Quelle Complex_Residues.thy Sprache: Isabelle

imports
begin

subsection imports Complex_Singularities

text\<open>Wenda Li and LC Paulson (2016). A Formal Proof of Cauchy's Residue Theorem.
 Interactive Theorem Proving\<close>

definition\<^marker>\<open>tag important\<close> residue :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" where
 "residue f z = (SOME int. \e>0. \\>0. \
 \<longrightarrow> (f has_contour_integral 2*pi* \ *int) (circlepath z \<epsilon>))"

lemma residue_cong:
 assumes eq: "eventually (\z. f z = g z) (at z)" and "z = z'"
 shows "residue f z = residue g z'"
proof -
 from assms have eq': "eventually (\z. g z = f z) (at z)"
 by (simp add: eq_commute)
 let ?P = "\f c e. (\\>0. \ < e \
 (f has_contour_integral of_real (2 * pi) * \ * c) (circlepath z \<epsilon>))"
 have "residue f z = residue g z" unfolding residue_def
 proof (rule Eps_cong)
 fix c :: complex
 have "\e>0. ?P g c e"
 if "\e>0. ?P f c e" and "eventually (\z. f z = g z) (at z)" for f g
 proof -
 from that(1) obtain e where e: "e > 0" "?P f c e"
 by blast
 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 unfolding eventually_at by blast
P c( ee)
 proof (intro
 case (1 \<epsilon>)
 hence"fhas_contour_integral (2 * pi
 using e(2) by auto eq (<> z=z at "z =z"
 thus ?case
 proof (rule has_contour_integral_eq)
 fix z' shows"residue f z = g z'java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 hence"distz' z < e'"and <>zjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
using auto: dist_commute
 with e' " f z =residue g z"
 qed

frome e' have " e e' >0
 ultimately show "
 qed
 from this[OF -
 show(
 bybyblast
 qed
with ?thesis
qed

lemma: "residue f z =residue (\x. f (z + x)) 0"
proof
d Q where "fhas_contour_integralof_real( pi < *)( z\)"
Q=\> \<epsilon>. (f has_contour_integral complex_of_real (2 * pi) * \ * r) (circlepath z \<epsilon>))"
 define has_contour_integral_eq
 z' z \ path_image (circlepath z \)"
 have path_eq:hence 'z z \<noteq> z"
bysimp o_def algebra_simps
 withe(of "fz' =gz'"bysimp
 qed
 m from eand" e e' > 0"by
 using _ zz]* _ \lambda. z+""java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
 by (introarg_cong[where=Eps iffI add:)
 thus ?thesis
 by( add: residue_def Q_def)
qed

lemma residue_shift_0': "NO_MATCH 0 z \ residue f z = residue (\x. f (z + x)) 0"
 by (rule residue_shift_0)

lemma contour_integral_circlepath_eq:
 assumes "open s" and f_holo:"f holomorphic_on (s-{z})" and "0 "
 and e2_cball:"cball z e2 \ s"
shows
 "f contour_integrable_on circlepath z e1"
 "f contour_integrable_on circlepath z e2"
 "contour_integral (circlepath z e2 "Q= (<lambda>r f z \<epsilon>. (f has_contour_integral complex_of_real (2 * pi) * \ * r) (circlepath z \<epsilon>))"
proof -
 define l where "l \ linepath (z+e2) (z+e1)"
 have []:"valid_path l""pathstartl=z+e2 pathfinish" unfolding l_def auto
 P \<
 have zl_img:"z have zl_img:"z\path_image l"epsilon+(w
 proof
 assume "z by (simp add: circlepath_def o_def part_circlepath_def algebra_simps)
 have
 using *: "P r f z" if "P r (\x. f (x + w)) (z - w)" for r w f z
 by (auto simp add:closed_segment_commuteusingthat (auto: P_defQ_def path_eq has_contour_integral_translate
 False
 apply (subst (asm) norm_of_real)
 by auto
qed
definewhere <equivz ++l ++ ( e1reversepath
 show [simp ?thesis
 proof -
 show
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
continuous_on_subset holomorphic_on_imp_continuous_on ]]])
 \<open>e2>0\<close> e2_cball by auto
 show "f "open"and f_holo:"f (s-}) 0e1\<le>e2"
 apply (intro ontour_integrable_continuous_circlepath
 continuous_on_subset holomorphic_on_imp_continuous_on f_holo fcontour_integrable_one1fcontour_integrable_on
 using \<open>e1>0\<close> \<open>e1\<le>e2\<close> e2_cball by auto l where

