a0 a1 star a2a3a4a5a6a7a8a9a10 4 a0a1a2 a3 a4 a5 a6 a71 a8 a9a10a11 a12 a13 a14 a15 §3.1 a16 a17 a18 a19 a20a21a22a23a24a20a25a26a27a28a29a30a22a23a31a32C a24a20a26a27a28a29a33a30a34a35a25f(z) a36C a28a37a38a39a31a40a33a30 C a41a42 a23a43a44n a45 a34a23a46a44 z0 = A,z1,z2,···,zn = B, ζk a24zk?1→zk a45 a28a29 a41a42a47 a46a34a48a49a25 nsummationdisplay k=1 f(ζk)(zk ?zk?1) = nsummationdisplay k=1 f(ζk)?zk, a50a51n→∞a34a52a53max|?z k| → 0a54 a34a55a49a25a29a56a57a58 a36 a34a59a60ζ k a29a61a62 a63a64a34a65a66a55a56a57a67a44a35a25 f(z) a68 a33a30C a29a22a23a34a69a44 integraldisplay C f(z)dz = lim max|?zk|→0 nsummationdisplay k=1 f(ζk)?zk. a703.1 a47a71 a20a21a22a23a72a73a28a24a74 a71 a72a21a30a22a23a29a37a75a76a77 integraldisplay C f(z)dz = integraldisplay C (u + iv)(dx + idy) = integraldisplay C (udx?vdy) + i integraldisplay C (vdx + udy). a78a55a34a79a80C a24a23 a45a81a82 a33a30a34 f(z)a24C a28a29a83a84a35a25a34a65a20a21a22a23 a47 a38a58 a36 a31 a20a21a22a23a29a85a86a87a88a89 1. a79a80a22a23 integraldisplay C f1(z)dz, integraldisplay C f2(z)dz, ···, integraldisplay C fn(z)dz a90a58a36a34a65 integraldisplay C bracketleftBig f1(z) + f2(z) +···+ fn(z) bracketrightBig dz = integraldisplay C f1(z)dz + integraldisplay C f2(z)dz +··· + integraldisplay C fn(z)dz; 2. a50C = C1 + C2 +···+ Cn a34a65integraldisplay C1 f(z)dz + integraldisplay C2 f(z)dz +··· + integraldisplay Cn f(z)dz = integraldisplay C f(z)dz; 3. integraldisplay C? f(z)dz = ? integraldisplay C f(z)dz a34a91a92C? a93a94C a29a95a96a97 4. integraldisplay C af(z)dz = a integraldisplay C f(z)dz a34a91a92aa44a98a25a97 5. vextendsinglevextendsingle vextendsingle integraldisplay C f(z)dz vextendsinglevextendsingle vextendsingle ≤ integraldisplay C |f(z)||dz|a97 6. vextendsinglevextendsingle vextendsingle integraldisplay C f(z)dz vextendsinglevextendsingle vextendsingle ≤ Mla34a91a92M a44 vextendsinglevextendsinglef(z)vextendsinglevextendsingle a36C a28a29a28a99a34la44C a29a100a101a31 a102a103a34a20a21a22a23a29a25a67a104a105a106 §3.1 a3 a4 a5 a6 a72a8 ? a107 a22a35a25a34 ? a108 a46a109a110a34a111a22a23a29 a112a28a113a57a114a34 ? a22a23a115a116a31 a117a106a118a38a29 a47a71a107 a22a35a25a34a51 a108 a46a119a38 a54 a34a117a106a120a121a29a22a23a115a116a34a22a23a67 a47a122 a24a120a121a29a31 a123 3.1 a124 integraldisplay C Rezdz a34 C a44 (i) a68 a72a125a1260 → 1 a34a127a26a128a106a129a125 1 → 1 + i a97 (ii) a68 a129a125a126 0 → ia34a127a26a128a106a72a125 i → 1 + ia97(iii)a68a130 a30 0 → 1 + ia31 a131 a117a106(i)a34 integraldisplay C Rezdz = integraldisplay 1 0 xdx + integraldisplay 1 0 idy = 12 + i; a117a106(ii)a34 integraldisplay C Rezdz = integraldisplay 1 0 xdx = 12; a117a106(iii)a34 integraldisplay C Rezdz = integraldisplay 1 0 (1 + i)tdt = 12(1 + i). a0a1a2 a3 a4 a5 a6 a73 a8 §3.2 a132a133a134a135a136a137 Cauchy a138a139 Cauchya38a140a141a142a29a24a22a23a67a60a22a23a115a116a143a144a29a64a145a31a60a146a147a29a148a149a37a64a31 a148a150a74a151a148a149a89 ?a152a153a154a155a156 a89a157a158a159a160a161a162a163a164a165a166a167a168a169a34a168a169a170a171a172a173a174a175a176a158a159a97 ?a177a153a154a155a156 a34a178a179a180a181a182a158a159a31 a703.2 a183a184a185a186a187a188a189a184a185a186a187 a152a153a154a155a156a190 Cauchy a191a192 a79a80a35a25f(z) a36a193 a83a194a148a149 G a92a195a196a34a65 a68G a92 a41a197a47a71 a23 a45a81a82 a29a198a77a199a200 C( a201a202 3.3)a37 contintegraldisplay C f(z)dz = 0, a203a204a29C a205a206a207 a24Ga29a208a99a31 a703.3 a183a184a185a186a187a209Cauchya210a211 a212 a44a213 a193a214a201 a34a113a27 a36a215a216 a29a217a218a113a219a220a203 a71 a38a140a31a221a222a29a217a218a24 fprime(z) a36G a92a83a84a223a31 a36 a55a217a218a113 a206a207a224a225 Greena226a227contintegraldisplay C bracketleftbigP(x,y)dx + Q(x,y)dybracketrightbig = integraldisplayintegraldisplay S parenleftbigg?Q ?x ? ?P ?y parenrightbigg dxdy a106 contintegraldisplay C f(z)dz = contintegraldisplay C bracketleftbigudx?vdybracketrightbig+ icontintegraldisplay C bracketleftbigvdx + udybracketrightbig, a223 a228a229f(z) a230Ga231a232a233a234 a235fprime(z) a236a230a234 a237fprimeprime(z) a238a236a230a234z ∈ Ga234 a239a240fprime(z) a184a241a234 a235a242a243a244a245a246?u/?x,?u/?y,?v/?x a247?v/?y a184a241a248 a249a250a2513.5a252 §3.2 a253a254a255a0a1a2 Cauchya3a4 a74 a8 a5a6a28a27a29a198a77a199a200a22a23a7a44a27a22a23 contintegraldisplay C parenleftbigudx?vdyparenrightbig = ?integraldisplayintegraldisplay S parenleftbigg?v ?x + ?u ?y parenrightbigg dxdy, contintegraldisplay C parenleftbigvdx + udyparenrightbig = integraldisplayintegraldisplay S parenleftbigg?u ?x ? ?v ?y parenrightbigg dxdy. a8a9Cauchy-Riemann a10a11 a34a12 a108 a74 a71 a22a23a92a29 a107 a22a35a25a13a44 0a34a14a37 contintegraldisplay C f(z)dz = 0. square a126a106Green a226a227 a29a15 a124 a34a203a204a16a17a29 a193 a83a194a148a149a34a18a19a24 a47a71 a37a99a148a149a34a111a120a19a24a20a21 ∞a46 a36a22 a29 (a63a99) a148a149a31 a23a24f(z)a157∞a172a25a26a34a27a28∞a172a29a30a171a31a32a33a34a35a36a37a38 0a31 Cauchy a38a140a39a47a71a40a27a41a42a43a195a196a35a25a29a47a71a85a86a44a87a89 a25a26a45a46a157a27a171a25a26a158a159a170a34a47 