w * (w-z) powr n ∧ g w≠0 ∧ f w = h w * (w - z) powr m ∧ h w≠0"
and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r"
shows "n=m"
proof -
have [simp]:"at z within ball z r ≠ bot" using ‹r>0›
by (auto simp add:at_within_ball_bot_iff)
have False when "n>m"
proof -
have "(h ⤏ 0) (at z within ball z r)"
proof (rule Lim_transform_within[OF _ ‹r>0›, where f="λw. (w - z) powr (n - m) * g w"])
have "∀w∈ball z r-{z}. h w = (w-z)powr(n-m) * g w"
using ‹n>m› asm ‹r>0›
apply (auto simp add:field_simps powr_diff)
by force
then show "⟦x' ∈ ball z r; 0 < dist x' z;dist x' z < r⟧
⟹ (x' - z) powr (n - m) * g x' = h x'" for x' by auto
next
define F where "F ≡ at z within ball z r"
define f' where "f' ≡ λx. (x - z) powr (n-m)"
have "f' z=0" using ‹n>m› unfolding f'_def by auto
moreover have "continuous F f'" unfolding f'_def F_def continuous_def
apply (subst Lim_ident_at)
using ‹n>m› by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
ultimately have "(f' ⤏ 0) F" unfolding F_def
by (simp add: continuous_within)
moreover have "(g ⤏ g z) F"
using holomorphic_on_imp_continuous_on[OF g_holo,unfolded continuous_on_def] ‹r>0›
unfolding F_def by auto
ultimately show " ((λw. f' w * g w) ⤏ 0) F" using tendsto_mult by fastforce
qed
moreover have "(h ⤏ h z) (at z within ball z r)"
using holomorphic_on_imp_continuous_on[OF h_holo]
by (auto simp add:continuous_on_def ‹r>0›)
ultimately have "h z=0" by (auto intro!: tendsto_unique)
thus False using ‹h z≠0› by auto
qed
moreover have False when "m>n"
proof -
have "(g ⤏ 0) (at z within ball z r)"
proof (rule Lim_transform_within[OF _ ‹r>0›, where f="λw. (w - z) powr (m - n) * h w"])
have "∀w∈ball z r -{z}. g w = (w-z) powr (m-n) * h w" using ‹m>n› asm
apply (auto simp add:field_simps powr_diff)
by force
then show "⟦x' ∈ ball z r; 0 < dist x' z;dist x' z < r⟧
⟹ (x' - z) powr (m - n) * h x' = g x'" for x' by auto
next
define F where "F ≡ at z within ball z r"
define f' where "f' ≡λx. (x - z) powr (m-n)"
have "f' z=0" using ‹m>n› unfolding f'_def by auto
moreover have "continuous F f'" unfolding f'_def F_def continuous_def
apply (subst Lim_ident_at)
using ‹m>n› by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
ultimately have "(f' ⤏ 0) F" unfolding F_def
by (simp add: continuous_within)
moreover have "(h ⤏ h z) F"
using holomorphic_on_imp_continuous_on[OF h_holo,unfolded continuous_on_def] ‹r>0›
unfolding F_def by auto
ultimately show " ((λw. f' w * h w) ⤏ 0) F" using tendsto_mult by fastforce
qed
moreover have "(g ⤏ g z) (at z within ball z r)"
using holomorphic_on_imp_continuous_on[OF g_holo]
by (auto simp add:continuous_on_def ‹r>0›)
ultimately have "g z=0" by (auto intro!: tendsto_unique)
thus False using ‹g z≠0› by auto
qed
ultimately show "n=m" by fastforce
qed
lemma holomorphic_factor_puncture:
assumes f_iso:"isolated_singularity_at f z"
and "not_essential f z"
and non_zero:"∃⇩Fw in (at z). f w≠0"
shows "∃!n::int. ∃g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z≠0
∧ (∀w∈cball z r-{z}. f w = g w * (w-z) powr n ∧ g w≠0)"
proof -
define P where "P = (λf n g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z≠0
∧ (∀w∈cball z r - {z}. f w = g w * (w-z) powr (of_int n) ∧ g w≠0))"
have imp_unique:"∃!n::int. ∃g r. P f n g r" when "∃n g r. P f n g r"
proof (rule ex_ex1I[OF that])
fix n1 n2 :: int
assume g1_asm:"∃g1 r1. P f n1 g1 r1" and g2_asm:"∃g2 r2. P f n2 g2 r2"
define fac where "fac ≡ λn g r. ∀w∈cball z r-{z}. f w = g w * (w - z) powr (of_int n) ∧ g w ≠ 0"
obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z≠0"
and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto
obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z≠0"
and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto
define r where "r ≡ min r1 r2"
have "r>0" using ‹r1>0› ‹r2>0› unfolding r_def by auto
moreover have "∀w∈ball z r-{z}. f w = g1 w * (w-z) powr n1 ∧ g1 w≠0
∧ f w = g2 w * (w - z) powr n2 ∧ g2 w≠0"
using ‹fac n1 g1 r1› ‹fac n2 g2 r2› unfolding fac_def r_def
by fastforce
ultimately show "n1=n2" using g1_holo g2_holo ‹g1 z≠0› ‹g2 z≠0›
apply (elim holomorphic_factor_unique)
by (auto simp add:r_def)
qed
have P_exist:"∃ n g r. P h n g r" when
"∃z'. (h ⤏ z') (at z)" "isolated_singularity_at h z" "∃⇩Fw in (at z). h w≠0"
for h
proof -
from that(2) obtain r where "r>0" "h analytic_on ball z r - {z}"
unfolding isolated_singularity_at_def by auto
obtain z' where "(h ⤏ z') (at z)" using ‹∃z'. (h ⤏ z') (at z)› by auto
define h' where "h'=(λx. if x=z then z' else h x)"
have "h' holomorphic_on ball z r"
apply (rule no_isolated_singularity'[of "{z}"])
subgoal by (metis LIM_equal Lim_at_imp_Lim_at_within ‹h ─z→ z'› empty_iff h'_def insert_iff)
subgoal using ‹h analytic_on ball z r - {z}› analytic_imp_holomorphic h'_def holomorphic_transform
by fastforce
by auto
have ?thesis when "z'=0"
proof -
have "h' z=0" using that unfolding h'_def by auto
moreover have "¬ h' constant_on ball z r"
using ‹∃⇩Fw in (at z). h w≠0› unfolding constant_on_def frequently_def eventually_at h'_def
apply simp
by (metis ‹0 < r› centre_in_ball dist_commute mem_ball that)
moreover note ‹h' holomorphic_on ball z r›
ultimately obtain g r1 n where "0 < n" "0 < r1" "ball z r1 ⊆ ball z r" and
g:"g holomorphic_on ball z r1"
"⋀w. w ∈ ball z r1 ⟹ h' w = (w - z) ^ n * g w"
"⋀w. w ∈ ball z r1 ⟹ g w ≠ 0"
using holomorphic_factor_zero_nonconstant[of _ "ball z r" z thesis,simplified,
OF ‹h' holomorphic_on ball z r› ‹r>0› ‹h' z=0› ‹¬ h' constant_on ball z r›]
by (auto simp add:dist_commute)
define rr where "rr=r1/2"
have "P h' n g rr"
unfolding P_def rr_def
using ‹n>0› ‹r1>0› g by (auto simp add:powr_nat)
then have "P h n g rr"
unfolding h'_def P_def by auto
then show ?thesis unfolding P_def by blast
qed
moreover have ?thesis when "z'≠0"
proof -
have "h' z≠0" using that unfolding h'_def by auto
obtain r1 where "r1>0" "cball z r1 ⊆ ball z r" "∀x∈cball z r1. h' x≠0"
proof -
have "isCont h' z" "h' z≠0"
by (auto simp add: Lim_cong_within ‹h ─z→ z'› ‹z'≠0› continuous_at h'_def)
then obtain r2 where r2:"r2>0" "∀x∈ball z r2. h' x≠0"
using continuous_at_avoid[of z h' 0 ] unfolding ball_def by auto
define r1 where "r1=min r2 r / 2"
have "0 < r1" "cball z r1 ⊆ ball z r"
using ‹r2>0› ‹r>0› unfolding r1_def by auto
moreover have "∀x∈cball z r1. h' x ≠ 0"
using r2 unfolding r1_def by simp
ultimately show ?thesis using that by auto
qed
then have "P h' 0 h' r1" using ‹h' holomorphic_on ball z r› unfolding P_def by auto
then have "P h 0 h' r1" unfolding P_def h'_def by auto
then show ?thesis unfolding P_def by blast
qed
ultimately show ?thesis by auto
qed
have ?thesis when "∃x. (f ⤏ x) (at z)"
apply (rule_tac imp_unique[unfolded P_def])
using P_exist[OF that(1) f_iso non_zero] unfolding P_def .
