a0 a1 star a2a3a4a5a6a7a8a9a10 2 star §4.5(a11a12a13a14a15a16a17a18a19a20a21) a22a6a7 a0a1a2 a3 a4 a5 a6 a71 a8 a9a10a11 a12 a13 a14 a15 a16a17a18a19a20a21a22a23a24a18a19a20a23a25a26a27a19a28a29a30a31a28a32a33a34a35a36a37a38 a39a40a41a42a27a19a43a21a44a27a19a45a23a46a24a18a19a47a48a28a38 a49a50a27a19a18a19a51a52a43a53a50a27a19a28a54a55a56a57a58a43a59a60a28a61a62a38 §4.1 a63 a64 a65 a64 a66a67 a68a69a70a69 u0 + u1 + u2 +···+ un +··· = ∞summationdisplay n=0 un. a71u n a72a73a74a75a76a74a77a78a79 αn a80 βn,a81 ∞summationdisplay n=0 un = ∞summationdisplay n=0 αn + i ∞summationdisplay n=0 βn. a82a83 a68a69a70a69summationtextu n a84a85a86a87a88a89 a83 a73 a69a70a69 summationtextα n a75 summationtextβ n a20a90a91a92a93a38 a94a95a96a95a97a98a99a100a101a102 a103a104a70a69 a72a74a77a75 Sn = u0 + u1 + u2 +···+ un a105a106a107 a72a108a109 {Sn}a110a111 a20 a81a112 a70a69summationtextu n a110a111 a20 a108a109 {Sn} a72a113a114 S = limn→∞Sn a20a112 a79 a70a69summationtextu n a72a75 ∞summationdisplay n=0 un = limn→∞Sn. a115 a81 a20a70a69summationtextu n a116a117a118a72 a38 a70a69 a72 a110a111a119 a20 a116a120a121a72a74a77a75a108a109a72 a110a111a119a122a123 a72 a38a124a125a20a126a127 a108a109 a110a111 a72a128a129a130a131 a20a132 a133a134a135a70a69 a110a111 a72a128a129a130a131 Cauchy a128a129a130a131 a56a136a137a138a47ε > 0a20a139a140a141a142a19na20a143a144a145a136a137 a141a142a19pa20a146 |un+1 + un+2 +···+ un+p| < ε. a147 a78a116 a20a71 p = 1a20a148a149a150a70a69 a110a111 a72a151a129a130a131 limn→∞un = 0. star a152a153a154a155a156 a75a157a108a72a158a159a160 a20a132a133a161 a110a111 a70a69a162a163a38 u1 + u2 + u3 + u4 +··· = (u1 + u2) + (u3 + u4) +···. §4.1 a164 a165 a166 a165 a1672a168 star a103a104a70a69 ∞summationtext n=0 |un|a110a111a20a81a112a70a69 ∞summationtext n=0 un a169a170 a110a111 a38 a169a170 a110a111 a72 a70a69 a82 a122 a116 a110a111 a72 a38 |un+1 + un+2 +···un+p| ≤ |un+1|+|un+2|+··· +|un+p|. a90a91a20 a82a83 a110a111 a72 a70a69a20a171 a153 a82 a122 a116a169a170 a110a111 a72 a38 star a169a170 a110a111 a70a69 a72a172a78a173 a38 stara174a175a176a177a178 a179|un|<vn a20a180 ∞summationtext n=0 vn a110a111 a20 a81 ∞summationtext n=0 |un|a110a111 parenleftbiga181 ∞summationtext n=0 un a169a170 a110a111 parenrightbiga38 a179|un| > vn,a180 ∞summationtext n=0 vn a117a118 a20 a81 ∞summationtext n=0 |un| a117a118 a38 stara174a182a176a177a178 a179a183a152 a80 na184a185 a72a186 a69 ρa20 a81 a187 vextendsinglevextendsingle vextendsingleun+1u n vextendsinglevextendsingle vextendsingle < ρ < 1a188a20a70a69 ∞summationtext n=0 un a169a170 a110a111a189 a187 vextendsinglevextendsingle vextendsingleun+1u n vextendsinglevextendsingle vextendsingle > ρ > 1a188a20a70a69 ∞summationtext n=0 |un| a117a118 