a172a171a45a46a48a49a50a51a52a53a171a31 ? Cauchy-Riemanna10a11 a24a203a151a64a54a29a55a23a56 a227 a34 ? Cauchya38a140a65a24a57a29a22a23a56a227a31 a126Cauchya38a140a58a111 a206a207 a53a59a113a27a29a60a142a89 a61a62 a50f(z) a36a193 a83a194a148a149G a92a195a196a34a65a20a21a22a23 integraldisplay C f(z)dz a60a115a116a63a64a31 (a131a63a64a65a190) a66a191a67a68 a69 a103 a36a193 a83a194a148a149a92a195a196a35a25a29a22a23a60a115a116a63a64a34a78a55a34a79a80a119a38 a214 a46z 0 a34a5a70a71a46z a44a21a46a34a65a48a44a22a23a28a57a29a35a25a34 integraldisplay z z0 f(z)dz = F(z) a24 a193 a83a194a148a149G a22 a29 a193 a67a35a25a34a66a44 f(z)a29a120a38a22a23a31 a191a192 3.1 a79a80a35a25f(z)a36a193 a83a194a148a149G a22 a195a196a34a65 F(z) = integraldisplay z z0 f(z)dz a205a36G a22 a195a196a34a72a59 Fprime(z) = ddz integraldisplay z z0 f(z)dz = f(z). a212 a18a15 a130a73a124a74F(z)a29a75a25a111a206 a31 a703.5 a76a77a78 a79 a80 a81 a82 a835 a84 a44a55a34a32z a24G a22a47 a46a34z +?z a24a57a29a85a46a34a79 a202 3.5a34a65 F(z) = integraldisplay z z0 f(ζ)dζ, F(z +?z) = integraldisplay z+?z z0 f(ζ)dζ. a78a44a22a23a60a115a116a63a64a34a16 a207 ?F ?z = F(z +?z)?F(z) ?z = 1 ?z integraldisplay z+?z z f(ζ)dζ. a126a55 a206 a53 vextendsinglevextendsingle vextendsinglevextendsingle?F ?z ?f(z) vextendsinglevextendsingle vextendsinglevextendsingle = vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle 1 ?z integraldisplay z+?z z f(ζ)dζ ?f(z) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle 1 ?z integraldisplay z+?z z bracketleftbigf(ζ)?f(z)bracketrightbigdζ vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle ≤ 1|?z| integraldisplay z+?z z vextendsinglevextendsinglef(ζ)?f(z)vextendsinglevextendsingle·vextendsinglevextendsingledζvextendsinglevextendsingle. a126a106f(z)a24a83a84a29a34a14a117a106 a41 a118a29 ε > 0a34a58 a36δ > 0a34a52a51|ζ?z| < δa54 a34|f(ζ)?f(z)| < εa34a16 a207vextendsingle vextendsinglevextendsingle vextendsingle ?F ?z ?f(z) vextendsinglevextendsingle vextendsinglevextendsingle ≤ 1 |?z| ·ε·|?z| = ε, a111a53 Fprime(z) = lim ?z→0 ?F ?z = f(z). a203a86a219a220a43F(z) a36G a22a206 a75a34a72a59Fprime(z) = f(z)a31 square a87a64a65 a79a80a35a25 Φ(z) a29a75a25 Φprime(z) = f(z) a34a65 Φ(z) a66a44 f(z) a29a88a35a25a31a28a27a38a39a29 f(z) a29a120a38a22a23a86a24f(z)a29 a47a71 a88a35a25a31a117a106a118a38a29 a47a71 a35a25f(z) a89 a17a34a88a35a25a120a24a90 a47 a29a31 a41a42 a74 a71 a88a35a25a143a144a18a91a92 a47a71 a98a25a31a203a24a78a44a34a79a80 Φ 1(z) a60Φ 2(z)a90 a24f(z)a29a88a35a25a34a65 Φprime1(z) = f(z), Φprime2(z) = f(z). a16 a207 a34bracketleftbigΦ 1(z)?