Wu Chong-shi a0a1a2( a3) Cauchya4a5a6a7 a81a9 a10a11a12 ( a13) Cauchy a14a15a16a17 §4.1 Cauchy a18a19a20a21 a22a23a24a25a26 Cauchy a27a28a29a30 a31f(z)a32a33a34 G a35a36 a37a38a39a40a41a42a43 G a36a44a45 C a32a46a47a48a49a50a51 a43 a a52 G a53a54 a55a43a56 f(a) = 12pii contintegraldisplay C f(z) z ?adz, a57 a35a58a46a59a51a60 C a36a61a62a63 a64 a65G a53 a66a67|z?a| < r( a68a694.1a43a70a71a67a72|z?a| = r a65G a53)a43a56a73a74a75a76a77a33a34a36 Cauchya78a79 a43a80 contintegraldisplay C f(z) z ?adz = contintegraldisplay |z?a|=r f(z) z ?adz, a814.1 a82a83a84a85a86Cauchya87a88a89a90 a91a92a93a94a95 r a36a96a97a98a99 a43a100a101a102 r → 0 a63a103a52 limz→a(z ?a) f(z)z ?a = f(a), a104a105 a79 3.1 a43a106a107a108 1 2pii contintegraldisplay C f(z) z ?adz = f(a). square Wu Chong-shi §4.1 Cauchya4a5a6a7 a82a9 a109a23a24a25a26 Cauchy a27a28a29a30 a110a111 a98a45a33a34 a43a112a113a114 a31f(z) a65a115a37a116a117a118a119C a120a121C a122 ( a123a124a98a125a126 a55) a37a38a39a40 a63a127a128a129 a43a130a65a131a132 1 2pii contintegraldisplay C f(z) z ?adz, a57 a35aa52C a122 a54 a55a43 a58a46a59a51C a36a133a62a32a134a135a136a137a62 a43a138a139 a98a125a126 a55 a36a61a62 a43a140 a69 4.2a63 a65C a122a141a66 a54a142a143a144 a55 a52 a67a145a43 R a52a146a147a36a96 a67C R a43 a148a149a43a110a111 C a150 CR a151 a123 a118 a36 a75a76a77 a33a34 a43a73a74a80 a45a33a34a36 Cauchya58a46a152a153 a43a80 a814.2 a154a83a84a85a86Cauchya87a88a89a90 1 2pii contintegraldisplay CR f(z) z ?adz + 1 2pii contintegraldisplay C f(z) z ?adz = f(a), a148a155 a58a46a59a51CR a36a133a62a32a156a135a136a137a62a63 a157a113 R a158a159a96 a43a148 a142 a92a93a160a161a106a95 R a36a162a163a96a97a98a99 a43 a111 a32 a43a101a102R → ∞a43a164a108a165 1 2pii contintegraldisplay C f(z) z ?adz = f(a)? limR→∞ bracketleftbigg 1 2pii contintegraldisplay CR f(z) z ?adz bracketrightbigg . a140a93f(z) a166a158a167a168a169 a105 a79 3.2a36 a113a170a43a56a101 a143 a131a132a171 a60a96 a67C R a36a58a46a36a172a173 a38a43 lim R→∞ bracketleftbigg 1 2pii contintegraldisplay CR f(z) z ?adz bracketrightbigg = K, K = limz→∞z · f(z)z ?a = f(∞). a103 a91a43 1 2pii contintegraldisplay C f(z) z ?adz = f(a)?K. a174a175a160K = 0 a135 a43a106a108a165a176 a109a23a24a25a26 Cauchy a27a28a29a30 a176a140a93 f(z) a65a115a37a116a117a118a119 C a120a121 C a122a39a40a43a177a160 z → ∞ a135 a43f(z) a54a178a129a179 a111 0a43a56 Cauchy a58a46a152a153 f(a) = 12pii contintegraldisplay C f(z) z ?adz a180a161a181a182a43a91a183 a a52C a122a54a55a43a58a46a59a51 C a52a134a135a136a137a62a63 Wu Chong-shi a0a1a2( a3) Cauchya4a5a6a7 a83a9 §4.2 a184a185a186a187a188a189a190a191a187 a192Cauchy a58a46a152a153 a43a101 a143a193a194 