a38 ? a54a190a191a22a60a28a192a193a56a144a145a39a40a194a46a18a19a20a195a35|un+1/un|a28a34a35a196a196a31a54un a28a34a35a197 a198a199a40a20a200a201a202a46a54a190a191a22a60a203a204a205a206a207a191a208 summationtext|u n| a28a209a210a211a38 ? ρa28a139a140a211a212 ? a213 a59a214a28a215a216a23a143a46a217a28a218a219a34a35a20a220d’Alemberta191a22a60a38 star d’Alembert a176a177a178 a103a104 lim n→∞|un+1/un| = l < 1,a81 ∞summationtext n=0 un a169a170 a110a111a189 a103a104 lim n→∞ |un+1/un| = l > 1,a81 ∞summationtext n=0 |un| a117a118 a38 ? d’Alemberta191a22a60a28a192a193a56a37a221a222a223a20a224a225a226a218a219a227a31a54a224a54a190a191a22a60a228a28 ρa223a199a197 a198a38 ? d’Alemberta191a22a60a28a229a193a56a230a46a231a62a28a232a233a191a22a18a19a28a209a210a43a234a235a20a220a46 limn→∞|un+1/un| a191a208a18a19 ∞summationtext n=0 |un| a28a209a210a20a236a46 lim n→∞ |un+1/un| a191a208a18a19a28a234a235a38a200a201a144a145 limn→∞|un+1/un| ≥ 1a237 lim n→∞ |un+1/un| ≤ 1a28a238a34a239a231a240a241a242a191a208a20a243a244 limn→∞|un+1/un| = lim n→∞ |un+1/un| = limn→∞|un+1/un|. ? Cauchya191a22a60a28a192a193a239a23a245a246a62a37a191a246 limn→∞|un|1/n a223a191a208a18a19a23a247a248a144a209a210a38 star Cauchy a176a177a178 a103a104 lim n→∞|un| 1/n < 1, a81 a70a69 ∞summationtext n=0 |un|a110a111a189 a103a104 lim n→∞|un| 1/n > 1, a81 a70a69 ∞summationtext n=0 un a117a118 a38 a0a1a2 a3 a4 a5 a6 a73 a8 star Gauss a176a177a178 a249a250a98a99a96a95a97a251a252a56 1. a154a253 a157a108 a38 u0 + u1+u2 + u3 + u4 +··· =u0 + u1 + u2 + u4 + u3 + u6 + u8 + u5 +···. 2. a132a133a254a82a83 a169a170 a110a111 a70a69a255a107a0 a83a1 a70a69a20a2 a83a1 a70a69a3 a169a170 a110a111 a38 ∞summationdisplay n=0 un = ∞summationdisplay n=0 u2n + ∞summationdisplay n=0 u2n+1. 3. a89 a83 a169a170 a110a111 a70a69a91a4a3a93 a169a170 a110a111 a20 summationdisplay k uk · summationdisplay l vl = summationdisplay k,l ukvl. a5a6 a72a7 a4 a116 a82a83a8a9 a70a69 u0v0 + u0v1 + u0v2 + u0v3 + ··· + u1v0 + u1v1 + u1v2 + u1v3 + ··· + u2v0 + u2v1 + u2v2 + u2v3 + ··· + u3v0 + u3v1 + u3v2 + u3v3 + ··· + ···. a10 a169a170 a110a111a119a11a12a13 a132a133a14a15a16 a11a17 a108 a156 a75 a20a10a18 a153a155 a38a19a103a132a14 k + l = n a72a20a21 a17 a108a22a109 a20 ∞summationdisplay k=0 uk · ∞summationdisplay l=0 vl = ∞summationdisplay n=0 wn, wn = nsummationdisplay k=0 ukvn?k. a103a104 a114a88 a5a23 a156 a75a157a108 (a5a23a156 a75a157a108a24a25 a147a26 a72 a9 a129 a119) a20a81 a7a173a72a130a131a27 a132a133a28a29a107a56 summationtextu k,summationtext vl a30 a110a111 a20a31a10a32a91 a82 a169a170 a110a111a189a33 summationtextu k, summationtextv l a75 summationtextw n a30 a110a111 a38 §4.2 a34 a6 a5 a6 a74 a8 §4.2 a35 a64 a65 a64 a36a95a96a95a97a98a99a251 a37u k(z) (k = 