Φ2(z) bracketrightbigprime = 0a34 Φ1(z)?Φ2(z) = C. a93a200a43 a107 a22a35a25a29a88a35a25a34 a206 a52a20a21a22a23a29a94a95a96a44a213a7a31a32 Φ(z)a44f(z)a29 a47a71 a88a35a25a34a65 f(z)a29a120a38a22a23 F(z) = integraldisplay z z0 f(z)dz = Φ(z) + C. a97a24a34a102a103a37 F(z0) = Φ(z0) + C = 0, C = ?Φ(z0). a16 a207 integraldisplay z z0 f(z)dz = Φ(z)?Φ(z0). §3.2 a253a254a255a0a1a2 Cauchya3a4 a76 a8 a123 3.2 a94a95a22a23 integraldisplay b a zndz a34na44a98a25a31 a131 a51na44a99a103a25 a54 a34zn a36a100 a26a27a195a196a34 1 n + 1z n+1 a24a57a29 a47a71 a88a35a25a31a78a55a34a117a106 z a26a27 a28a29 a41a42a47 a217a22a23a115a30a34a13a37 integraldisplay b a zndz = 1n + 1bracketleftbigbn+1 ?an+1bracketrightbig. a51n = ?2,?3,?4,··· a54 a34zn a36 a120a20a21z = 0a46 a36a22 a29 a41a42a47a71a193 a83a194a148a149 a22 a195a196a34a91a88a35a25 a101 a206 a62a44 1 n + 1z n+1 a31a78a55a34a101a37 integraldisplay b a zndz = 1n + 1bracketleftbigbn+1 ?an+1bracketrightbig. a97a126a106a113 a47a102a103 3a29a88a78a34a55a104a80a117a106a120a20a21 z = 0a46a36a22 a29 a41a42 a148a149a13a105a58a31 a51n = ?1 a54 a34z?1 a205 a24 a36 a120a20a21z = 0 a36a22 a29 a41a47 a148a149 a22 a195a196a34a97a91a88a35a25 a224 a44 lnza31a78a55a34 a36 a120a20a21z = 0a29 a41a47a193 a83a194a148a149 a22 a34 integraldisplay b a dz z = lnb?lna. a106a15a44a150a107 a42 a34 a36a47a71a193 a83a194a148a149 a22 a34a28a27a29a22a23a51a103a60a115a116a63a64a31a97a24 a108a175a37a109a171a165a181a182 a158a159a34a109a110a171a111a172a112a113a172a33a114a115a116a117a37a109a171a31a32a48a31a39a94a95a29a118 a11a119 a34a203a204a29a88a35a25a24a120a67a35a25a34 a78a55a22a23a67a60a126aa21a7a59ba29 a10a227 a37a64a31a51a57a121 a36 a120a21z = 0a29 a47a71a193 a83a194a148a149 a22a54 a34a86a24a40 lnz a57a121 a36a122a47a71a193 a67a23a123 a22 a34a14a22a23a67 lnb?lnaa24a90 a47a124 a38a29a31a5a117a106a120a121a29 a193 a83a194a148a149a34a86 a206 a19 a117 a224 a106 lnz a29a120a121 a193 a67a23a123a34a78a5a22a23a67 a205 a86 a206 a19a120a121a31 a0a1a2 a3 a4 a5 a6 a77 a8 §3.3 a16a133a134a135a136a137 Cauchy a138a139 a177a153a154a155a156a190 Cauchy a191a192 a79a80f(z)a24a20a83a194a148a149 G a92a29 a193 a67a195a196a35a25a34a65 contintegraldisplay C0 f(z)dz = nsummationdisplay i=1 contintegraldisplay Ci f(z)dz, a91a92C 0,C1,C2,···,Cn a24a125a105a20a83a194a148a149Ga29a208a99a29a126 a71 a23 a45a81a82 a198a77a33a30a34C 1,C2,···,Cn a90 a20 a21 a36C0 a29a22a127 a34a5a59a16a37a29a22a23a115a116a128a96a91a121a31 a703.6 a189a184a185a186a187a209Cauchya210a211 a212 a79 a2023.6a34a120a129a62C0, C1, C2, ···, Cn