moreover have ?thesis when "is_pole f z"
proof (rule imp_unique[unfolded P_def])
obtain e where [simp]:"e>0" and e_holo:"f holomorphic_on ball z e - {z}" and e_nz: "∀x∈ball z e-{z}. f x≠0"
proof -
have "∀⇩F z in at z. f z ≠ 0"
using ‹is_pole f z› filterlim_at_infinity_imp_eventually_ne unfolding is_pole_def
by auto
then obtain e1 where e1:"e1>0" "∀x∈ball z e1-{z}. f x≠0"
using that eventually_at[of "λx. f x≠0" z UNIV,simplified] by (auto simp add:dist_commute)
obtain e2 where e2:"e2>0" "f holomorphic_on ball z e2 - {z}"
using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by auto
define e where "e=min e1 e2"
show ?thesis
apply (rule that[of e])
using e1 e2 unfolding e_def by auto
qed
define h where "h ≡ λx. inverse (f x)"
have "∃n g r. P h n g r"
proof -
have "h ─z→ 0"
using Lim_transform_within_open assms(2) h_def is_pole_tendsto that by fastforce
moreover have "∃⇩Fw in (at z). h w≠0"
using non_zero
apply (elim frequently_rev_mp)
unfolding h_def eventually_at by (auto intro:exI[where x=1])
moreover have "isolated_singularity_at h z"
unfolding isolated_singularity_at_def h_def
apply (rule exI[where x=e])
using e_holo e_nz ‹e>0› by (metis open_ball analytic_on_open
holomorphic_on_inverse open_delete)
ultimately show ?thesis
using P_exist[of h] by auto
qed
then obtain n g r
where "0 < r" and
g_holo:"g holomorphic_on cball z r" and "g z≠0" and
g_fac:"(∀w∈cball z r-{z}. h w = g w * (w - z) powr of_int n ∧ g w ≠ 0)"
unfolding P_def by auto
have "P f (-n) (inverse o g) r"
proof -
have "f w = inverse (g w) * (w - z) powr of_int (- n)" when "w∈cball z r - {z}" for w
using g_fac[rule_format,of w] that unfolding h_def
apply (auto simp add:powr_minus )
by (metis inverse_inverse_eq inverse_mult_distrib)
then show ?thesis
unfolding P_def comp_def
using ‹r>0› g_holo g_fac ‹g z≠0› by (auto intro:holomorphic_intros)
qed
then show "∃x g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z ≠ 0
∧ (∀w∈cball z r - {z}. f w = g w * (w - z) powr of_int x ∧ g w ≠ 0)"
unfolding P_def by blast
qed
ultimately show ?thesis using ‹not_essential f z› unfolding not_essential_def by presburger
qed
lemma not_essential_transform:
assumes "not_essential g z"
assumes "∀⇩F w in (at z). g w = f w"
shows "not_essential f z"
using assms unfolding not_essential_def
by (simp add: filterlim_cong is_pole_cong)
lemma isolated_singularity_at_transform:
assumes "isolated_singularity_at g z"
assumes "∀⇩F w in (at z). g w = f w"
shows "isolated_singularity_at f z"
proof -
obtain r1 where "r1>0" and r1:"g analytic_on ball z r1 - {z}"
using assms(1) unfolding isolated_singularity_at_def by auto
obtain r2 where "r2>0" and r2:" ∀x. x ≠ z ∧ dist x z < r2 ⟶ g x = f x"
using assms(2) unfolding eventually_at by auto
define r3 where "r3=min r1 r2"
have "r3>0" unfolding r3_def using ‹r1>0› ‹r2>0› by auto
moreover have "f analytic_on ball z r3 - {z}"
proof -
have "g holomorphic_on ball z r3 - {z}"
using r1 unfolding r3_def by (subst (asm) analytic_on_open,auto)
then have "f holomorphic_on ball z r3 - {z}"
using r2 unfolding r3_def
by (auto simp add:dist_commute elim!:holomorphic_transform)
then show ?thesis by (subst analytic_on_open,auto)
qed
ultimately show ?thesis unfolding isolated_singularity_at_def by auto
qed
lemma not_essential_powr[singularity_intros]:
assumes "LIM w (at z). f w :> (at x)"
shows "not_essential (λw. (f w) powr (of_int n)) z"
proof -
define fp where "fp=(λw. (f w) powr (of_int n))"
have ?thesis when "n>0"
proof -
have "(λw. (f w) ^ (nat n)) ─z→ x ^ nat n"
using that assms unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
then have "fp ─z→ x ^ nat n" unfolding fp_def
apply (elim Lim_transform_within[where d=1],simp)
by (metis less_le powr_0 powr_of_int that zero_less_nat_eq zero_power)
then show ?thesis unfolding not_essential_def fp_def by auto
qed
moreover have ?thesis when "n=0"
proof -
have "fp ─z→ 1 "
apply (subst tendsto_cong[where g="λ_.1"])
using that filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def by auto
then show ?thesis unfolding fp_def not_essential_def by auto
qed
moreover have ?thesis when "n<0"
proof (cases "x=0")
case True
have "LIM w (at z). inverse ((f w) ^ (nat (-n))) :> at_infinity"
apply (subst filterlim_inverse_at_iff[symmetric],simp)
apply (rule filterlim_atI)
subgoal using assms True that unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
subgoal using filterlim_at_within_not_equal[OF assms,of 0]
by (eventually_elim,insert that,auto)
done
then have "LIM w (at z). fp w :> at_infinity"
proof (elim filterlim_mono_eventually)
show "∀⇩F x in at z. inverse (f x ^ nat (- n)) = fp x"
using filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def
apply eventually_elim
using powr_of_int that by auto
qed auto
then show ?thesis unfolding fp_def not_essential_def is_pole_def by auto
next
case False
let ?xx= "inverse (x ^ (nat (-n)))"
have "(λw. inverse ((f w) ^ (nat (-n)))) ─z→?xx"
using assms False unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
then have "fp ─z→?xx"
apply (elim Lim_transform_within[where d=1],simp)
unfolding fp_def by (metis inverse_zero nat_mono_iff nat_zero_as_int neg_0_less_iff_less
not_le power_eq_0_iff powr_0 powr_of_int that)
then show ?thesis unfolding fp_def not_essential_def by auto
qed
ultimately show ?thesis by linarith
qed
lemma isolated_singularity_at_powr[singularity_intros]:
assumes "isolated_singularity_at f z" "∀⇩F w in (at z). f w≠0"
shows "isolated_singularity_at (λw. (f w) powr (of_int n)) z"
proof -
obtain r1 where "r1>0" "f analytic_on ball z r1 - {z}"
using assms(1) unfolding isolated_singularity_at_def by auto
then have r1:"f holomorphic_on ball z r1 - {z}"
using analytic_on_open[of "ball z r1-{z}" f] by blast
obtain r2 where "r2>0" and r2:"∀w. w ≠ z ∧ dist w z < r2 ⟶ f w ≠ 0"
using assms(2) unfolding eventually_at by auto
define r3 where "r3=min r1 r2"
have "(λw. (f w) powr of_int n) holomorphic_on ball z r3 - {z}"
apply (rule holomorphic_on_powr_of_int)
subgoal unfolding r3_def using r1 by auto
subgoal unfolding r3_def using r2 by (auto simp add:dist_commute)
done
moreover have "r3>0" unfolding r3_def using ‹0 < r1› ‹0 < r2› by linarith
ultimately show ?thesis unfolding isolated_singularity_at_def
apply (subst (asm) analytic_on_open[symmetric])
by auto
qed
lemma non_zero_neighbour:
assumes f_iso:"isolated_singularity_at f z"
and f_ness:"not_essential f z"
and f_nconst:"∃⇩Fw in (at z). f w≠0"
shows "∀⇩F w in (at z). f w≠0"
proof -
obtain fn fp fr where [simp]:"fp z ≠ 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
"∀w∈cball z fr - {z}. f w = fp w * (w - z) powr of_int fn ∧ fp w ≠ 0"
using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
have "f w ≠ 0" when " w ≠ z" "dist w z < fr" for w
proof -
have "f w = fp w * (w - z) powr of_int fn" "fp w ≠ 0"
using fr(2)[rule_format, of w] using that by (auto simp add:dist_commute)
moreover have "(w - z) powr of_int fn ≠0"
unfolding powr_eq_0_iff using ‹w≠z› by auto
ultimately show ?thesis by auto
qed
then show ?thesis using ‹fr>0› unfolding eventually_at by auto