a171 a54a142a195 a113a92a196a176a140a93f(z) a65 G a35 a39a40a43a56a65 G a53f(z) a36a197a198a199a200 a42 f(n)(z) a201a202 a65a43 a203a177 f(n)(z) = n!2pii contintegraldisplay C f(ζ) (ζ ?z)n+1dζ, a57 a35C a32Ga36a61a62a44a45 a43 z a52G a53a197a204a54 a55a43a140 a69 4.3a63 a64 a205a206a170fprime(z) a63a103a52 f(z + h)?f(z) h = 1 2pii 1 h contintegraldisplay C bracketleftbigg f(ζ) ζ ?z ?h ? f(ζ) ζ ?z bracketrightbigg dζ = 12pii contintegraldisplay C f(ζ) (ζ ?z ?h)(ζ ?z)dζ, a814.3 a207a208a209a210a89a90 a211 a172a173h → 0a43a212a213a138a52fprime(z)a43a164a214a213a215a58 a41a42 a36a172a173a52 f(ζ)/(ζ ?z)2 a63a52a216 a107a217a65 a58a46a218a219 a170 a172a173 a117a220a43a221a222a223a224 contintegraldisplay C f(ζ)dζ (ζ ?z ?h)(ζ ?z) ? contintegraldisplay C f(ζ)dζ (ζ ?z)2 = h contintegraldisplay C f(ζ)dζ (ζ ?z ?h)(ζ ?z)2. a104a111f(ζ)a65C a120 a76a225a43a100a65C a120 a80|f(ζ)| ≤ M a43 a31za165C a36a226a227a228a229a52δa43la52C a36a230a231 a43a56a80 vextendsinglevextendsingle vextendsinglevextendsingle contintegraldisplay C f(ζ) (ζ ?z ?h)(ζ ?z)dζ ? contintegraldisplay C f(ζ) (ζ ?z)2dζ vextendsinglevextendsingle vextendsinglevextendsingle ≤ |h|· Ml δ2(δ ?|h|) → 0, a103 a91a43 a58a46a218a219 a170 a172a173 a117a220a43 fprime(z) = 12pii contintegraldisplay C f(ζ) (ζ ?z)2dζ. a232a149a43a101 a143 a170a171 fprimeprime(z) = lim h→0 fprime(z + h)?fprime(z) h = limh→0 1 2pii 1 h contintegraldisplay C bracketleftbigg f(ζ) (ζ ?z ?h)2 ? f(ζ) (ζ ?z)2 bracketrightbigg dζ = lim h→0 1 2pii contintegraldisplay C 2ζ ?2z ?h (ζ ?z ?h)2(ζ ?z)2f(ζ)dζ = 2! 2pii contintegraldisplay C f(ζ) (ζ ?z)3dζ. a140a91a233a225a43a138a101a170a171 f(n)(z) a63 square star a148a142 a92a93a234a217a43 a54a142 a75a235a41a42a43a157a113a65 a54a142a33a34a35a54a199a200 a42a183a183 a202 a65 ( a236a237a238a239a240a241a242a243a244 a245a246) a43a56a247 a36a197a198a199a200 a42a248 a202 a65a43a203a177a248 a32 a148 a142a33a34a53a36 a39a40a41a42 a63 star a65a249a235a41a42a35 a203a250a140a91 a63a251a252 a203a221a253a104fprime(x) a36a202 a65 a193a194 a171 fprimeprime(x) a36a202 a65 a63 star a75a235a41a42a35f(z) a65a54a33a34a35 a183a183a101 a200 (a138a39a40) a32a54a142a254a255a36 a113a170 a63 a249a235a41a42 a35fprime(x) a36a202 a65a157 a123a0 a160 xa65a42a1 a120 (a54a78a33a2a53) a235a3 a135 a110 f(x) a36 a113a170a43a164a75a235a41a42 a35fprime(z) a36a202 a65a56 a123a0a216 a65a4a5a6a7 a33a34a120 a110 f(z) a36 a113a170 a63 Wu Chong-shi §4.3 Cauchya8a4a5a9a10a11a12a4a5a13a14a15a16 a84a9 §4.3 Cauchy a17a18a19a18a19a20a21a18a19a188a184a185a22 a65 a120a54a23a99 a111a39a40a41a42 a255a199a200 a42 a152a153a36 a107a217a24a25 a35 a43f(z) a36 a39a40a26a157 a32a163 a130a65a176 (1) f(z)a101 a27 