a13a44a95a54a130a10 a96a31a48a131a51a29a43a30a40C 1, C2, ···, Cn a49 C0 a83a104a214a89 a34a39a5a53a59 a47a71a193 a83a194a148a149Gprime a34f(z) a36a193 a83a194a148a149Gprime a22 a24a195a196a29a34a78a5 a206a207a224a225a193 a83a194a148a149a29 Cauchya38a140a34 contintegraldisplay C0 f(z)dz + integraldisplay b1 a1 f(z)dz + contintegraldisplay C?1 f(z)dz + integraldisplay a1 b1 f(z)dz + integraldisplay b2 a2 f(z)dz + contintegraldisplay C?2 f(z)dz + integraldisplay a2 b2 f(z)dz +··· + integraldisplay bn an f(z)dz + contintegraldisplay C?n f(z)dz + integraldisplay an bn f(z)dz = 0. a126a106f(z) a36Gprime a22a193 a67a34a14 a68 a121 a47 a43a30a74a132a29a22a23a67a133a91a134a135a34 integraldisplay bi ai f(z)dz + integraldisplay ai bi f(z)dz = 0. a16 a207 contintegraldisplay C0 f(z)dz + nsummationdisplay i=1 contintegraldisplay C?i f(z)dz = 0, (3.1) contintegraldisplay C0 f(z)dz = ? nsummationdisplay i=1 contintegraldisplay C?i f(z)dz = nsummationdisplay i=1 contintegraldisplay Ci f(z)dz. square (3.2) a123 3.3 a94a95 contintegraldisplay C zndz a67a34na44a98a25a34C a29a128a96a44a95a54a130a10 a96a31 §3.3 a3a254a255a0a1a2 Cauchya3a4 a78 a8 a131 a51na44a99a103a25 a54 a34a102a103a34a136a137 a193 a83a194a148a149a29 Cauchya38a140 contintegraldisplay C zndz = 0. a51na44a138a98a25 a54 a34a79a80 C a22 a120a21z = 0a34a65 a205 a37 contintegraldisplay C zndz = 0. a79a80C a22 a21a37z = 0a34a65a136a20a83a194a148a149a29 Cauchya38a140a34a37 contintegraldisplay C zndz = contintegraldisplay |z|=1 zndz = integraldisplay 2pi 0 parenleftbigeiθparenrightbign eiθidθ = integraldisplay 2pi 0 ei(n+1)θidθ = ? ? ? 2pii, n = ?1; 0, n = ?2,?3,?4,···. a139a104a28a27a29a104a80a34a86a37 contintegraldisplay C zndz = ? ? ? 2pii, n = ?1, a59Ca22 a21a37z = 0; 0, a91a140a141a56. a142a143a34 a215a47a122a144 a34 contintegraldisplay C (z ?a)ndz = ? ? ? 2pii, n = ?1, a59Ca22 a21a37z = a; 0, a91a140a141a56. a0a1a2 a3 a4 a5 a6 a79 a8 §3.4 a145a146a147a148a137a149a139 a150 a192 3.1 a79a80a35a25f(z)a36z = aa46a29a85a149a22 a83a84a34a72a59a51 θ 1≤ arg(z?a)≤ θ2, |z?a| → 0 a54 a34(z ?a)f(z) a47a151a144a152a153 a106 ka34a65 lim δ→0 integraldisplay Cδ f(z)dz = ik(θ2 ?θ1), a91a92C δ a24 a207z = aa44a154a155a34δ a44a156a116a34a157a158a44θ2 ?θ1 a29a154a159a34|z ?a| = δ, θ1 ≤ arg(z ?a) ≤ θ2 a34 a201a2023.7 a31 a703.7 a212 a78a44 integraldisplay Cδ dz z ?a = i(θ2 ?θ1), a16 a207 vextendsingle vextendsinglevextendsingle vextendsingle integraldisplay Cδ f(z)dz ?ik(θ2 ?