qed
lemma non_zero_neighbour_pole:
assumes "is_pole f z"
shows "∀⇩F w in (at z). f w≠0"
using assms filterlim_at_infinity_imp_eventually_ne[of f "at z" 0]
unfolding is_pole_def by auto
lemma non_zero_neighbour_alt:
assumes holo: "f holomorphic_on S"
and "open S" "connected S" "z ∈ S" "β ∈ S" "f β ≠ 0"
shows "∀⇩F w in (at z). f w≠0 ∧ w∈S"
proof (cases "f z = 0")
case True
from isolated_zeros[OF holo ‹open S› ‹connected S› ‹z ∈ S› True ‹β ∈ S› ‹f β ≠ 0›]
obtain r where "0 < r" "ball z r ⊆ S" "∀w ∈ ball z r - {z}.f w ≠ 0" by metis
then show ?thesis unfolding eventually_at
apply (rule_tac x=r in exI)
by (auto simp add:dist_commute)
next
case False
obtain r1 where r1:"r1>0" "∀y. dist z y < r1 ⟶ f y ≠ 0"
using continuous_at_avoid[of z f, OF _ False] assms(2,4) continuous_on_eq_continuous_at
holo holomorphic_on_imp_continuous_on by blast
obtain r2 where r2:"r2>0" "ball z r2 ⊆ S"
using assms(2) assms(4) openE by blast
show ?thesis unfolding eventually_at
apply (rule_tac x="min r1 r2" in exI)
using r1 r2 by (auto simp add:dist_commute)
qed
lemma not_essential_times[singularity_intros]:
assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
shows "not_essential (λw. f w * g w) z"
proof -
define fg where "fg = (λw. f w * g w)"
have ?thesis when "¬ ((∃⇩Fw in (at z). f w≠0) ∧ (∃⇩Fw in (at z). g w≠0))"
proof -
have "∀⇩Fw in (at z). fg w=0"
using that[unfolded frequently_def, simplified] unfolding fg_def
by (auto elim: eventually_rev_mp)
from tendsto_cong[OF this] have "fg ─z→0" by auto
then show ?thesis unfolding not_essential_def fg_def by auto
qed
moreover have ?thesis when f_nconst:"∃⇩Fw in (at z). f w≠0" and g_nconst:"∃⇩Fw in (at z). g w≠0"
proof -
obtain fn fp fr where [simp]:"fp z ≠ 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
"∀w∈cball z fr - {z}. f w = fp w * (w - z) powr of_int fn ∧ fp w ≠ 0"
using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
obtain gn gp gr where [simp]:"gp z ≠ 0" and "gr > 0"
and gr: "gp holomorphic_on cball z gr"
"∀w∈cball z gr - {z}. g w = gp w * (w - z) powr of_int gn ∧ gp w ≠ 0"
using holomorphic_factor_puncture[OF g_iso g_ness g_nconst,THEN ex1_implies_ex] by auto
define r1 where "r1=(min fr gr)"
have "r1>0" unfolding r1_def using ‹fr>0› ‹gr>0› by auto
have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w≠0"
when "w∈ball z r1 - {z}" for w
proof -
have "f w = fp w * (w - z) powr of_int fn" "fp w≠0"
using fr(2)[rule_format,of w] that unfolding r1_def by auto
moreover have "g w = gp w * (w - z) powr of_int gn" "gp w ≠ 0"
using gr(2)[rule_format, of w] that unfolding r1_def by auto
ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w≠0"
unfolding fg_def by (auto simp add:powr_add)
qed
have [intro]: "fp ─z→fp z" "gp ─z→gp z"
using fr(1) ‹fr>0› gr(1) ‹gr>0›
by (meson open_ball ball_subset_cball centre_in_ball
continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on
holomorphic_on_subset)+
have ?thesis when "fn+gn>0"
proof -
have "(λw. (fp w * gp w) * (w - z) ^ (nat (fn+gn))) ─z→0"
using that by (auto intro!:tendsto_eq_intros)
then have "fg ─z→ 0"
apply (elim Lim_transform_within[OF _ ‹r1>0›])
by (metis (no_types, opaque_lifting) Diff_iff cball_trivial dist_commute dist_self
eq_iff_diff_eq_0 fg_times less_le linorder_not_le mem_ball mem_cball powr_of_int
that)
then show ?thesis unfolding not_essential_def fg_def by auto
qed
moreover have ?thesis when "fn+gn=0"
proof -
have "(λw. fp w * gp w) ─z→fp z*gp z"
using that by (auto intro!:tendsto_eq_intros)
then have "fg ─z→ fp z*gp z"
apply (elim Lim_transform_within[OF _ ‹r1>0›])
apply (subst fg_times)
by (auto simp add:dist_commute that)
then show ?thesis unfolding not_essential_def fg_def by auto
qed
moreover have ?thesis when "fn+gn<0"
proof -
have "LIM w (at z). fp w * gp w / (w-z)^nat (-(fn+gn)) :> at_infinity"
apply (rule filterlim_divide_at_infinity)
apply (insert that, auto intro!:tendsto_eq_intros filterlim_atI)
using eventually_at_topological by blast
then have "is_pole fg z" unfolding is_pole_def
apply (elim filterlim_transform_within[OF _ _ ‹r1>0›],simp)
apply (subst fg_times,simp add:dist_commute)
apply (subst powr_of_int)
using that by (auto simp add:field_split_simps)
then show ?thesis unfolding not_essential_def fg_def by auto
qed
ultimately show ?thesis unfolding not_essential_def fg_def by fastforce
qed
ultimately show ?thesis by auto
qed
lemma not_essential_inverse[singularity_intros]:
assumes f_ness:"not_essential f z"
assumes f_iso:"isolated_singularity_at f z"
shows "not_essential (λw. inverse (f w)) z"
proof -
define vf where "vf = (λw. inverse (f w))"
have ?thesis when "¬(∃⇩Fw in (at z). f w≠0)"
proof -
have "∀⇩Fw in (at z). f w=0"
using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
then have "∀⇩Fw in (at z). vf w=0"
unfolding vf_def by auto
from tendsto_cong[OF this] have "vf ─z→0" unfolding vf_def by auto
then show ?thesis unfolding not_essential_def vf_def by auto
qed
moreover have ?thesis when "is_pole f z"
proof -
have "vf ─z→0"
using that filterlim_at filterlim_inverse_at_iff unfolding is_pole_def vf_def by blast
then show ?thesis unfolding not_essential_def vf_def by auto
qed
moreover have ?thesis when "∃x. f─z→x " and f_nconst:"∃⇩Fw in (at z). f w≠0"
proof -
from that obtain fz where fz:"f─z→fz" by auto
have ?thesis when "fz=0"
proof -
have "(λw. inverse (vf w)) ─z→0"
using fz that unfolding vf_def by auto
moreover have "∀⇩F w in at z. inverse (vf w) ≠ 0"
using non_zero_neighbour[OF f_iso f_ness f_nconst]
unfolding vf_def by auto
ultimately have "is_pole vf z"
using filterlim_inverse_at_iff[of vf "at z"] unfolding filterlim_at is_pole_def by auto
then show ?thesis unfolding not_essential_def vf_def by auto
qed
moreover have ?thesis when "fz≠0"
proof -
have "vf ─z→inverse fz"
using fz that unfolding vf_def by (auto intro:tendsto_eq_intros)
then show ?thesis unfolding not_essential_def vf_def by auto
qed
ultimately show ?thesis by auto
qed
ultimately show ?thesis using f_ness unfolding not_essential_def by auto
qed
lemma isolated_singularity_at_inverse[singularity_intros]:
assumes f_iso:"isolated_singularity_at f z"
and f_ness:"not_essential f z"
shows "isolated_singularity_at (λw. inverse (f w)) z"
proof -
define vf where "vf = (λw. inverse (f w))"
have ?thesis when "¬(∃⇩Fw in (at z). f w≠0)"
proof -
have "∀⇩Fw in (at z). f w=0"
using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
then have "∀⇩Fw in (at z). vf w=0"
unfolding vf_def by auto
then obtain d1 where "d1>0" and d1:"∀x. x ≠ z ∧ dist x z < d1 ⟶ vf x = 0"
unfolding eventually_at by auto
then have "vf holomorphic_on ball z d1-{z}"
apply (rule_tac holomorphic_transform[of "λ_. 0"])
by (auto simp add:dist_commute)
then have "vf analytic_on ball z d1 - {z}"
by (simp add: analytic_on_open open_delete)
then show ?thesis using ‹d1>0› unfolding isolated_singularity_at_def vf_def by auto
qed
moreover have ?thesis when f_nconst:"∃⇩Fw in (at z). f w≠0"
proof -
have "∀⇩F w in at z. f w ≠ 0" using non_zero_neighbour[OF f_iso f_ness f_nconst] .