1,2,···)a152a38a39 G a32 a25 a122a123 a38a103a104 a170a88 G a32a82a40z0 a20a70 a69 ∞summationtext k=1 uk(z0)a110a111a20a81a112a70a69 ∞summationtext k=1 uk(z)a152z0 a40a110a111a38 a90a91a20a103a104 ∞summationtext k=1 vk(z0) a117a118 a20 a81a112 a70a69 ∞summationtext k=1 vk(z)a152z0 a40 a117a118 a38 a103a104a70a69 ∞summationtext k=1 uk(z)a152a38a39G a41a2a82a40 a30 a110a111 a20 a81a112 a70a69 a152 G a41a110a111 a38a10 a75a42 a69S(z) a116 G a41 a72a43 a18 a42 a69a38 a36a95a96a95a97a44a45a98a99a251 a103a104 a170a88 a16 a11a46a122 a72 ε > 0,a183a152 a82a83 a80 za184a185 a72 N(ε),a47 a187n > N(ε) a188 a20 vextendsinglevextendsingle vextendsingleS(z)? nsummationtext k=1 uk(z) vextendsinglevextendsingle vextendsingle < εa20a81a112a70a69 ∞summationtext k=1 uk(z)a152G a41 a82a48 a110a111 a38 a36a95a96a95a44a45a98a99a97 a176a177a178 (1)a49a50a51 a120 a122a123 a20 (2) Weierstrass a72 M a172a78a173 a38 Weierstrass a72 M a172a78a173 a56 a179a152a38a39Ga41|uk(z)| < ak a20ak a80 za184a185 a20a180 ∞summationtext k=1 ak a110a111 a20 a81 ∞summationtext k=1 uk(z) a152G a41 a169a170 a180a31 a82a48 a110a111 a38 a44a45a98a99a96a95a52a53a54a55a56a57a251a252a56 1. a58a59 a251 a103a104u k(z)a152Ga41 a60a61a20a70a69 ∞summationtext k=1 uk(z)a152Ga41 a82a48 a110a111 a20 a81 a10 a75a42 a69S(z) = ∞summationtext k=1 uk(z) a62 a152G a41 a60a61a38 a5 a83 a119a63a64a65a66a67 a20a103a104a70a69 a72 a2 a82 a163 a30a116 a60a61 a42 a69a20 a81 a82a48 a110a111 a70a69a132a133a68a163 a156 a113a114 (a33a69 a70a20 a71 a156 a113a114a72a80 a71 a156 a70a69 a75a72 a132a133a73 a253 a157a108 ) a20 limz→z 0 ∞summationdisplay k=1 uk(z) = ∞summationdisplay k=1 limz→z 0 uk(z). 2. a74a75a76a77a78 a37C a116 a38a39G a41 a72 a82 a130a77a79a80a81a82a83 a20a103a104u k(z) (k = 1,2,···)a116 C a84 a72 a60 a61 a42 a69a20 a81 a170a88 C a84 a82a48 a110a111 a72 a70a69 ∞summationtext k=1 uk(z)a132a133a68a163a156a4 a77 integraldisplay C ∞summationdisplay k=1 uk(z)dz = ∞summationdisplay k=1 integraldisplay C uk(z)dz. 3. a74a75a76a85a95 (Weierstrassa66a86) a37uk(z)(k = 1,2,···)a152 G a32 a43 a18a87a88a20 ∞summationtext k=1 uk(z)a152G a32 a82a48 a110a111 a20 a81 a125a70a69a91 a75 f(z) a116 G a41 a72 a87a88 a42 a69a20 f(z) a72a89a90a91 a69a132a133a92 ∞summationtext k=1 uk(z)a68a163a156 a91 a69 a0a1a2 a3 a4 a5 a6 a75 a8 a149a150a20 f(p)(z) = ∞summationdisplay k=1 u(p)k (z), a156 a91 a69a93 a72 a70a69 a152 G a41 a72 a16 a82a94 a38a39 a32 a82a48 a110a111 a38 a5a95 a119a63 a72a96a97a98a99a100a101 a38 §4.3 a102 a5 a6 a76 a8 §4.3 a103 a65 a64 a104a70a69a105 a186a116a106 a105a163 a79 a104 a42 a69 a72a42 a69a163a70a69a20 ∞summationdisplay n=0 cn(z ?a)n = c0 + c1(z ?a) + c2(z ?a)2 + ···+ cn(z ?a)n +···. a5 a116 a82 a23a147a26a107a108 a72a42 a69a163a70a69a20a62 