Cauchy a58a46a152a153a28a29a30 (2)f(z)a65C a120 a76a225 a63a103 a91a43 a195 a75 a120 a7 a36a31a32 a43a106a101 a143 a107a217a176a65 a54a33a46 a47a48a49a36 (a116a117a34a221a116a117) a50a51C a120 a76a225 a36 a41a42 φ(ζ) a151a35 a181 a36a58a46 f(z) = 12pii integraldisplay C φ(ζ) ζ ?zdζ (a36a52Cauchy a37a27a28) a32a50a51 a122a55 z a36 a39a40a41a42a43 fprime(z)a101a77a24 a58a46a218a219 a170 a200 a164a108a165a43 f(p)(z) = p!2pii integraldisplay C φ(ζ) (ζ ?z)p+1dζ. a38 a131a132 a58a46 f(z) = 12pii contintegraldisplay |ζ|=1 ζ? ζ ?zdζ, |z| negationslash= 1. a39 a148 a32a54a142 Cauchya40a58a46a63a103a52 a65|ζ| = 1 a120ζ? = 1/ζ a43a100 f(z) = 12pii contintegraldisplay |ζ|=1 1 ζ(ζ ?z)dζ. a160|z| > 1 a135 a43a91 a58a46 a101 a143 a27 Cauchy a58a46a152a153 a131a132a43 f(z) = 12pii contintegraldisplay |ζ|=1 1 ζ bracketleftbigg 1 ζ ?z bracketrightbigg dζ = ?1z. a1600 < |z| < 1 a135 a43 f(z) = 12pii contintegraldisplay |ζ|=1 1 z bracketleftbigg 1 ζ ?z ? 1 ζ bracketrightbigg dζ = 0. a41a42a43a171a43a91a92a93a110a111 z = 0a180a181a182 a63a44 a117 a143a120 a92a93a43a106a80 f(z) = 12pii contintegraldisplay |ζ|=1 ζ? ζ ?zdζ = ? ? ? ?1z, |z| > 1, 0, |z| < 1. a104a91a101 a68 a43 f(z)a65|z| negationslash= 1a183a39a40a43a45a46 ζ? a65a47a6a7a221a39a40 a63 Wu Chong-shi a0a1a2( a3) Cauchya4a5a6a7 a85a9 a48a27Cauchy a40a58a46 a43a106a101 a143a193 a171a49a50a51 a27a28 a26a39a52a53 a63 a54a55 4.2 a31 1. f(t,z)a238ta56z a242a57a58 a245a246a43 t ∈ [a,b]a43z ∈ Ga43 2. a59a60[a,b]a61a242a62a63ta64 a43f(t,z) a238Ga61a242a65a64a243a244 a245a246a43 a56F(z) = integraldisplay b a f(t,z)dta65G a53a32 a39a40 a36 a43a177 Fprime(z) = integraldisplay b a ?f(t,z) ?z dt. a64 a103a52f(t,z) a65G a120 a39a40a43a100a110a111 G a53a36a197a198a54 a55 z a43Cauchy a58a46a152a153 a181a182a43 f(t,z) = 12pii contintegraldisplay C f(t,ζ) ζ ?z dζ. a66a67F(z) a36a78a68 a43a203a69a70 a58a46a71a72 (a103a52f(t,z)a76a225) a43a108 F(z) = integraldisplay b a dt 2pii contintegraldisplay C f(t,ζ) ζ ?z dζ = 1 2pii contintegraldisplay C 1 ζ ?z bracketleftBiggintegraldisplay b a f(t,ζ)dt bracketrightBigg dζ. a148 a32a54a142 Cauchya40a58a46 a43 integraldisplay b a f(t,z)dta76a225a43a100F(z)a52G a53a36 a39a40a41a42a43a177 Fprime(z) = 12pii contintegraldisplay C 1 (ζ ?z)2 bracketleftBiggintegraldisplay b a f(t,ζ)dt bracketrightBigg dζ = integraldisplay b a bracketleftbigg 1 2pii contintegraldisplay C f(t,ζ) (ζ ?z)2dζ bracketrightbigg dt = integraldisplay b a ?f(t,z) ?z dt. square a73a161a43a148 a142 a92a196a74a75a27a111 integraldisplay C f(t,z)dta63a148a135a94a160a113a170 C a32a46a47a48a49a50a51a43a160 ta65C a120a235a76a43 z ∈ Ga135 a43f(t,z) a32 ta150z a36 a76a225a41a42 a63 a107a217 a36a137 