 have [simp]:"f contour_integrable_on l"
 proof -
 have zl_img
 have
 " (z +) (z +) \ s - {z}" using zl_img e2_cball unfolding l_def
 by auto
 ( simp:)
 contour_integrable_continuous_linepath
 subst) )
 by
qed
 let showsimpf contour_integrable_on e2 ( )
 have -
 "fcontour_integrable_on z e2"
 show ( contour_integrable_continuous_circlepath
showg unfolding l_defauto
 show "pathfinish g = pathstart g" unfolding g_def l_def by auto
 next
 have path_img:"path_image g \ cball z e2"
 proofusing \<open>e2>0\<close> e2_cball by auto
 closed_segmente2)( ) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
 by (intro[OFholomorphic_on_imp_continuous_onOF]]])
 moreover have "spherez \e1\ \ cball z e2" using \e2>0\ \e1\e2\ \e1>0\ by auto
 ultimately show ?thesis unfolding g_defqed
 by simp: path_image_join closed_segment_commute
d
 have +e2 )\<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>intro,auto add)
-
 have "z\path_image g" using zl_img
 unfolding g_def l_def by (auto simp add: path_image_join closed_segment_commute)
 moreovernote
 ultimately show ?thesis by auto
 qed
 show "winding_number g w = 0" when"w \ s - {z}" for w
 proof -
 "winding_number g =0" w\<notin>s" using that e2_cball
 java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 15
 by auto add:_ l_def
moreover "winding_number g z=0"
 proofshowvalid_path g_def by auto
Wz
 have "?Wz g = ?Wz (circlepath z e2) + ?Wz next
 + ?Wz (reversepath path_img \<subseteq> cball z e2"
 using have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>,auto add)
 by (subst winding_number_join,auto simp <open>e2>0\<close>
 lso ..=?(circlepath+?Wzreversepath e1
 using zl_img
 apply (substproof-
 have\<notin>path_image g" using zl_img
 have. "
 proof -
 have"Wz( e2 "using
 by (auto:winding_number_circlepath_centre
 moreoverhave "Wz(circlepathz e1)java.lang.StringIndexOutOfBoundsException: Range [70, 23) out of bounds for length 23
 apply ( ( simp:g_def)
 by (auto intro: winding_number_circlepath_centre -
 ultimately ? by auto
 qed
how
 qed
tely thesis by auto
 qed
 qed
 then have (ubst Theorem \<close
ig ze1
 + ?ig (reversepath l)"
 unfolding
 ( simp add apply (subst (2) winding_number_reversep
 also have ". y autosimpaddl_defclosed_segment_commute)
 by (auto simp:contour_integral_reversepath
 finally "contour_integral ( ze2 f= contour_integral(circlepath z e1) f"
 by simpbysimp)
 have?( )=1u\<open>e2>0\<close>

lemmabase_residue:
es "open s "\<in>s" "r>0" and f_holo:"f holomorphic_on (s - {z})"
 andr_cballzr\<subseteq> s"
 c : complex
proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 e where(winding_number_reversepath
 auto: winding_number_circlepath_centre
java.lang.StringIndexOutOfBoundsException: Range [9, 2) out of bounds for length 43
 that e e: e>"\And' z\ z \ dist z' z < e' \ f z' = g z'"
 have "(f java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 7
proof
 havealso "f of_real ( *pi) * c) (circlepath z \)"
 "f contour_integrable_on circlepath z \"
 " contour_integrable_on circlepath ze"
 using +?ig?case
tro[OF java.lang.StringIndexOutOfBoundsException: Range [0, 116) out of bounds for length 19
 then showhencedist <' also ". ig ( z )-?)"
 byauto:has_contour_integral_integral
 qed
 with)fz' "z=gz"by simp
 unfolding
 apply (rule_tac someI[of _ i],intro exI[where x=e])
 by auto add:java.lang.StringIndexOutOfBoundsException: Range [0, 28) out of bounds for length 3
 then obtain e' where "e'and ?hesis
 andd:\<forall>\<epsilon>>0. \<epsilon><e' \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)" this _

let"

 have
 have "(f has_contour_integral c * (residue f z) define i where "i\<equiv> contour_integral (circlepath z e) f / c"
usingdef, <><>0
 then -
 de Q here
 auto[ _"irclepathz\" "circlepath z r"])
qed" contour_integrable_on z \"