a116a109a110a99a111a109a186a120a72 a82 a23 a42 a69a163a70a69a38 a66a86 4.1 Abel( a112 a44) a66a86 a103a104a70a69 ∞summationtext n=0 cn(z ?a)n a152a113a40 z0 a110a111a20a81a152a133a a40 a79a114a115 a20 |z0 ?a| a79a116a117a72a114 a41 a169a170 a110a111 a20a180 a152 |z ?a| ≤ r(r < |z0 ?a|) a32a82a48a110a111a38 a118 a124 a79 ∞summationtext n=0 cn(z ?a)n a152z0 a110a111a20a119a82a122a120a121 a151a129a130a131 limn→∞cn(z0 ?a)n = 0. a124a125 a183a152a122 a69 q a20 a47|cn(z0 ?a) n| < q a38a105a133a20 |cn(z ?a)n| = |cn(z0 ?a)n|· vextendsinglevextendsingle vextendsinglevextendsingle z ?a z0 ?a vextendsinglevextendsingle vextendsinglevextendsingle n < q vextendsinglevextendsingle vextendsinglevextendsingle z ?a z0 ?a vextendsinglevextendsingle vextendsinglevextendsingle n . a124 a79 vextendsinglevextendsingle vextendsinglevextendsingle z ?a z0 ?a vextendsinglevextendsingle vextendsinglevextendsingle < 1a181|z ?a| < |z0 ?a| a188 a20 ∞summationtext n=0 vextendsinglevextendsingle vextendsinglevextendsingle z ?a z0 ?a vextendsinglevextendsingle vextendsinglevextendsingle n a110a111 a20a119 ∞summationdisplay n=0 cn(z ?a)na152 a114 |z ?a| < |z0 ?a|a41 a169a170 a110a111 a38 a180a187|z ?a| ≤ r < |z 0 ?a|a188 a20 |cn(z ?a)n| ≤ q r n |z0 ?a|n, a186 a69a163a70a69 ∞summationtext n=0 rn |z0 ?a|n a110a111 a20a119 ∞summationdisplay n=0 cn(z ?a)na152 a114 |z ?a| ≤ r (r < |z0 ?a|)a32a82a48a110a111a38 a123a124 a179 a70a69 ∞summationtext n=0 cn(z ?a)n a152a113 a40z1 a117a118 a20 a81a152 a114 |z ?a| = |z1 ?a|a125a126a126 a117a118 a38 a118 a120 a90 a96a173 a38 a179 a70a69 ∞summationtext n=0 cn(z ? a)n a152 a114 |z ? a| = |z1 ? a| a125a113 a82a40 z2 a110a111 a20 a81 a14 Abel a122 a127a20a70a69 a151 a93 a152 a114 |z ?a| = |z2 ?a|(|z2 ?a| > |z1 ?a|) a41a110a111 a20 a80a128 a37a129a130a38a119a70a69 ∞summationtext n=0 cn(z ?a)n a152 a114 |z ?a| = |z1 ?a|a125a126a126 a117a118 a38 square a98a99a131a132a98a99a133a134 a92 a88 a82a83 a70a69 a152za135a136a84 a72 a16 a11 a82a40 a20a137 a116a129a138 a110a111 a20 a129a138a117a118 a38a124a125a20 a170a88 a104a70a69a139a70a20a148a135a140a141a5a142 a72a143a144 a56 a152z a135a136a84 a82 a74a77 a40 a104a70a69 a110a111 a20 a152a145a125 a82 a74a77 a40 a104a70 a0a1a2 a3 a4 a5 a6 a77 a8 a69 a117a118 a38a5a95 a110a111 a40 a80a117a118 a40 a91a146 a183a152 a82a83 a77a147a83 a38 star a126a127 Abela122 a127a20a5 a83 a77a147a83 a82 a122 a116a114 a38a5 a83 a114 a20a148 a112 a79 a104a70a69 a72 a98a99a131a38 star a110a111 a114a72a114a115 a56 z = a a40 a38 star a110a111 a114a72a116a117 a112 a79 a98a99a133a134a38 a110a111 a116a117 a132a133 a116 0a38a5a188a20a110a111 a114a148a149a79 a82a83a40 a38a150z = a a40 a125 a20a104a70a69 a152 a85 a135a136a126a126 a117a118 a38 a110a111 a116a117 a62a132a133 