then obtain d1 where d1:"d1>0" "∀x. x ≠ z ∧ dist x z < d1 ⟶ f x ≠ 0"
unfolding eventually_at by auto
obtain d2 where "d2>0" and d2:"f analytic_on ball z d2 - {z}"
using f_iso unfolding isolated_singularity_at_def by auto
define d3 where "d3=min d1 d2"
have "d3>0" unfolding d3_def using ‹d1>0› ‹d2>0› by auto
moreover have "vf analytic_on ball z d3 - {z}"
unfolding vf_def
apply (rule analytic_on_inverse)
subgoal using d2 unfolding d3_def by (elim analytic_on_subset) auto
subgoal for w using d1 unfolding d3_def by (auto simp add:dist_commute)
done
ultimately show ?thesis unfolding isolated_singularity_at_def vf_def by auto
qed
ultimately show ?thesis by auto
qed
lemma not_essential_divide[singularity_intros]:
assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
shows "not_essential (λw. f w / g w) z"
proof -
have "not_essential (λw. f w * inverse (g w)) z"
apply (rule not_essential_times[where g="λw. inverse (g w)"])
using assms by (auto intro: isolated_singularity_at_inverse not_essential_inverse)
then show ?thesis by (simp add:field_simps)
qed
lemma
assumes f_iso:"isolated_singularity_at f z"
and g_iso:"isolated_singularity_at g z"
shows isolated_singularity_at_times[singularity_intros]:
"isolated_singularity_at (λw. f w * g w) z" and
isolated_singularity_at_add[singularity_intros]:
"isolated_singularity_at (λw. f w + g w) z"
proof -
obtain d1 d2 where "d1>0" "d2>0"
and d1:"f analytic_on ball z d1 - {z}" and d2:"g analytic_on ball z d2 - {z}"
using f_iso g_iso unfolding isolated_singularity_at_def by auto
define d3 where "d3=min d1 d2"
have "d3>0" unfolding d3_def using ‹d1>0› ‹d2>0› by auto
have "(λw. f w * g w) analytic_on ball z d3 - {z}"
apply (rule analytic_on_mult)
using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset)
then show "isolated_singularity_at (λw. f w * g w) z"
using ‹d3>0› unfolding isolated_singularity_at_def by auto
have "(λw. f w + g w) analytic_on ball z d3 - {z}"
apply (rule analytic_on_add)
using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset)
then show "isolated_singularity_at (λw. f w + g w) z"
using ‹d3>0› unfolding isolated_singularity_at_def by auto
qed
lemma isolated_singularity_at_uminus[singularity_intros]:
assumes f_iso:"isolated_singularity_at f z"
shows "isolated_singularity_at (λw. - f w) z"
using assms unfolding isolated_singularity_at_def using analytic_on_neg by blast
lemma isolated_singularity_at_id[singularity_intros]:
"isolated_singularity_at (λw. w) z"
unfolding isolated_singularity_at_def by (simp add: gt_ex)
lemma isolated_singularity_at_minus[singularity_intros]:
assumes f_iso:"isolated_singularity_at f z"
and g_iso:"isolated_singularity_at g z"
shows "isolated_singularity_at (λw. f w - g w) z"
using isolated_singularity_at_uminus[THEN isolated_singularity_at_add[OF f_iso,of "λw. - g w"]
,OF g_iso] by simp
lemma isolated_singularity_at_divide[singularity_intros]:
assumes f_iso:"isolated_singularity_at f z"
and g_iso:"isolated_singularity_at g z"
and g_ness:"not_essential g z"
shows "isolated_singularity_at (λw. f w / g w) z"
using isolated_singularity_at_inverse[THEN isolated_singularity_at_times[OF f_iso,
of "λw. inverse (g w)"],OF g_iso g_ness] by (simp add:field_simps)
lemma isolated_singularity_at_const[singularity_intros]:
"isolated_singularity_at (λw. c) z"
unfolding isolated_singularity_at_def by (simp add: gt_ex)
lemma isolated_singularity_at_holomorphic:
assumes "f holomorphic_on s-{z}" "open s" "z∈s"
shows "isolated_singularity_at f z"
using assms unfolding isolated_singularity_at_def
by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
subsubsection ‹The order of non-essential singularities (i.e. removable singularities or poles)›
definition zorder :: "(complex ⇒ complex) ⇒ complex ⇒ int" where
"zorder f z = (THE n. (∃h r. r>0 ∧ h holomorphic_on cball z r ∧ h z≠0
∧ (∀w∈cball z r - {z}. f w = h w * (w-z) powr (of_int n)
∧ h w ≠0)))"
definition zor_poly
::"[complex ⇒ complex, complex] ⇒ complex ⇒ complex" where
"zor_poly f z = (SOME h. ∃r. r > 0 ∧ h holomorphic_on cball z r ∧ h z ≠ 0
∧ (∀w∈cball z r - {z}. f w = h w * (w - z) powr (zorder f z)
∧ h w ≠0))"
lemma zorder_exist:
fixes f::"complex ⇒ complex" and z::complex
defines "n≡zorder f z" and "g≡zor_poly f z"
assumes f_iso:"isolated_singularity_at f z"
and f_ness:"not_essential f z"
and f_nconst:"∃⇩Fw in (at z). f w≠0"
shows "g z≠0 ∧ (∃r. r>0 ∧ g holomorphic_on cball z r
∧ (∀w∈cball z r - {z}. f w = g w * (w-z) powr n ∧ g w ≠0))"
proof -
define P where "P = (λn g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z≠0
∧ (∀w∈cball z r - {z}. f w = g w * (w-z) powr (of_int n) ∧ g w≠0))"
have "∃!n. ∃g r. P n g r"
using holomorphic_factor_puncture[OF assms(3-)] unfolding P_def by auto
then have "∃g r. P n g r"
unfolding n_def P_def zorder_def
by (drule_tac theI',argo)
then have "∃r. P n g r"
unfolding P_def zor_poly_def g_def n_def
by (drule_tac someI_ex,argo)
then obtain r1 where "P n g r1" by auto
then show ?thesis unfolding P_def by auto
qed
lemma
fixes f::"complex ⇒ complex" and z::complex
assumes f_iso:"isolated_singularity_at f z"
and f_ness:"not_essential f z"
and f_nconst:"∃⇩Fw in (at z). f w≠0"
shows zorder_inverse: "zorder (λw. inverse (f w)) z = - zorder f z"
and zor_poly_inverse: "∀⇩Fw in (at z). zor_poly (λw. inverse (f w)) z w
= inverse (zor_poly f z w)"
proof -
define vf where "vf = (λw. inverse (f w))"
define fn vfn where
"fn = zorder f z" and "vfn = zorder vf z"
define fp vfp where
"fp = zor_poly f z" and "vfp = zor_poly vf z"
obtain fr where [simp]:"fp z ≠ 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
"∀w∈cball z fr - {z}. f w = fp w * (w - z) powr of_int fn ∧ fp w ≠ 0"
using zorder_exist[OF f_iso f_ness f_nconst,folded fn_def fp_def]
by auto
have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))"
and fr_nz: "inverse (fp w)≠0"
when "w∈ball z fr - {z}" for w
proof -
have "f w = fp w * (w - z) powr of_int fn" "fp w≠0"
using fr(2)[rule_format,of w] that by auto
then show "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))" "inverse (fp w)≠0"
unfolding vf_def by (auto simp add:powr_minus)
qed
obtain vfr where [simp]:"vfp z ≠ 0" and "vfr>0" and vfr:"vfp holomorphic_on cball z vfr"
"(∀w∈cball z vfr - {z}. vf w = vfp w * (w - z) powr of_int vfn ∧ vfp w ≠ 0)"
proof -
have "isolated_singularity_at vf z"
using isolated_singularity_at_inverse[OF f_iso f_ness] unfolding vf_def .
moreover have "not_essential vf z"
using not_essential_inverse[OF f_ness f_iso] unfolding vf_def .