lemmaresidue_holo
sumes s "z\ s" and f_holo: "f holomorphic_on s"
 have: " (z -w)\ = (+) (-w) \ circlepath z \" for z w \
roof
 then ?thesisi_defc_def
 obtain e where "e>0" andby( intro)
 using open_contains_cball_eq byauto
 have simp
usingthen'"'"
 by (auto intro: base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
 moreover "(fhas_contour_integral 0 (circlepath ze)"
 using ?="\e. contour_integral (circlepath z e) f"
 by ( add: residue_def Q_def)
atelyc* 0
 using has_contour_integral_unique e'
 thusthesis c_def byauto
qed

lemma residue_const:"residue (\_. c) z = 0"
 by (intro contour_integral_circlepath_eq

lemmajava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 s" <>s andf_holo:" holomorphic_on{}java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
 andg_holo:"g holomorphic_on -z}"
 shows "residue (\z. f z + g z) z= residue f z + residue g z"
proof e2_cball \<subseteq> s"
 define wherejava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
 define "(f has_contour_integral c*residue " circlepath"
 contour_integral ze1
-
 havefg c*residue( )java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
 unfolding fg_def by(uto: base_residue <open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
 apply( base_residue \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
 by auto:holomorphic_introsjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 by autointroCauchy_theorem_convex_simple _ "cball z e"java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
 segment_furthest_le "+ "z+e1e2 \<open>e1>0\<close> \<open>e2>0\<close> unfolding l_def
 ( intro:has_contour_integral_add[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
ultimately"*residuefz residue z)=c *esidue "
 using by (auto add:distrib_left
 thus qed
 ( simp add)
qed

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
assumes z\<in> s" and f_holo: "f holomorphic_on s - {z}"
 showsresidue_add:
proof (cases "="
 casejava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 ? r (<lambda>z. f z + g z) z= residue f z + residue g z"
next
 case appl contour_integrable_continuous_circlepath
 definecontinuous_on_subset ewhereandcball< "
 define \<open>e1>0\<close> \<open>e1\<le>e2\<close> e2_cball by auto
 obtainwhere"ande_cball:"ball z e\<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close> c* ) -
 using open_contains_cball_eq fg_def g_holoclosed_segment_subset,simp)
 have" (base_residue[OF\java.lang.StringIndexOutOfBoundsException: Range [121, 39) out of bounds for length 121
 unfoldingf by( introholomorphic_intros
apply( [ <open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])continuous_on_subset holomorphic_on_imp_continuous_on f_holo])
 ( java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
ntegral z)circlepathjava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
 unfoldingf'def using f_holo
 ( fg_def
 (auto( Cauchy_theorem_globalf_holo
 ultimatelyhavejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 "open s "\inf g_def byauto
 thus ?thesisshows\<lambda>z. c * (f z)) z= c * residue f z" "pathfinish = pathstartproof (cases c0)
 by( java.lang.StringIndexOutOfBoundsException: Range [17, 11) out of bounds for length 11
qed

lemma (intro,auto simp False
 assumes "open s" "z \ s" and f_holo: "f holomorphic_on s - {z}"
 shows "residue (\z. (f z) * c) z= residue f z * c"
using residue_lmul[OF assms,of c] by (auto simp add:algebra_simps)

lemma residue_div:
 assumes "open s" "z \ s" and f_holo: "f holomorphic_on s - {z}"
 shows "residue (\z. (f z) / c) z= residue f z / c "
using residue_lmul[OF assms,of "1/c"] by (auto simp add:algebra_simps)

lemma residue_neg:
 assumes "open s" "z \ s" and f_holo: "f holomorphic_on s - {z}"
 shows "residue (\z. - (f z)) z= - residue f z"
using residue_lmul[OF assms,of "-1"] by auto

lemma residue_diff:
 assumes "open s" "z \ s" and f_holo: "f holomorphic_on s - {z}"
 and g_holo:"g holomorphic_on s - {z}"
 shows "residue (\z. f z - g z) z= residue f z - residue g z"
using residue_add[OF assms(1,2,3),of "\z. - g z"] residue_neg[OF assms(1,2,4)]
by (auto intro:holomorphic_intros g_holo)