a116 ∞a38a5a188a110a111 a114 a148 a116a85 a135a136 a38a104a70a69 a152 a85 a135a136a110a111 a20a151 a152∞ a40 a132a152 a110a111 a20 a62a132a152 a117a118 a38 a152a153a154 a104a70a69 a72 a119a63a188 a20a155a156a157a187 a156 a135 a110a111 a114 (a110a111 a116a117 )a38 a156 a104a70a69 a72 a110a111 a116a117a72a158a173 a20 a186a120a72a25a89 a83 a56 1. a126a127Cauchy a172a78a173 a20a187 limn→∞|cn(z ?a)n|1/n < 1 a181 |z ?a| < 1lim n→∞|cn| 1/n a188 a70a69 a169a170 a110a111a189 a187 limn→∞|cn(z ?a)n|1/n > 1 a181 |z ?a| > 1lim n→∞|cn| 1/n a188 a70a69 a117a118 a38a124a125a20a104a70a69 ∞summationtext n=0 cn(z ?a)n a72 a110a111 a116a117a116 R = 1lim n→∞|cn| 1/n = limn→∞ vextendsinglevextendsingle vextendsinglevextendsingle 1 cn vextendsinglevextendsingle vextendsinglevextendsingle 1/n . 2. a126a127d’Alembert a172a78a173 a20a103a104 limn→∞ vextendsinglevextendsingle vextendsinglevextendsinglecn+1(z ?a)n+1 cn(z ?a)n vextendsinglevextendsingle vextendsinglevextendsingle = |z ?a| lim n→∞ vextendsinglevextendsingle vextendsinglevextendsinglecn+1 cn vextendsinglevextendsingle vextendsinglevextendsingle a183a152 a20 a81 a187 limn→∞ vextendsinglevextendsingle vextendsinglevextendsinglecn+1(z ?a)n+1 cn(z ?a)n vextendsinglevextendsingle vextendsinglevextendsingle < 1 a181 |z ?a| < lim n→∞ vextendsinglevextendsingle vextendsinglevextendsingle cn cn+1 vextendsinglevextendsingle vextendsinglevextendsingle a188 a70a69 a169a170 a110a111a189 a187 limn→∞ vextendsinglevextendsingle vextendsinglevextendsinglecn+1(z ?a)n+1 cn(z ?a)n vextendsinglevextendsingle vextendsinglevextendsingle > 1 a181 |z ?a| > lim n→∞ vextendsinglevextendsingle vextendsinglevextendsingle cn cn+1 vextendsinglevextendsingle vextendsinglevextendsingle a188 a70a69 a117a118 a38a124a125a20a104a70a69 ∞summationtext n=0 cn(z ?a)n a72 a110a111 a116a117a116 R = limn→∞ vextendsinglevextendsingle vextendsingle cnc n+1 vextendsinglevextendsingle vextendsingle. §4.3 a102 a5 a6 a78 a8 a159a160a161a224a209a210a162a163a28a164a35a165a146a192a229a193a38Cauchya164a35a23a166a167a168a169a28a20a236d’Alemberta164a35 a170a23a146a171a172a28(a31a224a218a219 lim n→∞|cn/cn+1| a139a140)a38a173a215a174a175a240a176a46a177a20a196a196a178a179 a213 a197a198a180a38 a92 a88 a104a70a69 ∞summationtext n=0 cn(z ?a)n a72 a2 a82 a163 a30a116 z a72 a87a88 a42 a69a20 Abel a122 a127 a64a65a66a67 a20a104a70a69 a152 a10 a110 a111 a114 a41 a16 a82a94 a38a39 a32 a82a48 a110a111 a20a124a125a20a126a127 4.2 a181 a20 a152a110a111 a114 a41 a20a104a70a69a182a183a141 a82a83 a87a88 a42 a69 ( a33 a69 a70a20a104a70a69 a72a75a42 a69 a152a110a111 a114 a41 a87a88) a20a132a133 a170 a104a70a69a68a163a4 a77 a33 a68a163 a156 a91 a69a20 integraldisplay z z0 ∞summationdisplay n=0 cn(z ?a)n dz = ∞summationdisplay n=0 cn integraldisplay z z0 (z ?a)n dz = ∞summationdisplay n=0 cn n + 1 bracketleftbig(z ?a)n+1 ?