moreover have "∃⇩F w in at z. vf w ≠ 0"
using f_nconst unfolding vf_def by (auto elim:frequently_elim1)
ultimately show ?thesis using zorder_exist[of vf z, folded vfn_def vfp_def] that by auto
qed
define r1 where "r1 = min fr vfr"
have "r1>0" using ‹fr>0› ‹vfr>0› unfolding r1_def by simp
show "vfn = - fn"
apply (rule holomorphic_factor_unique[of r1 vfp z "λw. inverse (fp w)" vf])
subgoal using ‹r1>0› by simp
subgoal by simp
subgoal by simp
subgoal
proof (rule ballI)
fix w assume "w ∈ ball z r1 - {z}"
then have "w ∈ ball z fr - {z}" "w ∈ cball z vfr - {z}" unfolding r1_def by auto
from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)]
show "vf w = vfp w * (w - z) powr of_int vfn ∧ vfp w ≠ 0
∧ vf w = inverse (fp w) * (w - z) powr of_int (- fn) ∧ inverse (fp w) ≠ 0" by auto
qed
subgoal using vfr(1) unfolding r1_def by (auto intro!:holomorphic_intros)
subgoal using fr unfolding r1_def by (auto intro!:holomorphic_intros)
done
have "vfp w = inverse (fp w)" when "w∈ball z r1-{z}" for w
proof -
have "w ∈ ball z fr - {z}" "w ∈ cball z vfr - {z}" "w≠z" using that unfolding r1_def by auto
from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)] ‹vfn = - fn› ‹w≠z›
show ?thesis by auto
qed
then show "∀⇩Fw in (at z). vfp w = inverse (fp w)"
unfolding eventually_at using ‹r1>0›
apply (rule_tac x=r1 in exI)
by (auto simp add:dist_commute)
qed
lemma
fixes f g::"complex ⇒ complex" and z::complex
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
and f_ness:"not_essential f z" and g_ness:"not_essential g z"
and fg_nconst: "∃⇩Fw in (at z). f w * g w≠ 0"
shows zorder_times:"zorder (λw. f w * g w) z = zorder f z + zorder g z" and
zor_poly_times:"∀⇩Fw in (at z). zor_poly (λw. f w * g w) z w
= zor_poly f z w *zor_poly g z w"
proof -
define fg where "fg = (λw. f w * g w)"
define fn gn fgn where
"fn = zorder f z" and "gn = zorder g z" and "fgn = zorder fg z"
define fp gp fgp where
"fp = zor_poly f z" and "gp = zor_poly g z" and "fgp = zor_poly fg z"
have f_nconst:"∃⇩Fw in (at z). f w ≠ 0" and g_nconst:"∃⇩Fw in (at z).g w≠ 0"
using fg_nconst by (auto elim!:frequently_elim1)
obtain fr where [simp]:"fp z ≠ 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
"∀w∈cball z fr - {z}. f w = fp w * (w - z) powr of_int fn ∧ fp w ≠ 0"
using zorder_exist[OF f_iso f_ness f_nconst,folded fp_def fn_def] by auto
obtain gr where [simp]:"gp z ≠ 0" and "gr > 0"
and gr: "gp holomorphic_on cball z gr"
"∀w∈cball z gr - {z}. g w = gp w * (w - z) powr of_int gn ∧ gp w ≠ 0"
using zorder_exist[OF g_iso g_ness g_nconst,folded gn_def gp_def] by auto
define r1 where "r1=min fr gr"
have "r1>0" unfolding r1_def using ‹fr>0› ‹gr>0› by auto
have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w≠0"
when "w∈ball z r1 - {z}" for w
proof -
have "f w = fp w * (w - z) powr of_int fn" "fp w≠0"
using fr(2)[rule_format,of w] that unfolding r1_def by auto
moreover have "g w = gp w * (w - z) powr of_int gn" "gp w ≠ 0"
using gr(2)[rule_format, of w] that unfolding r1_def by auto
ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w≠0"
unfolding fg_def by (auto simp add:powr_add)
qed
obtain fgr where [simp]:"fgp z ≠ 0" and "fgr > 0"
and fgr: "fgp holomorphic_on cball z fgr"
"∀w∈cball z fgr - {z}. fg w = fgp w * (w - z) powr of_int fgn ∧ fgp w ≠ 0"
proof -
have "fgp z ≠ 0 ∧ (∃r>0. fgp holomorphic_on cball z r
∧ (∀w∈cball z r - {z}. fg w = fgp w * (w - z) powr of_int fgn ∧ fgp w ≠ 0))"
apply (rule zorder_exist[of fg z, folded fgn_def fgp_def])
subgoal unfolding fg_def using isolated_singularity_at_times[OF f_iso g_iso] .
subgoal unfolding fg_def using not_essential_times[OF f_ness g_ness f_iso g_iso] .
subgoal unfolding fg_def using fg_nconst .
done
then show ?thesis using that by blast
qed
define r2 where "r2 = min fgr r1"
have "r2>0" using ‹r1>0› ‹fgr>0› unfolding r2_def by simp
show "fgn = fn + gn "
apply (rule holomorphic_factor_unique[of r2 fgp z "λw. fp w * gp w" fg])
subgoal using ‹r2>0› by simp
subgoal by simp
subgoal by simp
subgoal
proof (rule ballI)
fix w assume "w ∈ ball z r2 - {z}"
then have "w ∈ ball z r1 - {z}" "w ∈ cball z fgr - {z}" unfolding r2_def by auto
from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)]
show "fg w = fgp w * (w - z) powr of_int fgn ∧ fgp w ≠ 0
∧ fg w = fp w * gp w * (w - z) powr of_int (fn + gn) ∧ fp w * gp w ≠ 0" by auto
qed
subgoal using fgr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
subgoal using fr(1) gr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
done
have "fgp w = fp w *gp w" when "w∈ball z r2-{z}" for w
proof -
have "w ∈ ball z r1 - {z}" "w ∈ cball z fgr - {z}" "w≠z" using that unfolding r2_def by auto
from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] ‹fgn = fn + gn› ‹w≠z›
show ?thesis by auto
qed
then show "∀⇩Fw in (at z). fgp w = fp w * gp w"
using ‹r2>0› unfolding eventually_at by (auto simp add:dist_commute)
qed
lemma
fixes f g::"complex ⇒ complex" and z::complex
assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
and f_ness:"not_essential f z" and g_ness:"not_essential g z"
and fg_nconst: "∃⇩Fw in (at z). f w * g w≠ 0"
shows zorder_divide:"zorder (λw. f w / g w) z = zorder f z - zorder g z" and
zor_poly_divide:"∀⇩Fw in (at z). zor_poly (λw. f w / g w) z w
= zor_poly f z w / zor_poly g z w"
proof -
have f_nconst:"∃⇩Fw in (at z). f w ≠ 0" and g_nconst:"∃⇩Fw in (at z).g w≠ 0"
using fg_nconst by (auto elim!:frequently_elim1)
define vg where "vg=(λw. inverse (g w))"
have "zorder (λw. f w * vg w) z = zorder f z + zorder vg z"
apply (rule zorder_times[OF f_iso _ f_ness,of vg])
subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] .
subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] .
subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1)
done
then show "zorder (λw. f w / g w) z = zorder f z - zorder g z"
using zorder_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
by (auto simp add:field_simps)
have "∀⇩F w in at z. zor_poly (λw. f w * vg w) z w = zor_poly f z w * zor_poly vg z w"
apply (rule zor_poly_times[OF f_iso _ f_ness,of vg])
subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] .
subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] .
subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1)
done
then show "∀⇩Fw in (at z). zor_poly (λw. f w / g w) z w = zor_poly f z w / zor_poly g z w"
using zor_poly_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
apply eventually_elim
by (auto simp add:field_simps)
qed
lemma zorder_exist_zero:
fixes f::"complex ⇒ complex" and z::complex
defines "n≡zorder f z" and "g≡zor_poly f z"
assumes holo: "f holomorphic_on s" and
"open s" "connected s" "z∈s"
and non_const: "∃w∈s. f w ≠ 0"
shows "(if f z=0 then n > 0 else n=0) ∧ (∃r. r>0 ∧ cball z r ⊆ s ∧ g holomorphic_on cball z r
∧ (∀w∈cball z r. f w = g w * (w-z) ^ nat n ∧ g w ≠0))"
proof -
obtain r where "g z ≠ 0" and r: "r>0" "cball z r ⊆ s" "g holomorphic_on cball z r"
"(∀w∈cball z r - {z}. f w = g w * (w - z) powr of_int n ∧ g w ≠ 0)"
proof -
have "g z ≠ 0 ∧ (∃r>0. g holomorphic_on cball z r
∧ (∀w∈cball z r - {z}. f w = g w * (w - z) powr of_int n ∧ g w ≠ 0))"
proof (rule zorder_exist[of f z,folded g_def n_def])
show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
using holo assms(4,6)
by (meson Diff_subset open_ball analytic_on_holomorphic holomorphic_on_subset openE)
show "not_essential f z" unfolding not_essential_def
using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
by fastforce
have "∀⇩F w in at z. f w ≠ 0 ∧ w∈s"
proof -
obtain w where "w∈s" "f w≠0" using non_const by auto
then show ?thesis
by (rule non_zero_neighbour_alt[OF holo ‹open s› ‹connected s› ‹z∈s›])
qed
then show "∃⇩F w in at z. f w ≠ 0"
apply (elim eventually_frequentlyE)
by auto
qed
then obtain r1 where "g z ≠ 0" "r1>0" and r1:"g holomorphic_on cball z r1"
"(∀w∈cball z r1 - {z}. f w = g w * (w - z) powr of_int n ∧ g w ≠ 0)"
by auto
obtain r2 where r2: "r2>0" "cball z r2 ⊆ s"
using assms(4,6) open_contains_cball_eq by blast
define r3 where "r3=min r1 r2"
have "r3>0" "cball z r3 ⊆ s" using ‹r1>0› r2 unfolding r3_def by auto
moreover have "g holomorphic_on cball z r3"
using r1(1) unfolding r3_def by auto
moreover have "(∀w∈cball z r3 - {z}. f w = g w * (w - z) powr of_int n ∧ g w ≠ 0)"
using r1(2) unfolding r3_def by auto
ultimately show ?thesis using that[of r3] ‹g z≠0› by auto
qed
have if_0:"if f z=0 then n > 0 else n=0"
proof -
have "f─ z → f z"
by (metis assms(4,6,7) at_within_open continuous_on holo holomorphic_on_imp_continuous_on)
then have "(λw. g w * (w - z) powr of_int n) ─z→ f z"
apply (elim Lim_transform_within_open[where s="ball z r"])
using r by auto
moreover have "g ─z→g z"
by (metis (mono_tags, lifting) open_ball at_within_open_subset
ball_subset_cball centre_in_ball continuous_on holomorphic_on_imp_continuous_on r(1,3) subsetCE)
ultimately have "(λw. (g w * (w - z) powr of_int n) / g w) ─z→ f z/g z"
apply (rule_tac tendsto_divide)
using ‹g z≠0› by auto
then have powr_tendsto:"(λw. (w - z) powr of_int n) ─z→ f z/g z"
apply (elim Lim_transform_within_open[where s="ball z r"])
using r by auto
have ?thesis when "n≥0" "f z=0"
proof -
have "(λw. (w - z) ^ nat n) ─z→ f z/g z"
using powr_tendsto
apply (elim Lim_transform_within[where d=r])
by (auto simp add: powr_of_int ‹n≥0› ‹r>0›)
then have *:"(λw. (w - z) ^ nat n) ─z→ 0" using ‹f z=0› by simp
moreover have False when "n=0"
proof -
have "(λw. (w - z) ^ nat n) ─z→ 1"
using ‹n=0› by auto
then show False using * using LIM_unique zero_neq_one by blast
qed
ultimately show ?thesis using that by fastforce
qed
moreover have ?thesis when "n≥0" "f z≠0"
proof -
have False when "n>0"
proof -
have "(λw. (w - z) ^ nat n) ─z→ f z/g z"
using powr_tendsto
apply (elim Lim_transform_within[where d=r])
by (auto simp add: powr_of_int ‹n≥0› ‹r>0›)
moreover have "(λw. (w - z) ^ nat n) ─z→ 0"
using ‹n>0› by (auto intro!:tendsto_eq_intros)
ultimately show False using ‹f z≠0› ‹g z≠0› using LIM_unique divide_eq_0_iff by blast
qed
then show ?thesis using that by force
qed
moreover have False when "n<0"
proof -
have "(λw. inverse ((w - z) ^ nat (- n))) ─z→ f z/g z"
"(λw.((w - z) ^ nat (- n))) ─z→ 0"
subgoal using powr_tendsto powr_of_int that
by (elim Lim_transform_within_open[where s=UNIV],auto)
subgoal using that by (auto intro!:tendsto_eq_intros)
done
from tendsto_mult[OF this,simplified]
have "(λx. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) ─z→ 0" .