a220a95 a120 a7a77a232 a63 Wu Chong-shi §4.4 a78 a79 a80 a79 a86a9 a10a11a12 ( a81) a82 a83 a84 a85 a86a87a88a246a43a89a90 a238a91 a88a246a43 a238a243a244 a245a246 a242a92a93a94a242a95a96a97a98a99a100a63 a101a102a103a104a245a246 a56 a89a105a245a246a106 a238a107a91 a88a246a108a109 a242a63 a110a111a245a246a88a246a112a113 a56a114 a111a245a246 a242a115a116 a176a117a118 a56a119a120a242a121a122a63 §4.4 a123 a187 a124 a187 a54a125 a75a42a126a42 u0 + u1 + u2 +···+ un +··· = ∞summationdisplay n=0 un. a102u n a36 a249a127 a150a128 a127 a46 a175 a52 αn a95βn,a56 ∞summationdisplay n=0 un = ∞summationdisplay n=0 αn + i ∞summationdisplay n=0 βn. a54a142 a75a42a126a42summationtextu n a129 a47a130a131a111a132 a142 a249a42a126a42summationtextα n a150 summationtextβ n a43a133a134a135a161 a63 a136a137a138a137a26a139a140a141a142a143 a140a93a126a42 a36 a127 a46a150 Sn = u0 + u1 + u2 +···+ un a151a35 a181 a36a72a144{Sn}a145a146 a43a56 a36 a126a42summationtextu n a145a146 a43 a72a144{Sn}a36a172a173 S = limn→∞Sn a43a36a52 a126a42summationtextu n a36a150 ∞summationdisplay n=0 un = limn→∞Sn. a147a56a43a126a42summationtextu n a32a148a149a36a63 a126a42 a36a145a146 a26a43 a32 a27a247 a36 a127 a46a150a72a144a36a145a146 a26 a78a68a36a63a103 a91a43a73a74 a72a144a145a146a36a150 a113 a33a151 a43a101 a143a152 a171a126a42 a145a146a36a150 a113 a33a151 Cauchya150 a113 a33a151 a176 a62a153a154 a108ε > 0a43a155a156a157a158a246na43a159 a59a60a62a153 a157a158a246pa43a160 |un+1 + un+2 +···+ un+p| < ε. a174a175 a32 a43a102 p = 1a43a106a108a165a126a42 a145a146a36a161 a113 a33a151 limn→∞un = 0. star a65a221a162a235a170a150a71a72a36a163a164a219 a43a101 a143a165a145a146 a126a42a203a166 a63 u1 + u2 + u3 + u4 +··· = (u1 + u2) + (u3 + u4) +···. Wu Chong-shi a0a1a2( a167) a168 a169 a80 a79 a87a9 star a140a93a126a42 ∞summationtext n=0 |un|a145a146a43a56a36a126a42 ∞summationtext n=0 un a170 a110 a145a146a63 a170 a110 a145a146a36 a126a42 a54a78a32a145a146a36a63 |un+1 + un+2 +···un+p| ≤ |un+1|+|un+2|+··· +|un+p|. a133a134a43 a54a142a145a146a36 a126a42a43a171a221 a54a78a32 a170 a110 a145a146a36a63 star a170 a110 a145a146 a126a42 a36a172 a175a220 a63 stara173a174a175a176a177 a178|un|<vn a43a164 ∞summationtext n=0 vn a145a146 a43a56 ∞summationtext n=0 |un|a145a146 parenleftbiga138 ∞summationtext n=0 un a170 a110 a145a146 parenrightbig a63 a178|un| > vn,a164 ∞summationtext n=0 vn a148a149 a43a56 ∞summationtext n=0 |un|a148a149a63 stara173a179a175a176a177 a178a202 a65a95 n a98a99a36a180 a42ρa43a56 a160 vextendsinglevextendsingle vextendsingleun+1u n vextendsinglevextendsingle vextendsingle < ρ < 1a135a43a126a42 ∞summationtext n=0 un a170 a110 a145a146a30 a160 