(z 0 ?a)n+1 bracketrightbig, d dz bracketleftbigg ∞summationdisplay n=0 cn(z ?a)n bracketrightbigg = ∞summationdisplay n=0 cnd(z ?a) n dz = ∞summationdisplay n=0 cn+1(n + 1)(z ?a)n. star a104a70a69a152a110a111 a114 a84 a72 a110a111a119 a212 ? a132a133a126a126a110a111 a20 ? a132a133a126a126 a117a118 a20 ? a62a132a133a152a82 a74a77 a40 a110a111 a20 a152a145 a82 a74a77 a40 a117a118 a38 z2 2 +···+ zn n(n?1) +··· a152|z|= 1a84a126a126a110a111a189 1 + z +···+ zn +··· a152|z|= 1a84a126a126 a117a118 a189 1 + z1 +···+ z n n +··· a152|z| = 1 a84 a150 z = 1 a125a184a110 a111 a20a180 a152z = 1 a40 a117a118 a38 a153a154a185 a23 a143a144 a20a104a70a69 a72 a110a111 a114 a84 a137a186 a122 a25a187 a40 a38 a151a181 a47a152 a187 a40 a20a104a70a69a3a93a132a152 a116 a110a111 a72 (a181 a25a188 a122 a72a42 a69a18) a38 a37a104a70a69 ∞summationdisplay n=0 cn(z ?a)n a152a110a111 a114 a41a110a111 a150f(z)a20a103a104a70 a69 a152a110a111 a114a189 a84a113 a40 z0 a62 a110a111 a20 a75a79 S(z0)a20a81 a190a191a192a112a193 a66a86( a153 a96 ) a64a65a66a67 a20a187 z a92 a110a111 a114 a41a194 a88 z0 a188 a20a195 a129a196a197 a152 a133z 0 a79a198 a40 a111 a199a200 a79 2φ < pi a72a201a202 a41( a98a203 4.1)a20f(z)a148 a82 a122a194 a88 S(z0)a38 a2044.1 a205a206a207 a7a208a209a210 a0a1a2 a3 a4 a5 a6 a79 a8 §4.4 a211a212a213a214a215a216a217a218a214a219a220a221 4.3a181 a32 a25 a185 a42 a69a70a69a87a88 a119 a72a222 a154 a20a62a132a133 a120 a139 a153a154a223a224a225 a72 a90 a186 a4 a77a72 a87a88 a119 a38 a66a86 4.2 a37 1. f(t,z)a23ta43z a28a226a227a27a19a20 t > aa20z ∈ Ga20 2. a144a145a136a228t ≥ aa20f(t,z)a23Ga225a28a198a190a25a26a27a19a20 3. a229 a195 integraldisplay ∞ a f(t,z)dta140Ga225a37a230a209a210a20a220?ε > 0a20?T(ε)a20a215T2 > T1 > T(ε)a177a20a146 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle integraldisplay T2 T1 f(t,z)dt vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle < ε, a81F(z) = integraldisplay ∞ a f(t,z)dta152G a41 a116 a87a88 a72 a20a31 Fprime(z) = integraldisplay ∞ a ?f(t,z) ?z dt. a118 a16a231 a82a83a232a233 a108a109 {an} a0 = a < a2 < a3 < ··· < an < an+1 < ···, limn→∞an = ∞. a71u n(z) = integraldisplay an+1 an f(t,z)dta20a81a126a127 3.7a181a185 a88 a223a224a225 a72 a122 a4 a77a72 a87a88 a119 a72 a122 a127a20a132a234u n(z)a116 G a41 a72a43 a18a87a88 a42 a69a38a235a124 a79 F(z) = ∞summationdisplay n=0 un(z) a152Ga84 a82a48 a110a111 a20a119a126a127 Weierstrass a122 a127a20a234 F(z) = ∞summationdisplay n=0 un(z) = integraldisplay ∞ a f(t,z)dt a152G a41 a87a88a20a31 Fprime(z) = ∞summationdisplay n=0 uprimen(z) = integraldisplay ∞ a ?f(t,z) ?z dt. square a170a88 a223a224a225 a72a236 