lemma residue_simple:
 assumes "open s" "z\s" and f_holo:"f holomorphic_on s"
 shows "residue (\w. f w / (w - z)) z = f z"
proof -
 define c where "c \ 2 * pi * \"
 define f' where "f' \<equiv> \<lambda>w. f w / (w - z)"
 obtain e where "e>0" and e_cball:"cball z e \ s" using \open s\ \z\s\
 using open_contains_cball_eq by blast
 have "(f' has_contour_integral c * f z) (circlepath z e)"
 unfolding f'_def c_def using \e>0\ f_holo e_cball
 by (auto intro!: Cauchy_integral_circlepath_simple holomorphic_intros)
 moreover have "(f' has_contour_integral c * residue f'theory Complex_Residues
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 intro[
 by (auto intro!:holomorphic_intros)
 ultimatelyresidueSOME

 thus: eq eventually
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 3

residue_simple
 assumes openjava.lang.NullPointerException
 : (
 shows " Eps_congjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
e> e_cball
 define
 showjava.lang.NullPointerException
 ( intro
P
 (fof_real*
 finally havebyautoholomorphic_intros

 note
 also have "(\w. f w * (w - z)) \z\ c \ g \z\ g z"moreover \<open>cball z e2 \<subseteq> s\<close> and path_img
 intro simp_allabs_defproofhas_contour_integral_eq
ve\<midarrow>z\<rightarrow> g z" .

 have g_holo: " have" g w = 0 when
 rule'where K ="z"ue f )
 (assmswith[ '"z gz"bysimp
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 show "(\<exis.P )\<longleftrightarrow> (\<exists>e>0. ?P g c e)"
unfolding!exI1simpg_def
 hence "residue (
 by( residue_cong + Wzl"
 show
 by (simp with show ?thesis(substauto add closed_segment_commute+
qed

lomorphic_over_power
 assumesproofby(add)
Q where
proofproof
 ? "\z. f z / (z - z0) ^ Suc n"
java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
 by (auto simp
 byauto " s" zjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
 byaddpart_circlepath_def
moreover "f )
assms thatauto (: g_holo
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
thus
 ultimatelyqed
 (rule)
 thus ?thesis show
qedqed

lemma residue_holomorphic_over_power
 assumesopenthen whereequiv
showsjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
residue_holomorphic_over_power] by simp

theorem residue_fps_expansion_over_power_at_0:
assumeshas_fps_expansion
shows"residueunfolding '_defc_defu \e>0\ f_holo e_cball
proof shows
from[auto
 ( simphave'java.lang.StringIndexOutOfBoundsException: Range [43, 41) out of bounds for length 77
 java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 with assms havezl_img
 by (esidue:
e_holomorphic_over_power) (uto)
 also from have "\ = fps_nth F n"
bysubstauto
 finally show False\opene2
qed

lemma residue_pole_ordershowsh 'def byjava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 fixes f::" open_contains_cball[of s \open s\ \z\s\ by auto
 defines "n \ nat (- zorder f z)" and "h \ zor_poly f z"
 assumes"isolated_singularity_at f z"
 pole residue_simple
 shows where <> contour_integral z e) f/cjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
proof
 definewhere [F [f_holo proof -
 obtain[imp"and :fholomorphic_on -z}
 using shows "resi"f z \"
 obtain r where g where\<open>\<epsilon><e\<close>
 and:"(\w\cball z r. (w\z \ f w = h w / (w - z) ^ n) \ h w \ 0)"
 byintro
 obtain have"P\ by uto introhas_contour_integral_integral)
 (java.lang.StringIndexOutOfBoundsException: Range [89, 80) out of bounds for length 58
 using
 haven0 using
 moreover have "(\w\cball z r. (w\z \ f w = h w / (w - z) ^ n) \ h w \ 0)"
> r6b
 ultimately -
 qedunfoldingresidue_def
java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
 using h_divide contour_integrable_continuous_linepath
 finally *: "\z\ g z" .
 define have "(f has_contour_integral 0) g"
 define have where{]
/n ) deriv )h z circlepath
 h'def
 ( Cauchy_has_contour_integral_higher_derivative_circlepathassms'[, \\>0\ \\] .
 folded rule
show -
showholomorphic_onauto[ ]: field_simps
 y (c,, simp:dist_norm
 qed
 then have "(h' has_contour_integral c * der_f) (circlepath z r)" unfolding der_f_def show
 then have "(f java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 25
 proof open_contains_cball_eq
 fix x assume g_def by ( simp: assumes "have "fh c*residue ( z e)"
 hence"xjava.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 7
 then show ver(
 qed
reover "(f has_contour_integral c * residue f z)(circlepath z r)"
using ( intro Cauchy_theorem_convex_simple" z e"])
unfoldingby
 ultimately have has_contour_integral_uniqueblast
 ence residueunfoldingc_def
 thusthesis der_f_def by auto have "?fhas_contour_integral2* pi*\ java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 3

lemmaresidue_simple_pole
 assumes "isolated_singularity_at f z0"
 assumes "is_pole java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 "open s""z s" and f_holo: "f holomorphic_on s - {z}"
 using assms by (subst residue_pole_order) simp_all

lemmaresidue_simple_pole_limit
assumes f z0java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 assumes cwhereapplys 2winding_number_reversepath
 assumes e theorem:
 " g ( also have ". =0java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

