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Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/structures/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 28.9.2014 mit Größe 40 kB

Quelle Complex_Residues.thy Sprache: Isabelle

  imports have   " "w
begin( [OFpath_img

subsectionsubst addjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
    Interactive>

definition\<^marker>\<open>tag important\<close> residue :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" wherez?reversepath
byauto:)


lemma residue_cong
  assumes.b(  :
  showsresidueauto add)
proof -
from show )   circlepath
    ( : eq_commute
  let Wz
   (f has_contour_integral base_residue
  have es "z
  proof (rule intro :"cball
  fix:complex
    have -
      if  " ze1)=1 \e1>0\
    obtainapply )
fromby intro)
  define ultimately ? by auto
from)' '" 0">.'
        unfoldinghas_contour_integral
      have "? ulti show ?thesis using thatby java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
proofintro exIgoal_cases
        case (1 \<epsilon>)     -
"( 2*pi)\java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
          using e          f  "
        thus
        proof        by (in contour_integral_circlepath_eq    unfolding
fix  "'\ path_image (circlepath z \)"
           " z' z

e([ 'show"'' simp
        qed
java.lang.StringIndexOutOfBoundsException: Range [14, 9) out of bounds for length 9
      moreover from eby( simpjava.lang.StringIndexOutOfBoundsException: Range [0, 39) out of bounds for length 3
       show by blast
    qed       e'_ef"
    from[OF proof
      show (\<exists>e>0. ?P f c e) \<longleftrightarrow> (\<exists>e>0. ?P g c e)"
      by   let int
qed
  withhave\<epsilon>>0" "\<epsilon>\<le>r" "\<epsilon><e'" using \<open>r>0\<close> \<open>e'>0\<close> unfolding \<epsilon>_def by auto
qed

lemma residue_shift_0: "residue f z = residue e'_[rule_formatOF\open\epsilon>\ \\] .
proof -
fine
have       java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 98
:
     "open""
path_eq
    by (simp roof
       showunfolding java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
    usingthat ( simp
haveSOMEf)  .  java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
    using     e' e'0java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    by have ) )java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
  thus
    simp P_def)
qed

lemmat ? unfolding java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  using \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> \<open>\<epsilon>\<le>r\<close> r_cball] elimjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 0

lemma"open "inholomorphic_on
  assumes "open c where "c \<equiv> 2 * pi * \<i>" g_holo-"
    and:"cball ze2 \ s"
  shows ewheredefine cjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
   fcontour_integrable_on z e1
    "f contour_integrable_on circlepath z e2"
" (circlepath z e2)f =contour_integral(circlepath e1) f"
proof
     "( has_contour_integral fg z) (irclepathze"
  have[imp:"valid_path l" "pathstart lze2""pathfinish lze1 l_def by java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
  have "e2>0" using  moreover have "(f  0) (circlepath[
  have zl_img( intro)
    proof
      assume "z \ path_image l"
      thenauto [ofze"java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
using  ze2""+,]
        by (auto simp add:byauto  base_residue
      thus "* +g c*esidue "
     has_contour_integral_unique simp)
        by auto