a4 a77 a62a132a133a237a238a239 a126 a127a38 a152 a157 a120 a5 a83 a122 a127 a188 a20a240 a129a172a241 a184a242 a4 a77 (a33 a236 a4 a77 ) a116 a115 a82a48 a110a111 a38 a186a120a72a172a78a173a116 a56a243a244a139 a140a27a19φ(t)a20a143a199|f(t,z)| < φ(t)a20z ∈ Ga20a236a245 integraldisplay ∞ a φ(t)dta209a210a20a170 integraldisplay ∞ a f(t,z)dta140Ga225a248a144a236a245 a37a230a209a210a38 a246 a79 a223a224a225 a72 a184a242 a4 a77a72 a82a83 a19 a1 a20 a160 a136a153a154 a4 a77 F(z) = integraldisplay ∞ 0 e?t2 cos2ztdt. (4.1) a5 a83 a4 a77 a32 a72a247 a4 a42 a69a248a93 a120a121a122 a127 a72a158a89 a83 a130a131 a20a180a31a124 a79a170a88 a68a69 z = x + iy a20 a25 |cos2zt| = radicalBig cosh22yt?cos22xt ≤ cosh2|yt| ≤ e2|yt|. §4.4 a249a250a251a252a253a254a255a0a252a1a2a3 a710 a8 a105a133a20 a170a88 z a135a136a84 a72 a16 a11 a82a83a94 a38a39a84 a20 |Imz| < y 0 a20 a88a116 a20 vextendsinglevextendsingle vextendsinglee?t2 cos2zt vextendsinglevextendsingle vextendsingle < e?t2+2y0t, a180a4 a77 integraldisplay ∞ 0 e?t2+2y0tdt a110a111a20a105a133a223a224a225 a72 a184a242 a4 a77 (4.1)a82a48a110a111 a20a124a125a20a5 a83 a4 a77 a246 a79 z a72a42 a69a20 a152z a135a136a84 a72 a16 a11 a82a83 a38a39a41 a87a88a38a4a5 a82a6 a20a148 a25 Fprime(z) = ? integraldisplay ∞ 0 e?t22t sin2ztdt = e?t2 sin2zt vextendsinglevextendsingle vextendsingle ∞ 0 ?2z integraldisplay ∞ 0 e?t2 cos2ztdt = ?2zF(z). a87a5 a83a7 a77a8a9 a20a148a132a133a149a150 F(z) = Ce?z2 a20a10a32 a186 a69C a116 C = F(0) = integraldisplay ∞ 0 e?t2dt = 12√pi, a5a142a20 a109 a93a148a149a150 integraldisplay ∞ 0 e?t2 cos2ztdt = 12√pie?z2. a10a11a12 a13 a14 a15 a16 a1711 a18 §4.5 a19a20a21a22a23a24a25a26a27a28a29 1. a30a31a32 a33a34 uk(z) a35 G a36a37a38a39a40a41 ∞summationtext k=1 uk(z) a35 G a36a42a43a44a45a39a46a47a48a49a41 S(z) = ∞summationtext k=1 uk(z)a50a35G a36a37a38a51 a52 a53z 0 a54 G a36a42a55a39 |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)|. a56a57 a40a41 summationtextu k(z)a58a42a43a44a45a59a60a61a39a62a63 ε > 0a39a64a60a65a66a67 N(ε)a39a68a69n > N(ε)a70a39 |S(z)?Sn(z)| < ε3, |S(z0) ?Sn(z0)| < ε3. a71a72S n(z) = nsummationtext k=1 uk(z)a73G a36a58a37a38a49a41a39a74a75a76a77a42a78 ε > 0a39a64a79a35 δ > 0a39a68a69|z ?z0| < δ a70a39 |Sn(z)?Sn(z0)| < ε3. a80a81 a65a82a83a34a39a84a85a67a86a75a76a62a63a58 ε > 0a39a79a35δ > 0a39a87a88|z ?z0| < δ a39a84a89 |S(z)?S(z0)| < ε. S(z)a35z0 a55a37a38a51a90a76z0 ∈ Ga62a91a39a92a65S(z)a35G a36a37a38a51 square 2. a93a94a95a96a97 a53C a73a98a99G a36a58a42a100a101a102a103a104a105a106a39a33a34 uk(z) (k = 1,2,···)a73C a82a58a37 a38a49a41a39a46a75a76 C a82a42a43a44a45a58a40a41 ∞summationtext k=1 uk(z)a60a65a107a108a109a110a101 integraldisplay C ∞summationdisplay k=1 uk(z)dz = ∞summationdisplay k=1 