-
 haveby( intro)
idue_simple_pole
 also have"\ = c"
applyrule)
 using assms by "c ultimatelyshow? by auto
 finally show has_fps_expansion_imp_holomorphic] obtains
qed

java.lang.StringIndexOutOfBoundsException: Range [37, 5) out of bounds for length 5
af_holo:" s g_holo:g holomorphic_ons"
 and "open s" "connected s" "z \ s"
 qed
assumesfz \<noteq> 0" "g z = 0" "g' \<noteq> 0"
 assumes " byintro residue_holomorphic_over_power s] autosimp:)
and:" w. f w / g w) z = f z / g'"
proof - also from assms "\ = fps_nth F n"
havesimpsolated_singularity_at "g z"
 True
(circlepath define "'\<equiv> 2 * pi * \"
 have [simp f::"complex \ complex" and z::complex
 unfolding
nuous_on_eq_continuous_atapply(ntro[OF\<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
 have g_nconst\exists\<^sub>F w in at z. g w \<noteq>0 "
 proofule"fhas_contour_integral ('* )( e"
 assumew []:"e>0"and:"f holomorphic_on ball ze -z"
 have "\<^sub>F w in nhds z. g w = 0"
 unfolding eventually_atproof-
by( usingf_iso unfolding by blast
then" (l>.0 zjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
 by(ntro) auto
 then have residue_rmul
 then have "g' = 0" using "residue (\z. (f z) * c) z= residue f z * c"
 then show False using \<open>g'\<noteq>0\<close> by auto
 qed

 have "zorder (\w. f w / g w) z = zorder f z - zorder g z"
 proof-
lemma:
 apply (rule non_zero_neighbour_alt java.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 124
 using ? unfoldingjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
 with g_nconst have "\\<^sub>F w in at z. f w * g w \ 0"
 by( frequently_rev_mp"
show using[of] auto
 qed
 moreover have "zorder f z
ruleOF \<open>open s\<close> \<open>z\<in>s\<close>])
 \<open>f z\<noteq>0\<close> by auto'java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 93
moreoverzorder
 ( zorder_zero_eqI_def<open>n>0\<close>]])
 subgoal letshow" h_holobysimp
 DERIV_imp_deriv9g_deriv
 usingresidue_add[ z \<in> ball z r" using \<open>r>0\<close> by auto
 done
 ultimatelyhave" ave\epsilonjava.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134

 show "residueproof-
 proofhen unfolding c_def
 show f where\<equiv> \<lambda>w. f w / (w - z)"
 showjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 by (auto intro: have "(f' has_contou c * f z) ( z e)"
 "is_pole(
 proofis_pole_divide
\<forall>\<^sub>F x in at z. g x \<noteq> 0"
 apply(ule showsz 0java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
 usingg_nconst auto ?thesis der_f_def by auto
 moreover have" using open_contains_cball_eq by blastapply( base_residueOF <>open s<> \s\ \e>0\ _ e_cball,folded c_def])
 using DERIV_isCont uresidue_simple_pole
 "isolated_singularity_at f z0"
 isCont
 using have"java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 by auto
 showf <noteq> 0" by factusing by (subst) simp_all
 qed
 show "filterlim id (at z) (at z)" by (simp add: filterlim_iff)
 (<lambda>w. (f w * (w - z)) / g w) \<longlongrightarrow> f z / g') (at z)"
 ?thesis c_def
showjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 assms "filterlim proof-
 show "g has_field_derivative g)( ) by fact
 qed (insert -
 fz0 "
by and: holomorphic_intros
 qed
qed

n\<open>Poles and residues of some well-known functions\<close>where

(* TODO: add more material here for other functions *)
lemma is_pole_Gammafinally java.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds for length 0
unfolding usingjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 3

lemma Gamma_residue have finally have **: "g \<midarrow
( **,java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 65

 open<>zaz za ajava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
 closed_subset_Ints
 henceesidue
 lemmresidue_lmul
show
 using Gamma_residues finally?java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
qed True

end

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