vextendsinglevextendsingle vextendsingleun+1u n vextendsinglevextendsingle vextendsingle > ρ > 1a135a43a126a42 ∞summationtext n=0 |un|a148a149a63 ? a115a64a181 a90 a120a242a182a183 a176 a59a60 a101a102a184 a107 a88a246a43a185 a98|un+1/un|a242a97a98a186a186a94a115un a242a97a98a187 a65a188 a102a43 a236a237a189a107a115a64a181 a90 a120a190a191a192a193a194a181a195 summationtext|u n|a242a196a197a198a63 ? ρa242 a155a156 a198a199 ? a200a119a201a242a202a203a238 a159 a107a204a242a205a206a97a98 a43a207d’Alembert a181 a90 a120a63 star d’Alembert a175a176a177 a140a93 lim n→∞|un+1/un| = l < 1, a56 ∞summationtext n=0 un a170 a110 a145a146a30 a140a93 lim n→∞ |un+1/un| = l > 1,a56 ∞summationtext n=0 |un|a148a149a63 ? d’Alemberta181 a90 a120a242a182a183 a176 a100a208a209a210 a43a211 a61a212a205a206a213a94a115 a211 a115a64a181 a90 a120a214a242 ρa210a188a187 a65a63 ? d’Alemberta181 a90 a120a242a215a183 a176a216 a107a217a122a242a218a219a181 a90a88a246 a242a196a197a56a220a221 a43a207 a107 limn→∞|un+1/un| a181a195 a88a246 ∞summationtext n=0 |un| a242a196a197 a43a222 a107 lim n→∞ |un+1/un| a181a195 a88a246 a242a220a221a63a236a237a59a60 limn→∞|un+1/un| ≥ 1a223 lim n→∞ |un+1/un| ≤ 1a242a224a97a225a217a226a227a228a181a195 a43a229a230 limn→∞|un+1/un| = lim n→∞ |un+1/un| = limn→∞|un+1/un|. ? Cauchya181 a90 a120a242a182a183a225a238a231a232a122a100a181a232 limn→∞|un|1/n a210a181a195 a88a246 a238a233a234a59a196a197a63 star Cauchy a175a176a177 a140a93 lim n→∞|un| 1/n < 1,a56a126a42 ∞summationtext n=0 |un|a145a146a30 a140a93 lim n→∞|un| 1/n > 1,a56a126a42 ∞summationtext n=0 un a148a149a63 Wu Chong-shi §4.4 a78 a79 a80 a79 a88a9 a235a236a139a140a138a137a26a53a237a176 1. a162a70a71a72a63a238a140 u0 + u1 + u2 + u3 + u4 +··· = u0 + u1 + u2 + u4 + u3 + u6 + u8 + u5 +···. 2. a101a143a239a54a142 a170 a110 a145a146 a126a42a240a181a241 a142a242 a126a42a43a243 a142a242 a126a42a180 a170 a110 a145a146a63 ∞summationdisplay n=0 un = ∞summationdisplay n=0 u2n + ∞summationdisplay n=0 u2n+1. 3. a132a142 a170 a110 a145a146 a126a42a134 a58 a180a161 a170 a110 a145a146 a43 summationdisplay k uk · summationdisplay l vl = summationdisplay k,l ukvl. a148a155 a36a244a58a32a54a142 a4 a195 a126a42 u0v0 + u0v1 + u0v2 + u0v3 + ··· + u1v0 + u1v1 + u1v2 + u1v3 + ··· + u2v0 + u2v1 + u2v2 + u2v3 + ··· + u3v0 + u3v1 + u3v2 + u3v3 + ··· + ···. a57 a170 a110 a145a146 a26 a204a245a246 a101 a143a247a248a197a204a134a72 a170 a150 a43a57a38a221a235 a63a238 a140a101 a247 k + l = na36a96a97a134a72a249a144 a43 ∞summationdisplay k=0 uk · ∞summationdisplay l=0 vl = ∞summationdisplay n=0 wn, wn = nsummationdisplay k=0 ukvn?k. a140a93 a173 a111a148a250a170 a150a71a72 ( a148a250a170 a150a71a72a162 