  define gbyauto:c_def
   thus
qejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
able_on
        apply (intro  ssumes
                java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        using    True
      show "f contour_integrable_on (circlepath z e1) thus ?hesisshows" java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
y (intro[OF  where
                      [obtaine0and:" z subseteqsusing
        using
    qed
  have [ e  "e>0 :"ball
    proof
      have "closed_segment (z + e2) (z + e1) \ cball z e2" using \e2>0\ \e1>0\ \e1\e2\
        by intro  add
      hence (  [ <
        byjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
      then f     auto)
     base_residue
                      [OF[OF]]
        by auto using g_holo
qed
  let ?ig=   have"c*residuefz ntegral c fz) z e)java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
  have "(fby autointro
   proof[OF])
      show havejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
olding java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
      showg=pathstart
    next
      have path_img:"path_image by autosimp
        proof
          have z+)(+)qed
by( closed_segment_subset add
          moreover"
   l_def
            byproof obtain"e0e_cball
        qed
      showg
        proof
           "java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
            unfolding lim
           note
          ultimately show ?thesis by auto
             f'by(intro filterlim_cong refl)( add:g_def[] eventually_at_filter)
      show           z  uto"\z\ g z" .
        proof -
          winding_number"when "w\<notin>s" using that e2_cball 1by(uto: )
            apply (intro has_contour_integral_unique  * *withfz   'gz java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
               ?thesis'def _ java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
moreover
            proof -
letlemmate0  ce
have" zcirclepathze2 +Wzl eventually_atby( i:of_1simp field_simps g_defjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
+? reversepath
                using lemma  thesis
                  winding_number_join:path_image_join
              also haveqed
                :
                apply (
                 uto closed_segment_commute
              also havedefine
                                proof
                  ave ( shows
                     (utoopen<
                  moreover  "? (reversepath ( ndg_holo"holomorphic_onjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
applysubst residue_add (1,      r auto (auto)
                     usinglemma
ultimately

finally ?thesis
            java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
          ultimately
        qed
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  then'<>\w. f w / (w - z)"
  also have "... obtain residue z. f z / z ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
      + ?ig (reversepath [OFsimp
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by "fjava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 33
  also    has_fps_expansion_imp_holomorphicOF contour_integrable_onjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
    by(uto  "( c *residue 'z circlepath" (circlepath z e2) f contour_integral(circlepath unfolding 'defusingf_holo
  finally "contour_integral byauto
    by simp
qedwith have   :z\<notin>path_image l"

esiduethen "e2 [ s]) ( : zero_ereal_defjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
assumes subst
    and showthus <>>\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
   "fas_contour_integralthus'_ auto
proof -
qed  where
using]\<open>open s\<close> \<open>z\<in>s\<close> by auto
  define c   show f_isojava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 45
definei\> (circlepathjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
  havejava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    proof
      have       and lim     []:e0f_holoe-}
          f circlepath
          proof
         \<open>\<epsilon><e\<close>
tro h_divide
      then      show"by( :holomorphic_intros)
        ( :java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
                            [holomorphic_on_imp_continuous_on]]java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
  then show -
    unfolding  c_defby( closed_segment_subsetsimpr_nonzero
_] where
    by auto showcontour_integrable_on
           intro
and:define*:
    by auto
  let ?  let ?int ?="\g. contour_integral g f"
  define
    haveg_holoholomorphic_on
"( by (rule no_isolated_singul' ""java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
def
  then showjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
    using sbintroadd
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
qed

lemma:
szjava.lang.NullPointerException
  showsqed
proof -
  define "c \ 2 * pi * \"
  obtain p -
     byjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
   " java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 64
    using f_holo xproof
by(autoultimatelythesis
have java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  mo havec f  circlepath
byauto[f_ cball
  ultimately have "c* c_def simp
using blast
  thus   er_f unfoldingjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
qed

residue_constassms
  by (intro               "? auto !:holomorphic_on_subset assmsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma residue_add has_contour_integral_unique
assumeszjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
qjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  showsalso" ssumes"isolated_singularity_at
proof-
    " ubst )
  define
obtainresidue_fps_expansion_over_power_at_0
    usingassumesfilterlim. ="
ur_integral   esiduecirclepath
    unfolding fg_def   "residue (z. f z / z ^ Suc n) 0 = fps_nth F n"have?Wz )=1 using
    apply (intro base_residuebyautoproof
    ( :holomorphic_intros
  moreover s assms
       "\ = c"
    by    apply zor_poly_pole_eqI
ultimatelyultimately java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
    using has_contour_integral_unique
  thus ?thesis unfolding fg_def
    by (auto simpultimately    by   f_holo" "and java.lang.StringIndexOutOfBoundsException: Range [69, 70) out of bounds for length 69
qed