then have "(λx. 1::complex) ─z→ 0"
by (elim Lim_transform_within_open[where s=UNIV],auto)
then show False using LIM_const_eq by fastforce
qed
ultimately show ?thesis by fastforce
qed
moreover have "f w = g w * (w-z) ^ nat n ∧ g w ≠0" when "w∈cball z r" for w
proof (cases "w=z")
case True
then have "f ─z→f w"
using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on by fastforce
then have "(λw. g w * (w-z) ^ nat n) ─z→f w"
proof (elim Lim_transform_within[OF _ ‹r>0›])
fix x assume "0 < dist x z" "dist x z < r"
then have "x ∈ cball z r - {z}" "x≠z"
unfolding cball_def by (auto simp add: dist_commute)
then have "f x = g x * (x - z) powr of_int n"
using r(4)[rule_format,of x] by simp
also have "... = g x * (x - z) ^ nat n"
apply (subst powr_of_int)
using if_0 ‹x≠z› by (auto split:if_splits)
finally show "f x = g x * (x - z) ^ nat n" .
qed
moreover have "(λw. g w * (w-z) ^ nat n) ─z→ g w * (w-z) ^ nat n"
using True apply (auto intro!:tendsto_eq_intros)
by (metis open_ball at_within_open_subset ball_subset_cball centre_in_ball
continuous_on holomorphic_on_imp_continuous_on r(1) r(3) that)
ultimately have "f w = g w * (w-z) ^ nat n" using LIM_unique by blast
then show ?thesis using ‹g z≠0› True by auto
next
case False
then have "f w = g w * (w - z) powr of_int n ∧ g w ≠ 0"
using r(4) that by auto
then show ?thesis using False if_0 powr_of_int by (auto split:if_splits)
qed
ultimately show ?thesis using r by auto
qed
lemma zorder_exist_pole:
fixes f::"complex ⇒ complex" and z::complex
defines "n≡zorder f z" and "g≡zor_poly f z"
assumes holo: "f holomorphic_on s-{z}" and
"open s" "z∈s"
and "is_pole f z"
shows "n < 0 ∧ g z≠0 ∧ (∃r. r>0 ∧ cball z r ⊆ s ∧ g holomorphic_on cball z r
∧ (∀w∈cball z r - {z}. f w = g w / (w-z) ^ nat (- n) ∧ g w ≠0))"
proof -
obtain r where "g z ≠ 0" and r: "r>0" "cball z r ⊆ s" "g holomorphic_on cball z r"
"(∀w∈cball z r - {z}. f w = g w * (w - z) powr of_int n ∧ g w ≠ 0)"
proof -
have "g z ≠ 0 ∧ (∃r>0. g holomorphic_on cball z r
∧ (∀w∈cball z r - {z}. f w = g w * (w - z) powr of_int n ∧ g w ≠ 0))"
proof (rule zorder_exist[of f z,folded g_def n_def])
show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
using holo assms(4,5)
by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
show "not_essential f z" unfolding not_essential_def
using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
by fastforce
from non_zero_neighbour_pole[OF ‹is_pole f z›] show "∃⇩F w in at z. f w ≠ 0"
apply (elim eventually_frequentlyE)
by auto
qed
then obtain r1 where "g z ≠ 0" "r1>0" and r1:"g holomorphic_on cball z r1"
"(∀w∈cball z r1 - {z}. f w = g w * (w - z) powr of_int n ∧ g w ≠ 0)"
by auto
obtain r2 where r2: "r2>0" "cball z r2 ⊆ s"
using assms(4,5) open_contains_cball_eq by metis
define r3 where "r3=min r1 r2"
have "r3>0" "cball z r3 ⊆ s" using ‹r1>0› r2 unfolding r3_def by auto
moreover have "g holomorphic_on cball z r3"
using r1(1) unfolding r3_def by auto
moreover have "(∀w∈cball z r3 - {z}. f w = g w * (w - z) powr of_int n ∧ g w ≠ 0)"
using r1(2) unfolding r3_def by auto
ultimately show ?thesis using that[of r3] ‹g z≠0› by auto
qed
have "n<0"
proof (rule ccontr)
assume " ¬ n < 0"
define c where "c=(if n=0 then g z else 0)"
have [simp]:"g ─z→ g z"
by (metis open_ball at_within_open ball_subset_cball centre_in_ball
continuous_on holomorphic_on_imp_continuous_on holomorphic_on_subset r(1) r(3) )
have "∀⇩F x in at z. f x = g x * (x - z) ^ nat n"
unfolding eventually_at_topological
apply (rule_tac exI[where x="ball z r"])
using r powr_of_int ‹¬ n < 0› by auto
moreover have "(λx. g x * (x - z) ^ nat n) ─z→c"
proof (cases "n=0")
case True
then show ?thesis unfolding c_def by simp
next
case False
then have "(λx. (x - z) ^ nat n) ─z→ 0" using ‹¬ n < 0›
by (auto intro!:tendsto_eq_intros)
from tendsto_mult[OF _ this,of g "g z",simplified]
show ?thesis unfolding c_def using False by simp
qed
ultimately have "f ─z→c" using tendsto_cong by fast
then show False using ‹is_pole f z› at_neq_bot not_tendsto_and_filterlim_at_infinity
unfolding is_pole_def by blast
qed
moreover have "∀w∈cball z r - {z}. f w = g w / (w-z) ^ nat (- n) ∧ g w ≠0"
using r(4) ‹n<0› powr_of_int
by (metis Diff_iff divide_inverse eq_iff_diff_eq_0 insert_iff linorder_not_le)
ultimately show ?thesis using r(1-3) ‹g z≠0› by auto
qed
lemma zorder_eqI:
assumes "open s" "z ∈ s" "g holomorphic_on s" "g z ≠ 0"
assumes fg_eq:"⋀w. ⟦w ∈ s;w≠z⟧ ⟹ f w = g w * (w - z) powr n"
shows "zorder f z = n"
proof -
have "continuous_on s g" by (rule holomorphic_on_imp_continuous_on) fact
moreover have "open (-{0::complex})" by auto
ultimately have "open ((g -` (-{0})) ∩ s)"
unfolding continuous_on_open_vimage[OF ‹open s›] by blast
moreover from assms have "z ∈ (g -` (-{0})) ∩ s" by auto
ultimately obtain r where r: "r > 0" "cball z r ⊆ s ∩ (g -` (-{0}))"
unfolding open_contains_cball by blast
let ?gg= "(λw. g w * (w - z) powr n)"
define P where "P = (λn g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z≠0
∧ (∀w∈cball z r - {z}. f w = g w * (w-z) powr (of_int n) ∧ g w≠0))"
have "P n g r"
unfolding P_def using r assms(3,4,5) by auto
then have "∃g r. P n g r" by auto
moreover have unique: "∃!n. ∃g r. P n g r" unfolding P_def
proof (rule holomorphic_factor_puncture)
have "ball z r-{z} ⊆ s" using r using ball_subset_cball by blast
then have "?gg holomorphic_on ball z r-{z}"
using ‹g holomorphic_on s› r by (auto intro!: holomorphic_intros)
then have "f holomorphic_on ball z r - {z}"
apply (elim holomorphic_transform)
using fg_eq ‹ball z r-{z} ⊆ s› by auto