θ1) vextendsinglevextendsingle vextendsinglevextendsingle = vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay Cδ bracketleftbigg f(z)? kz ?a bracketrightbigg dz vextendsinglevextendsingle vextendsinglevextendsingle ≤ integraldisplay Cδ |(z ?a)f(z)?k| |dz||z ?a|. a126a106a51θ 1 ≤ arg(z ?a) ≤ θ2 a34z ?a → 0 a54 a34(z ?a)f(z) a47a151a144a152a153 a106ka34a203 a42a160a161?ε > 0a34?(a60 arg(z ?a)a63a64a29) r(ε) > 0a34a52a51|(z ?a)| = δ < ra54 a34|(z ?a)f(z)?k| < εa31a16 a207vextendsingle vextendsinglevextendsingle vextendsingle integraldisplay Cδ f(z)dz ?ik(θ2 ?θ1) vextendsinglevextendsingle vextendsinglevextendsingle ≤ ε(θ2 ?θ1), a111 lim δ→0 integraldisplay Cδ f(z)dz = ik(θ2 ?θ1). square a150 a192 3.2 a32f(z)a36∞a46a29a85a149a22 a83a84a34a51θ 1 ≤ argz ≤ θ2 a34z → ∞ a54 a34zf(z) a47a151a144a152 a153 a106K a34a65 lim R→∞ integraldisplay CR f(z)dz = iK(θ2 ?θ1), a91a92C R a24 a207 a88a46a44a154a155a34Ra44a156a116a162a157a158a44θ 2?θ1 a29a154a159a34|z| = R, θ 1 ≤ argz ≤ θ2 (a201a2023.8) a31 §3.4 a163a164a165a166a2a167a4 a710 a8 a703.8 a212 a55a168a140a29a219a220a49a150 a192 3.1 a29a219a220a91a169a31a78a44 integraldisplay CR dz z = i parenleftbigθ 2 ?θ1 parenrightbiga34a16 a207 vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay CR f(z)dz ?iK(θ2 ?θ1) vextendsinglevextendsingle vextendsinglevextendsingle = vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay CR bracketleftbigg f(z)? Kz bracketrightbigg dz vextendsinglevextendsingle vextendsinglevextendsingle = vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay CR bracketleftbigzf(z)?Kbracketrightbigdz z vextendsinglevextendsingle vextendsinglevextendsingle ≤ integraldisplay CR vextendsinglevextendsinglezf(z)?Kvextendsinglevextendsingle· |dz| |z| . a126a106a51θ 1 ≤ argz ≤ θ2 a34z → ∞ a54 a34zf(z) a47a151a144a152a153 a106K a34a203 a42a160a161?ε > 0 a34?(a60 argz a63a64 a29)M(ε) > 0a34a52a51|z| = R > M a54 a34|zf(z)?K| < εa31a16 a207vextendsingle vextendsinglevextendsingle vextendsingle integraldisplay CR f(z)dz ?iKparenleftbigθ2 ?θ1parenrightbig vextendsinglevextendsingle vextendsinglevextendsingle ≤ ε(θ2 ?θ1), a111 lim R→∞ integraldisplay CR f(z)dz = iKparenleftbigθ2 ?θ1parenrightbig. square a0a1a2 a3 a4 a5 a6 a711 a8 §3.5 Cauchy a18a19a170a171 a172a173 a155a156a190 Cauchy a67a68a174a175 a32f(z) a24a148a149 G a92a29 a193 a67a195a196a35a25a34 Ga29a208a99C a24a23 a45 a81a82 a33a30a34aa44G a22a47 a46a34a65 f(a) = 12pii contintegraldisplay C f(z) z ?adz, a91a92a22a23a115a30 a68C a29a176a96a31 a703.9 a177a178a186a187a209Cauchya179a180a181a182 a212 a36 G a22 a48a154|z ? a| < r( a201a202 3.9 a34a183a184a154a185 |z ? a| = r a36 G a22) a34a65a8a9a20a83a194a148a149a29 