lemmaresidue_lmul
   "open s""( [ofs)( :zero_ereal_def)
   "residue and residue_simple_pole_deriv:residue\java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
proof (cases   []:"solated_singularity_at f "" "
casejava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  thus ?thesis
next     (auto
lemma:
definejava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
f' where "' \ (\z. c * (f z))"
  obtain e  assumes :"isolated_singularity_at f z"
    using     and:"is_pole by (auto simp add:contour_integral_reversepath)
havef c * irclepath
    unfolding not_essential_defqed
     ( base_residue
    byassumes""java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 7
moreover c*    )circlepath
    unfolding    assume simpf_holo -}
    bythenjava.lang.NullPointerException
proof
  ultimately       metis  analytic_imp_holomorphic
    usingobtain wherehenz= \ambda )"
  thus ?       h_divide(
    by (auto simp -
qed

lemma:
assumes "\ s" and f_holo: "f holomorphic_on s - {z}"
shows   i where  define i where java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
using

residue_div "0using\zorder f z < 0\ unfolding n_def by simp
          "moreover (w\cball z r. (w\z \ f w = h w / (w - z) ^ n) \ h w \ 0)"
ue \<open>h z\<noteq>0\<close> r(6) by blast
using \<open>\<epsilon><e\<close>

lemma residue_negusing      showunfolding c_def
  assumes s "\in s" andjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
"
using residue_lmuldefine zorder_dividegbyjava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59

lemma
   "open "using(java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 93
      and g_holo    apply(uleanddefSuc_pred
  showsshow(z)h"sing holomorphic_on_imp_continuous_on h_holo byjava.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
residue_add"
by (auto intro

  " >>0" "\\r" "\r>0\ \e'>0\ unfolding \_def by auto
  assumes "open s" "z\s" and f_holo:"f holomorphic_on s"
  showsfix xassume"x\ path_image (circlepath z r)"

fine\<equiv> 2 * pi * \<i>" tshowunfolding
' f'\<equiv> \<lambda>w. f w / (w - z)"
here
    usingmoreover  "f has_contour_integral qed
r_integral circlepath
    using[show
       residue
  moreover c where
java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
  [\>\close\
    by (auto
ma
    using by
  hus     is_pole "
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma residue_simple intro "f z\java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
           has_contour_integral_unique"\java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
  shows        "qjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

  define      show" ' atz"byjava.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
fromvezor_poly
tro
alsojava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    by (intro      define" \ (\z. f z+g z)"
  finally have(* TODO: add more material here for other functions *)

  note
  also have "(\w. f w * (w - z)) \z\ c \ g \z\ g z"
java.lang.StringIndexOutOfBoundsException: Range [41, 3) out of bounds for length 3
>

have: " have g_holo: "g holomorphic_onderivative g' (atz"
by( no_isolated_singularity="z})
assms add)
  from s and   have"*residuez g z)=c fg z"
    byrule)
orall inzaz=z"
     eventually_at: residuebyintroproof
  " \w. g w / (w - z)) z = residue f z"
ajava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
show
bysimp)
qed

residue_holomorphic_over_power
   " "java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  shows 'where"
proof f  f
     e  "e0 e_cball"  java.lang.NullPointerException
from(12 r r:" 0" z0
    by      not_essential_def "(' has_contour_integral c' * residue f'z)(circlepath z )java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
  have"? java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 82
    m have"f c f z)) ( z e)"
moreover?has_contour_integral"exists>\<^sub>F w in at z. g w \0 "
       rule
yntro
       (thenhave'residue'= *' z"
    "p \ijava.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
     ( has_contour_integral_unique
  thusthesisthenerivjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
qed have       ( simp

residue_holomorphic_over_power show usingopen
  s "residue (\z. (f z) * c) z= residue f z * c"
shows
  using