then show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
using analytic_on_open open_delete r(1) by blast
next
have "not_essential ?gg z"
proof (intro singularity_intros)
show "not_essential g z"
by (meson ‹continuous_on s g› assms(1) assms(2) continuous_on_eq_continuous_at
isCont_def not_essential_def)
show " ∀⇩F w in at z. w - z ≠ 0" by (simp add: eventually_at_filter)
then show "LIM w at z. w - z :> at 0"
unfolding filterlim_at by (auto intro:tendsto_eq_intros)
show "isolated_singularity_at g z"
by (meson Diff_subset open_ball analytic_on_holomorphic
assms(1,2,3) holomorphic_on_subset isolated_singularity_at_def openE)
qed
then show "not_essential f z"
apply (elim not_essential_transform)
unfolding eventually_at using assms(1,2) assms(5)[symmetric]
by (metis dist_commute mem_ball openE subsetCE)
show "∃⇩F w in at z. f w ≠ 0" unfolding frequently_at
proof (rule,rule)
fix d::real assume "0 < d"
define z' where "z'=z+min d r / 2"
have "z' ≠ z" " dist z' z < d "
unfolding z'_def using ‹d>0› ‹r>0›
by (auto simp add:dist_norm)
moreover have "f z' ≠ 0"
proof (subst fg_eq[OF _ ‹z'≠z›])
have "z' ∈ cball z r" unfolding z'_def using ‹r>0› ‹d>0› by (auto simp add:dist_norm)
then show " z' ∈ s" using r(2) by blast
show "g z' * (z' - z) powr of_int n ≠ 0"
using P_def ‹P n g r› ‹z' ∈ cball z r› calculation(1) by auto
qed
ultimately show "∃x∈UNIV. x ≠ z ∧ dist x z < d ∧ f x ≠ 0" by auto
qed
qed
ultimately have "(THE n. ∃g r. P n g r) = n"
by (rule_tac the1_equality)
then show ?thesis unfolding zorder_def P_def by blast
qed
lemma simple_zeroI:
assumes "open s" "z ∈ s" "g holomorphic_on s" "g z ≠ 0"
assumes "⋀w. w ∈ s ⟹ f w = g w * (w - z)"
shows "zorder f z = 1"
using assms(1-4) by (rule zorder_eqI) (use assms(5) in auto)
lemma higher_deriv_power:
shows "(deriv ^^ j) (λw. (w - z) ^ n) w =
pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)"
proof (induction j arbitrary: w)
case 0
thus ?case by auto
next
case (Suc j w)
have "(deriv ^^ Suc j) (λw. (w - z) ^ n) w = deriv ((deriv ^^ j) (λw. (w - z) ^ n)) w"
by simp
also have "(deriv ^^ j) (λw. (w - z) ^ n) =
(λw. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))"
using Suc by (intro Suc.IH ext)
also {
have "(… has_field_derivative of_nat (n - j) *
pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)"
using Suc.prems by (auto intro!: derivative_eq_intros)
also have "of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j =
pochhammer (of_nat (Suc n - Suc j)) (Suc j)"
by (cases "Suc j ≤ n", subst pochhammer_rec)
(insert Suc.prems, simp_all add: algebra_simps Suc_diff_le pochhammer_0_left)
finally have "deriv (λw. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w =
… * (w - z) ^ (n - Suc j)"
by (rule DERIV_imp_deriv)
}
finally show ?case .
qed
lemma zorder_zero_eqI:
assumes f_holo:"f holomorphic_on s" and "open s" "z ∈ s"
assumes zero: "⋀i. i < nat n ⟹ (deriv ^^ i) f z = 0"
assumes nz: "(deriv ^^ nat n) f z ≠ 0" and "n≥0"
shows "zorder f z = n"
proof -
obtain r where [simp]:"r>0" and "ball z r ⊆ s"
using ‹open s› ‹z∈s› openE by blast
have nz':"∃w∈ball z r. f w ≠ 0"
proof (rule ccontr)
assume "¬ (∃w∈ball z r. f w ≠ 0)"
then have "eventually (λu. f u = 0) (nhds z)"
using ‹r>0› unfolding eventually_nhds
apply (rule_tac x="ball z r" in exI)
by auto
then have "(deriv ^^ nat n) f z = (deriv ^^ nat n) (λ_. 0) z"
by (intro higher_deriv_cong_ev) auto
also have "(deriv ^^ nat n) (λ_. 0) z = 0"
by (induction n) simp_all
finally show False using nz by contradiction
qed
define zn g where "zn = zorder f z" and "g = zor_poly f z"
obtain e where e_if:"if f z = 0 then 0 < zn else zn = 0" and
[simp]:"e>0" and "cball z e ⊆ ball z r" and
g_holo:"g holomorphic_on cball z e" and
e_fac:"(∀w∈cball z e. f w = g w * (w - z) ^ nat zn ∧ g w ≠ 0)"
proof -
have "f holomorphic_on ball z r"
using f_holo ‹ball z r ⊆ s› by auto
from that zorder_exist_zero[of f "ball z r" z,simplified,OF this nz',folded zn_def g_def]
show ?thesis by blast
qed
from this(1,2,5) have "zn≥0" "g z≠0"
subgoal by (auto split:if_splits)
subgoal using ‹0 < e› ball_subset_cball centre_in_ball e_fac by blast
done
define A where "A = (λi. of_nat (i choose (nat zn)) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z)"
have deriv_A:"(deriv ^^ i) f z = (if zn ≤ int i then A i else 0)" for i
proof -
have "eventually (λw. w ∈ ball z e) (nhds z)"
using ‹cball z e ⊆ ball z r› ‹e>0› by (intro eventually_nhds_in_open) auto
hence "eventually (λw. f w = (w - z) ^ (nat zn) * g w) (nhds z)"
apply eventually_elim
by (use e_fac in auto)
hence "(deriv ^^ i) f z = (deriv ^^ i) (λw. (w - z) ^ nat zn * g w) z"
by (intro higher_deriv_cong_ev) auto
also have "… = (∑j=0..i. of_nat (i choose j) *
(deriv ^^ j) (λw. (w - z) ^ nat zn) z * (deriv ^^ (i - j)) g z)"
using g_holo ‹e>0›
by (intro higher_deriv_mult[of _ "ball z e"]) (auto intro!: holomorphic_intros)
also have "… = (∑j=0..i. if j = nat zn then
of_nat (i choose nat zn) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z else 0)"
proof (intro sum.cong refl, goal_cases)
case (1 j)
have "(deriv ^^ j) (λw. (w - z) ^ nat zn) z =
pochhammer (of_nat (Suc (nat zn) - j)) j * 0 ^ (nat zn - j)"
by (subst higher_deriv_power) auto
also have "… = (if j = nat zn then fact j else 0)"
by (auto simp: not_less pochhammer_0_left pochhammer_fact)
also have "of_nat (i choose j) * … * (deriv ^^ (i - j)) g z =
(if j = nat zn then of_nat (i choose (nat zn)) * fact (nat zn)
* (deriv ^^ (i - nat zn)) g z else 0)"
by simp
finally show ?case .
qed
also have "… = (if i ≥ zn then A i else 0)"
by (auto simp: A_def)
finally show "(deriv ^^ i) f z = …" .