a80a174a251 a36a195 a113a26) a43a56 a244 a220 a36a33a151a252 a101 a143a253a254 a181a176 summationtextu k,summationtext vl a248a145a146 a43a177a57 a35 a134 a54 a170 a110 a145a146a30 a34 summationtextu k, summationtextv l a150 summationtextw n a248 a145a146a63 Wu Chong-shi a0a1a2( a167) a168 a169 a80 a79 a89a9 §4.5 a186 a187 a124 a187 a255a137a138a137a26a139a140a53 a31uk(z) (k = 1,2,···) a65 a33a34 G a35 a80 a78a68a63 a140a93a110a111G a35a54 a55z 0 a43a126 a42 ∞summationtext k=1 uk(z0)a145a146 a43a56 a36 a126a42 ∞summationtext k=1 uk(z)a65z0 a55a145a146a63 a133a134a43a140a93 ∞summationtext k=1 vk(z0)a148a149a43a56a36a126a42 ∞summationtext k=1 vk(z)a65z0 a55a148a149a63 a140a93a126a42 ∞summationtext k=1 uk(z)a65a33a34G a53 a243 a54 a55a248 a145a146 a43a56 a36 a126a42a65 G a53a145a146a63 a57 a150 a41a42S(z) a32G a53 a36 a37a38a41a42 a63 a255a137a138a137a26a0a1a139a140a53 a140a93a110a111 a197a204a2a78a36ε > 0,a202 a65 a54a142 a95z a98a99a36N(ε),a3 a160n > N(ε) a135 a43 vextendsinglevextendsingle vextendsingleS(z)? nsummationtext k=1 uk(z) vextendsinglevextendsingle vextendsingle < εa43a56a36a126a42 ∞summationtext k=1 uk(z)a65G a53a54a178a145a146a63 a255a137a138a137a0a1a139a140a26 a175a176a177 (1)a4a5a6 a27 a78a68 a43 (2) Weierstrass a36M a172 a175a220 a63 Weierstrassa36M a172 a175a220a176 a178 a65 a33a34Ga53|uk(z)| < ak a43aka95za98a99 a43a164 ∞summationtext k=1 ak a145a146a43a56 ∞summationtext k=1 uk(z) a65G a53 a170 a110a164a177 a54a178a145a146a63 a0a1a139a140a138a137a7a22a8a9a10a11a53a237a12 1. a13a14a15 a16a17 uk(z) a18 G a19a20a21a22a23a24 ∞summationtext k=1 uk(z) a18 G a19a25a26a27a28a22a29a30a31a32a24 S(z) = ∞summationtext k=1 uk(z)a33a18G a19a20a21a34 a35a36a37a38a39a40a41a42 a22a16a17a23a24a43a44a25a45a46a47a20a21a32a24a22a29a25a26a27a28a23a24a48a49a50a45a51a52a53(a54a55 a56 a22 a57a51a52a53a58a59 a57a51a23a24a31a58a48a49a60a61a62a63) a22 limz→z 0 ∞summationdisplay k=1 uk(z) = ∞summationdisplay k=1 limz→z 0 uk(z). 2. a64a65a66a67a68 a69C a47a70a71G a19a43a25a72a73a74a75a76a77a78a22a16a17uk(z) (k = 1,2,···)a47C a79a43a20 a21a32a24a22a29a80a81 C a79a25a26a27a28a43a23a24 ∞summationtext k=1 uk(z)a48a49a50a45a51a82a73 integraldisplay C ∞summationdisplay k=1 uk(z)dz = ∞summationdisplay k=1 integraldisplay C uk(z)dz. 3. a64a65a66a83a84(Weierstrassa85a86) a69uk(z)(k = 1,2,···)a18 G a87a88a89a90a91a22 ∞summationtext k=1 uk(z)a18G a87 a25a26a27a28a22a29a92a23a24a93a31 f(z)a47G a19a43a90a91a32a24a22 f(z)a43a94a95a96a24a48a49a97 ∞summationtext k=1 uk(z)a50a45a51a96a24 Wu Chong-shi §4.5 a98 a99 a100 a99 a10110a102 a103a104 a22 f(p)(z) = ∞summationdisplay k=1 u(p)k (z), a51a96a24a105a43a23a24a18 G a19a43a106a25a107a70a71a87a25a26a27a28a34 