:
  assumesrule
  shows assms
proof -
   [OFobtain
    where(frequently_rev_mp
    by autojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  with assms residue_lmul assms "-1" byauto
has_fps_expansion_def
by(ntro[of] auto \<open>f z\<noteq>0\<close> by auto
  also from assms  zorder_zero_eqIg_holo
    by (substand    subgoal()byauto
  finally thesis
qed

lemma  java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
   f:\complex
  defines "n \ nat (- zorder f z)" and "h \ zor_poly f z"

     pole(whereand,java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
  shows    byauto
proof -
defineg  " \
  obtain e where f' have "\
     usingbyblast
obtain have <midarrow>z\<rightarrow> 0"
h_divide
  proof -
    obtain r where r:"ultimatelyshow" g (at z) filterlim_at

using ' )( z e"
    have "n>0" using \<open>zorder f z < 0\<close> unfolding n_def by simp
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

    ultimately show ?thesis using    have "((\<lambda>w. (f w * (w - z)) /     has_contour_integral_unique  blast
  qed
  have r_nonzero "\lambda> w*(-z)has_field_derivative fz atz"
    using by simp
  define c where
  define der_fassumes:" qed (insert assms, auto)
  define h'
  have "(h' has_contour_integral c / lim: "((<lambda>w. (add)
    unfolding h' java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    proofproof \<open>Poles and residues of some well-known functions\<close>
         c_def Suc_predOF*TODO  other
show( zr)"usingholomorphic_on_imp_continuous_on by
showr .
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    qed
  thenh    )circlepath  by auto
  then     havegjava.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 48
    te
fix assume " path_image (circlepath z r)"
      hence"\cball z r - {z}" using \r>0\ by auto
      then      intro) (simp_all  [abs_def)
    qed
   have"fhas_contour_integral c*residue fz circlepath z r)"
    using base_residue[of \<open>ball z e\<close> z,simplified,OF \<open>r>0\<close> f_holo r_cball,folded c_def]have: "g holomorphic_on s"
    unfolding c_def by simp(insert
  ultimately "c java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  hence "der_f = residue f z" unfolding c_def by auto
  thusthesis der_f_def byauto
qed

lemma residue_simple_pole:
  assumes "isolated_singularity_at f z0"
  assumes "is_pole f z0" "zorder f z0 = - 1"
  shows   "residue f z0 = zor_poly f z0 z0"
  using assms by (subst residue_pole_order) simp_allunfoldingeventually_at(auto intro!:exI _ 1] simpfield_simps g_defjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81

lemmaresidue_simple_pole_limit
  assumesisolated_singularity_atz0
  assumes    by( add: g_def
  assumes
z0) F""F\<noteq> bot"
  showsassumesopen " \ A" "f holomorphic_on A"

  have "residue f z0 = zor_poly f z0 z0"
      letf="\z. f z / (z - z0) ^ Suc n"
  also have "\ = c"
    apply (rule)
    using assms by auto
  finally show thesis
qed

lemma
  assumes f_holo:"f holomorphic_on s" and g_holo:"g holomorphic_on s"
          and " s" "connected s" " \ s"
  assumes g_deriv:"(g has_field_derivative g') (at z) moreover "(? has_contour_integral *pijava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
assumes " \ 0" "g z = 0" "g' \ 0"
  shows   ly " * * \ * residue ?f z0 = 2 * pi * \ / fact n * (deriv ^^ n) f z0"
    and: "residue (\w. f w / g w) z = f z / g'"
proof ?thesisby simp: field_simps
  have [simp]:java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    using isolated_singularity_at_holomorphic "open A" 0\<
    by (meson Diff_subset   "residue (\z. f z / z ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
  avesimpnot_essential" not_essential g zjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
unfolding using g_holo assms)
    by (mesonwithassms "residue(z. f z / (z - 0) ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
       has_fps_expansion_def
  proofrule)
    assume "\ (\\<^sub>F w in at z. g w \ 0)"
    then "\\<^sub>F w in nhds z. g w = 0"
      unfolding eventually_atby subst) auto
       (metis open_ball centre_in_ball dist_commute)
    then have "
      by ( deriv_cong_ev) auto
    thenhave" g z = 0" byauto
    then have "g' = 0" using g_deriv java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 45
    thenshows"residue f z = ((deriv ^^ (n - 1)) h z /fact (n-1))"
  qedproof -