qed
have False when "n<zn"
proof -
have "(deriv ^^ nat n) f z = 0"
using deriv_A[of "nat n"] that ‹n≥0› by auto
with nz show False by auto
qed
moreover have "n≤zn"
proof -
have "g z ≠ 0" using e_fac[rule_format,of z] ‹e>0› by simp
then have "(deriv ^^ nat zn) f z ≠ 0"
using deriv_A[of "nat zn"] by(auto simp add:A_def)
then have "nat zn ≥ nat n" using zero[of "nat zn"] by linarith
moreover have "zn≥0" using e_if by (auto split:if_splits)
ultimately show ?thesis using nat_le_eq_zle by blast
qed
ultimately show ?thesis unfolding zn_def by fastforce
qed
lemma
assumes "eventually (λz. f z = g z) (at z)" "z = z'"
shows zorder_cong:"zorder f z = zorder g z'" and zor_poly_cong:"zor_poly f z = zor_poly g z'"
proof -
define P where "P = (λff n h r. 0 < r ∧ h holomorphic_on cball z r ∧ h z≠0
∧ (∀w∈cball z r - {z}. ff w = h w * (w-z) powr (of_int n) ∧ h w≠0))"
have "(∃r. P f n h r) = (∃r. P g n h r)" for n h
proof -
have *: "∃r. P g n h r" if "∃r. P f n h r" and "eventually (λx. f x = g x) (at z)" for f g
proof -
from that(1) obtain r1 where r1_P:"P f n h r1" by auto
from that(2) obtain r2 where "r2>0" and r2_dist:"∀x. x ≠ z ∧ dist x z ≤ r2 ⟶ f x = g x"
unfolding eventually_at_le by auto
define r where "r=min r1 r2"
have "r>0" "h z≠0" using r1_P ‹r2>0› unfolding r_def P_def by auto
moreover have "h holomorphic_on cball z r"
using r1_P unfolding P_def r_def by auto
moreover have "g w = h w * (w - z) powr of_int n ∧ h w ≠ 0" when "w∈cball z r - {z}" for w
proof -
have "f w = h w * (w - z) powr of_int n ∧ h w ≠ 0"
using r1_P that unfolding P_def r_def by auto
moreover have "f w=g w" using r2_dist[rule_format,of w] that unfolding r_def
by (simp add: dist_commute)
ultimately show ?thesis by simp
qed
ultimately show ?thesis unfolding P_def by auto
qed
from assms have eq': "eventually (λz. g z = f z) (at z)"
by (simp add: eq_commute)
show ?thesis
by (rule iffI[OF *[OF _ assms(1)] *[OF _ eq']])
qed
then show "zorder f z = zorder g z'" "zor_poly f z = zor_poly g z'"
using ‹z=z'› unfolding P_def zorder_def zor_poly_def by auto
qed
lemma zorder_nonzero_div_power:
assumes "open s" "z ∈ s" "f holomorphic_on s" "f z ≠ 0" "n > 0"
shows "zorder (λw. f w / (w - z) ^ n) z = - n"
apply (rule zorder_eqI[OF ‹open s› ‹z∈s› ‹f holomorphic_on s› ‹f z≠0›])
apply (subst powr_of_int)
using ‹n>0› by (auto simp add:field_simps)
lemma zor_poly_eq:
assumes "isolated_singularity_at f z" "not_essential f z" "∃⇩F w in at z. f w ≠ 0"
shows "eventually (λw. zor_poly f z w = f w * (w - z) powr - zorder f z) (at z)"
proof -
obtain r where r:"r>0"
"(∀w∈cball z r - {z}. f w = zor_poly f z w * (w - z) powr of_int (zorder f z))"
using zorder_exist[OF assms] by blast
then have *: "∀w∈ball z r - {z}. zor_poly f z w = f w * (w - z) powr - zorder f z"
by (auto simp: field_simps powr_minus)
have "eventually (λw. w ∈ ball z r - {z}) (at z)"
using r eventually_at_ball'[of r z UNIV] by auto
thus ?thesis by eventually_elim (insert *, auto)
qed
lemma zor_poly_zero_eq:
assumes "f holomorphic_on s" "open s" "connected s" "z ∈ s" "∃w∈s. f w ≠ 0"
shows "eventually (λw. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)) (at z)"
proof -
obtain r where r:"r>0"
"(∀w∈cball z r - {z}. f w = zor_poly f z w * (w - z) ^ nat (zorder f z))"
using zorder_exist_zero[OF assms] by auto
then have *: "∀w∈ball z r - {z}. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)"
by (auto simp: field_simps powr_minus)
have "eventually (λw. w ∈ ball z r - {z}) (at z)"
using r eventually_at_ball'[of r z UNIV] by auto
thus ?thesis by eventually_elim (insert *, auto)
qed
lemma zor_poly_pole_eq:
assumes f_iso:"isolated_singularity_at f z" "is_pole f z"
shows "eventually (λw. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)) (at z)"
proof -
obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
obtain r where r:"r>0"
"(∀w∈cball z r - {z}. f w = zor_poly f z w / (w - z) ^ nat (- zorder f z))"
using zorder_exist_pole[OF f_holo,simplified,OF ‹is_pole f z›] by auto
then have *: "∀w∈ball z r - {z}. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)"
by (auto simp: field_simps)
have "eventually (λw. w ∈ ball z r - {z}) (at z)"
using r eventually_at_ball'[of r z UNIV] by auto
thus ?thesis by eventually_elim (insert *, auto)
qed
lemma zor_poly_eqI:
fixes f :: "complex ⇒ complex" and z0 :: complex
defines "n ≡ zorder f z0"
assumes "isolated_singularity_at f z0" "not_essential f z0" "∃⇩F w in at z0. f w ≠ 0"
assumes lim: "((λx. f (g x) * (g x - z0) powr - n) ⤏ c) F"
assumes g: "filterlim g (at z0) F" and "F ≠ bot"
shows "zor_poly f z0 z0 = c"
proof -
from zorder_exist[OF assms(2-4)] obtain r where
r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
"⋀w. w ∈ cball z0 r - {z0} ⟹ f w = zor_poly f z0 w * (w - z0) powr n"
unfolding n_def by blast
from r(1) have "eventually (λw. w ∈ ball z0 r ∧ w ≠ z0) (at z0)"
using eventually_at_ball'[of r z0 UNIV] by auto
hence "eventually (λw. zor_poly f z0 w = f w * (w - z0) powr - n) (at z0)"
by eventually_elim (insert r, auto simp: field_simps powr_minus)
moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
using r by (intro holomorphic_on_imp_continuous_on) auto
with r(1,2) have "isCont (zor_poly f z0) z0"
by (auto simp: continuous_on_eq_continuous_at)
hence "(zor_poly f z0 ⤏ zor_poly f z0 z0) (at z0)"
unfolding isCont_def .
ultimately have "((λw. f w * (w - z0) powr - n) ⤏ zor_poly f z0 z0) (at z0)"
by (blast intro: Lim_transform_eventually)
hence "((λx. f (g x) * (g x - z0) powr - n) ⤏ zor_poly f z0 z0) F"
by (rule filterlim_compose[OF _ g])
from tendsto_unique[OF ‹F ≠ bot› this lim] show ?thesis .
qed
lemma zor_poly_zero_eqI:
fixes f :: "complex ⇒ complex" and z0 :: complex
defines "n ≡ zorder f z0"
assumes "f holomorphic_on A" "open A" "connected A" "z0 ∈ A" "∃z∈A. f z ≠ 0"
assumes lim: "((λx. f (g x) / (g x - z0) ^ nat n) ⤏ c) F"
assumes g: "filterlim g (at z0) F" and "F ≠ bot"
shows "zor_poly f z0 z0 = c"
proof -
from zorder_exist_zero[OF assms(2-6)] obtain r where
r: "r > 0" "cball z0 r ⊆ A" "zor_poly f z0 holomorphic_on cball z0 r"
"⋀w. w ∈ cball z0 r ⟹ f w = zor_poly f z0 w * (w - z0) ^ nat n"
unfolding n_def by blast
from r(1) have "eventually (λw. w ∈ ball z0 r ∧ w ≠ z0) (at z0)"
using eventually_at_ball'[of r z0 UNIV] by auto
hence "eventually (λw. zor_poly f z0 w = f w / (w - z0) ^ nat n) (at z0)"
by eventually_elim (insert r, auto simp: field_simps)
moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
using r by (intro holomorphic_on_imp_continuous_on) auto
with r(1,2) have "isCont (zor_poly f z0) z0"
by (auto simp: continuous_on_eq_continuous_at)
hence "(zor_poly f z0 ⤏ zor_poly f z0 z0) (at z0)"
unfolding isCont_def .
ultimately have "((λw. f w / (w - z0) ^ nat n) ⤏ zor_poly f z0 z0) (at z0)"
by (blast intro: Lim_transform_eventually)
hence "((λx. f (g x) / (g x - z0) ^ nat n) ⤏ zor_poly f z0 z0) F"
by (rule filterlim_compose[OF _ g])
from tendsto_unique[OF ‹F ≠ bot› this lim] show ?thesis .
qed
lemma zor_poly_pole_eqI:
fixes f :: "complex ⇒ complex" and z0 :: complex
defines "n ≡ zorder f z0"
assumes f_iso:"isolated_singularity_at f z0" and "is_pole f z0"
assumes lim: "((λx. f (g x) * (g x - z0) ^ nat (-n)) ⤏ c) F"
assumes g: "filterlim g (at z0) F" and "F ≠ bot"
shows "zor_poly f z0 z0 = c"
proof -
obtain r where r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
proof -
have "∃⇩F w in at z0. f w ≠ 0"
using non_zero_neighbour_pole[OF ‹is_pole f z0›] by (auto elim:eventually_frequentlyE)
moreover have "not_essential f z0" unfolding not_essential_def using ‹is_pole f z0› by simp
ultimately show ?thesis using that zorder_exist[OF f_iso,folded n_def] by auto
qed
from r(1) have "eventually (λw. w ∈ ball z0 r ∧ w ≠ z0) (at z0)"
using eventually_at_ball'[of r z0 UNIV] by auto
have "eventually (λw. zor_poly f z0 w = f w * (w - z0) ^ nat (-n)) (at z0)"
using zor_poly_pole_eq[OF f_iso ‹is_pole f z0›] unfolding n_def .
moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
using r by (intro holomorphic_on_imp_continuous_on) auto
with r(1,2) have "isCont (zor_poly f z0) z0"
by (auto simp: continuous_on_eq_continuous_at)
hence "(zor_poly f z0 ⤏ zor_poly f z0 z0) (at z0)"
unfolding isCont_def .
ultimately have "((λw. f w * (w - z0) ^ nat (-n)) ⤏ zor_poly f z0 z0) (at z0)"
by (blast intro: Lim_transform_eventually)
hence "((λx. f (g x) * (g x - z0) ^ nat (-n)) ⤏ zor_poly f z0 z0) F"
by (rule filterlim_compose[OF _ g])
from tendsto_unique[OF ‹F ≠ bot› this lim] show ?thesis .
qed
end