a35a108a37a38 a43a109a110a111a112a113a114a34 Wu Chong-shi a115a116a117( a118) a119 a120 a100 a99 a10111a102 §4.6 a121a122a123a124a125a126a127a128a129a130a131 1. a13a14a15 a16a17 uk(z) a18 G a19a20a21a22a23a24 ∞summationtext k=1 uk(z) a18 G a19a25a26a27a28a22a29a30a31a32a24 S(z) = ∞summationtext k=1 uk(z)a33a18G a19a20a21a34 a132 a69z0 a133 G a19a25a134a22 |S(z)?S(z0)| = |S(z)?Sn(z) + Sn(z)?Sn(z0) + Sn(z0)?S(z0)| ≤ |S(z)?Sn(z)| +|Sn(z)?Sn(z0)| +|S(z0)?Sn(z0)|. a135a136 a23a24 summationtextu k(z)a43a25a26a27a28 a37 a48a137a22a106a138 ε > 0a22a139a48a49a140 a104 N(ε) a22a141a142n > N(ε)a143a22 |S(z)?Sn(z)| < ε3, |S(z0)?Sn(z0)| < ε3. a144a145S n(z) = nsummationtext k=1 uk(z)a47G a19a43a20a21a32a24a22a146a80a81a147a25 a36 ε > 0 a22a139a148a18 δ > 0a22a141a142|z ?z0| < δ a143a22 |Sn(z)?Sn(z0)| < ε3. a149a150 a49a79a151a17a22a152 a103a104a12 a80a81a106a138a43 ε > 0a22a148a18δ > 0a22a153a154|z ?z0| < δ a22a152a155 |S(z)?S(z0)| < ε. S(z)a18z0 a134a20a21a34a97a81z0 ∈ Ga106a156a22a157a49S(z)a18G a19a20a21a34 square 2. a64a65a66a67a68 a69 C a47a70a71 G a19a43a25a72a73a74a75a76a77a78a22a16a17 uk(z) (k = 1,2,···) a47 C a79a43 a20a21a32a24a22a29a80a81 C a79a25a26a27a28a43a23a24 ∞summationtext k=1 uk(z)a48a49a50a45a51a82a73 integraldisplay C ∞summationdisplay k=1 uk(z)dz = ∞summationdisplay k=1 integraldisplay C uk(z)dz. a132 a135a136a37a38 1 a137 ∞summationtext k=1 uk(z)a47a77a78C a79a43a20a21a32a24a22a146a82a73 integraldisplay C ∞summationtext k=1 uk(z)dz a148a18a22a158a155 integraldisplay C ∞summationdisplay k=1 uk(z)dz = integraldisplay C nsummationdisplay k=1 uk(z)dz + integraldisplay C Rn(z)dz = nsummationdisplay k=1 integraldisplay C uk(z)dz + integraldisplay C Rn(z)dz, a30a87Rn(z) = S(z)?Sn(z) = ∞summationtext k=1 uk(z)a34 a135a136 a23a24a43a25a26a27a28 a37 a22a80a81a106a138a43ε > 0a22a148a18N(ε) > 0a22 a153a154n > N(ε)a22a152a155 |S(z)?Sn(z)| = |Rn(z)| < ε. Wu Chong-shi §4.6 a159a160a161a162a100a99a163a164a165a166a167 a10112a102 a145 a92 vextendsingle vextendsinglevextendsingle vextendsingle integraldisplay C Rn(z)dz vextendsinglevextendsingle vextendsinglevextendsingle ≤ integraldisplay c |Rn(z)|·|dz| < εl, la47C a43a168a169a34a170a171a172a52a53a22a173a103 limn→∞ integraldisplay C Rn(z)dz = 0. a157a49 integraldisplay C ∞summationdisplay k=1 uk(z)dz = limn→∞ nsummationdisplay k=1 integraldisplay C uk(z)dz = ∞summationdisplay k=1 integraldisplay C uk(z)dz. square 3. a64a65a66a83a84(Weierstrassa85a86) a69uk(z)(k = 1,2,···)a18 G a87a88a89a90a91a22 ∞summationtext k=1 uk(z)a18G a87 a25a26a27a28a22a29a92a23a24a93a31f(z)a47G a19 a43a90a91a32a24a22f(z)a43a94a95a96a24a48a49a97 ∞summationtext k=1 uk(z)a50a45a51a96a24 