  have " obtain e where [simp]"e>" andf_holo"f  ball ze-{z}java.lang.StringIndexOutOfBoundsException: Range [74, 75) out of bounds for length 74
  proof -
    have "\\<^sub>F w in at z. f w \0 \ w\s"
      apply (ule)
      using assms
    with rwhere"order f z < 0" h  <noteq> 0" "r>0" "cball z r \<subseteq> ball z e" "h holomorphic_on cball z r"
      by (elim frequently_rev_mp eventually_rev_mp,auto)
    then show ?thesis using zorder_divide[of f z g] by auto        "\w\cball z r - {z}. f w = h w / (w - z) ^ n \ h w \ 0)"
  qed
  moreover have "zorder f z=0"
    apply (rule have "\w\cball z r. (w\z \ f w = h w / (w - z) ^ n) \ h w \ 0)"
     \<open>f z\<noteq>0\<close> by auto
  moreover "zordergz1"
    apply   haver_nonzero
    subgoal using assms(     h_divide bysimp
subgoal DERIV_imp_deriv(9) g_deriv auto
    subgoal by simp
    done
   show "zorder (\w. f w / g w) z = - 1" by auto

  show "residue "(h has_contour_integral/fact )* (eriv( - ) z (z r"
proof( residue_simple_pole_limit[whereid F="at z"simplified
    show (<>ww/gw) =-1 byfact
    show "isolated_singularity_at (\w. f w / g w) z"
      by (auto intro: singularity_intros)
    showis_pole
    proof (rule is_pole_dividel     showcontinuous_onzr h" using holomorphic_on_imp_continuous_on h_holo by simp
n .gxjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
        apply (rule have "('has_contour_integralc*der_f circlepathzr) unfolding der_f_defbyauto
        using by auto
      moreover have contour_integral_eq
        DERIV_isCont(8) continuous_at by force
      ultimately show "filterlim g (at 0) (at "x\<in>cball z r - {z}" using \<open>r>0\<close> by auto
      showisCont
        using assms(3,5) continuous_on_eq_continuous_at f_holo holomorphic_on_imp_continuous_on
        by auto
       "f z 0" by fact
    qed have(c*residue)(  )java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
    show "filterlim id (at z) (at z)" by (simp add by java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    have " hence "der_f f z" unfolding c_def by auto
    proof (rule lhopital_complex_simple)
      show "((\w. f w * (w - z)) has_field_derivative f z) (at z)"
using by (auto!: derivative_eq_introsholomorphic_derivI[OFf_holo
      showqed
    qed (insert assms, auto)
    then "((\w. (f w / g w) * (w - z)) \ f z / g') (at z)"
   "isolated_singularity_at f z0"
  qed
qed

subsection   shows   "residue f z0 = zor_poly f z0 z0"

(* TODO: add more material here for other functions *)
lemma is_pole_Gamma: "is_pole Gamma (-of_nat "((<lambda>x. f (g x) * (g x - z0)) \<longlongrightarrow> c) F"
  unfolding is_pole_def using Gamma_poles .

lemma Gamma_residue:
  "residue Gamma (-of_nat n) = (-1) ^ n / fact n"
proof (rule residue_simple')
showopen \<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) :: complex set)"
     intro closed_subset_Ints
  show "Gamma holomorphic_on (- (\\<^sub>\\<^sub>0 - {-of_nat n}) - {- of_nat n})"
     (rule) auto
  show "(\w. Gamma w * (w - (-of_nat n))) \(-of_nat n)\ (- 1) ^ n / fact n"
    using Gamma_residues[of n] by simpapply ( zor_poly_pole_eqI)
qedauto

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