integraldisplay C uk(z)dz. a52 a56a57 a59a111 1a61 ∞summationtext k=1 uk(z)a73a105a106C a82a58a37a38a49a41a39a74a110a101 integraldisplay C ∞summationtext k=1 uk(z)dz a79a35a39a112a89 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, a47a113Rn(z) = S(z)?Sn(z) = ∞summationtext k=1 uk(z)a51 a56a57 a40a41a58a42a43a44a45a59a39a75a76a62a63a58ε > 0a39a79a35N(ε) > 0a39 a87a88n > N(ε)a39a84a89 |S(z)?Sn(z)| = |Rn(z)| < ε. §4.5 a114a115a116a117 a15a16a118a119a120a121a122 a1712 a18 a72a123 vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay C Rn(z)dz vextendsinglevextendsingle vextendsinglevextendsingle ≤ integraldisplay c |Rn(z)|·|dz| < εl, la73C a58a124a125a51a126a127a128a129a130a39a131a85 limn→∞ integraldisplay C Rn(z)dz = 0. a92a65 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. a93a94a95a132a133 (Weierstrassa134a135) a53uk(z)(k = 1,2,···)a35 G a113a136a137a138a139a39 ∞summationtext k=1 uk(z)a35G a113 a42a43a44a45a39a46 a123 a40a41a140a48f(z)a73G a36 a58a138a139a49a41a39f(z)a58a141a142a143a41a60a65a90 ∞summationtext k=1 uk(z)a107a108a109a143a41 a85a67a39 f(p)(z) = ∞summationdisplay k=1 u(p)k (z), a109a143a41a144a58a40a41a35 G a36a58a62a42a145a98a99a113a42a43a44a45a51 a52 a146a147a148 a101a149a150a101a51 (1)a146a147f(z) = ∞summationtext k=1 uk(z)a35G a36a138a139a51 a72 a54 uk(z)a138a139a39a74 a56a57 Cauchy a110a101a151a152 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. a90a59a111 1a39a61f(ζ)a73C a82a58a37a38a49a41a39a74 f(z)a138a139a51 (2)a146a147a60a65a107a108a109a143a41a51 a53Gprime a54 G a36a62a42a145a98a99a39 Cprime a73a153a58a154a155a39a46 f(z)a35Gprime a113a138a139a39a74a89 f(p)(z) = p!2pii contintegraldisplay Cprime f(ζ) (ζ ?z)p+1dζ a156a157a158 a159 a14 a15 a16 a1713 a18 = 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)a146a147a40a41 ∞summationtext k=1 u(p)k a35Gprime a113a42a43a44a45a51 a33a160a92a161a39a35G a36a64a60a65a162a128a42a78a145a98a99Gprimeprime a39a68a85Gprime a163a164a35Gprimeprime a36a51 a165Gprime a58a154a155Cprime a67Gprimeprime a58a154a155Cprimeprime a58a166a167a168a169 a54 δ a51a128z a54 Gprime a36a42a55a39ζ a54 Cprimeprime a82a58a170a55a51 a171 Weierstrass a172a173 a90a82a174a58a59a1112a39a60a65a175a176a39 ∞summationtext k=1 uk(ζ)a35a154a155Cprimeprime a82a42a43a44a45a39a131a75a76a62a91a63a176a58ε > 0a39a64 a79a35N(ε)a39a68a69n > N(ε)a70a39a177a89 vextendsingle vextendsinglevextendsingle vextendsinglevextendsingle n+qsummationdisplay k=n+1 uk(ζ) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle < ε, q a73a62a91a178a179a41a51a76a73a39vextendsingle 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 a54 Cprimeprimea58a125a180. a92a65 ∞summationtext k=1 u(p)k (z)a35(G a36a58a62a42a145a98a99)Gprime a113a42a43a44a45a51 square