a103a104 a22 f(p)(z) = ∞summationdisplay k=1 u(p)k (z), a51a96a24a105a43a23a24a18 G a19a43a106a25a107a70a71a87a25a26a27a28a34 a132 a109a110a174a73a175a176a73a34 (1)a109a110f(z) = ∞summationtext k=1 uk(z)a18G a19a90a91a34 a145 a133 uk(z)a90a91a22a146 a135a136 Cauchy a82a73a177a178 uk(z) = 12pii contintegraldisplay C uk(ζ) ζ ?z dζ, f(z) = ∞summationdisplay k=1 uk(z) = 12pii ∞summationdisplay k=1 contintegraldisplay C uk(ζ) ζ ?zdζ = 12pii contintegraldisplay C ∞summationdisplay k=1 uk(ζ) dζζ ?z = 12pii contintegraldisplay C f(ζ) ζ ?zdζ. a97 a37a38 1 a22a137f(ζ)a47C a79a43a20a21a32a24a22a146 f(z)a90a91a34 (2)a109a110a48a49a50a45a51a96a24a34 a69Gprime a133 G a19a106a25a107a70a71a22 Cprime a47a179a43a180a181a22a29 f(z)a18Gprime a87a90a91a22a146a155 f(p)(z) = p!2pii contintegraldisplay Cprime f(ζ) (ζ ?z)p+1dζ Wu Chong-shi a115a116a117( a118) a119 a120 a100 a99 a10113a102 = p!2pii contintegraldisplay Cprime ∞summationdisplay k=1 uk(ζ) dζ(ζ ?z)p+1 = ∞summationdisplay k=1 p! 2pii contintegraldisplay Cprime uk(ζ) (ζ ?z)p+1dζ = ∞summationdisplay k=1 u(p)k (z). (3)a109a110a23a24 ∞summationtext k=1 u(p)k a18Gprime a87a25a26a27a28a34 a16a182a157a183a22a18G a19a139a48a49a184a172a25 a36 a107a70a71Gprimeprime a22a141 a103Gprime a185a186 a18G primeprime a19a34 a187Gprime a43a180a181Cprime a104Gprimeprime a43a180a181Cprimeprime a43a188a189a190a191 a133 δ a34a172z a133 Gprime a19a25a134a22ζ a133 Cprimeprime a79a43a192a134a34 a193 Weierstrass a194a195 a97a79a196a43 a37a382 a22a48a49a197a198a22 ∞summationtext k=1 uk(ζ)a18a180a181Cprimeprime a79a25a26a27a28a22a173a80a81a106a156a138a198a43ε > 0a22a139 a148a18N(ε)a22a141a142n > N(ε)a143a22a199a155 vextendsingle vextendsinglevextendsingle vextendsinglevextendsingle n+qsummationdisplay k=n+1 uk(ζ) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle < ε, q a47a106a156a200a201a24a34a81a47a22vextendsingle vextendsinglevextendsingle vextendsinglevextendsingle n+qsummationdisplay k=n+1 u(p)k (ζ) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle n+qsummationdisplay k=n+1 p! 2pii contintegraldisplay Cprimeprime uk(ζ) (ζ ?z)p+1dζ vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle p! 2pii contintegraldisplay Cprimeprime n+qsummationdisplay k=n+1 uk(ζ) dζ(ζ ?z)p+1 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle ≤ p!2pi contintegraldisplay Cprimeprime vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle n+qsummationdisplay k=n+1 uk(ζ) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle |dζ| |ζ ?z|p+1 < p!2pi εδp+1lprimeprime, lprimeprime a133 Cprimeprimea43a169a202. a157a49 ∞summationtext k=1 u(p)k (z)a18(G a19a43a106a25a107a70a71)Gprime a87a25a26a27a28a34 square