Cauchya38a140a34a37 contintegraldisplay C f(z) z ?adz = contintegraldisplay |z?a|=r f(z) z ?adz, a55a104a80 a224 a60ra29a96a186a63a64a34a14 a206 a70 r → 0a31a78a44 limz→a(z ?a) f(z)z ?a = f(a), a126a28 a102 a29a150 a192 3.1 a34a86a219a53 1 2pii contintegraldisplay C f(z) z ?adz = f(a). square a63a99a148a149a29 Cauchya22a23 a226a227 a117a106a63a99a148a149a34a106a15a187a32 f(z) a36 a213 a193 a198a77a199a200 C a28a147C a188 (a20a189a63a190a191a46) a193 a67a195a196a31a192a193 a144 a34a194 a36 a94a95 1 2pii contintegraldisplay C f(z) z ?adz, a91a92aa44C a188a47 a46a34a22a23a115a30C a29a128a96a24a195 a54a130a10 a96a34a111a196a63a190a191a46a29a176a96a34a79 a202 3.10a31 a703.10 a197a178a186a187a209Cauchya179a180a181a182 §3.5 Cauchya5a6a198a199 a712a8 a36C a188 a127a48 a47a71a207 a88a46a44a154a155a34Ra44a156a116a29a96a154C R a34a203a200a34a117a106C a49C R a16a20a199a29a20a83a194 a148a149a34a8a9a37a99a148a149a29 Cauchya22a23 a226a227 a34a37 1 2pii contintegraldisplay CR f(z) z ?adz + 1 2pii contintegraldisplay C f(z) z ?adz = f(a), a203a204a22a23a115a30C R a29a128a96a24a95 a54a130a10 a96a31a18a15R a201a202 a96a34a203 a71 a104a80a51a103a86a60Ra29a203a204a96a186a63a64a34 a106a24a34 a206 a70R → ∞a34a5a53a59 1 2pii contintegraldisplay C f(z) z ?adz = f(a)? limR→∞ bracketleftbigg 1 2pii contintegraldisplay CR f(z) z ?adz bracketrightbigg . a44a43a94a95 a68 a96a154C R a29a22a23a29a56a57a67a34a106a15 a225 a59 3.4 a102 a92a29a150 a192 3.2 a34 1 2pii contintegraldisplay C f(z) z ?adz = f(a)?K, K = limz→∞z · f(z)z ?a = f(∞). a44a150a51K = 0 a54 a34a86a53a59a89 a205a173 a155a156a190 Cauchy a67a68a174a175 a89a79a80 f(z) a36 a213 a193 a198a77a199a200 C a28a147 C a188 a195a196a34a59a51 z → ∞ a54 a34f(z) a47a151a144a152 a106 0a34a65 Cauchya22a23 a226a227 f(a) = 12pii contintegraldisplay C f(z) z ?adz a101a103a105a58a34a55a206aa44C a188a47 a46a34a22a23a115a30C a44a195 a54a130a10 a96a31 a0a1a2 a3 a4 a5 a6 a713 a8 §3.6 a207a208a209a210a137a211a212a213a210 a39Cauchya22a23 a226a227 a34 a206a207 a60a214 a74a47a71a215 a15a104a142a89a79a80f(z) a36G a92a195a196a34a65a36G a22f(z)a29a41 a197a216 a75a25f(n)(z)a13a58 a36 a34a72a59 f(n)(z) = n!2pii contintegraldisplay C f(ζ) (ζ ?z)n+1dζ, a91a92C a24Ga29a176a96a208a99a34z a44G a22a41a42a47 a46a34a79 a202 3.11a31 a703.11 a217a218 a245a246 a181a182 a212 a219a220 a124f prime(z)a31a78a44 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, a62a56a57h → 0a34a221 a108 a111a44fprime(z)a34a5a12 a108a107 a22a35a25a29a56a57a44f(ζ)/(ζ ?z)2 a31a44a43a219a220 a36 a22a23a222a113 a124 a56a57a77a223a34a120a129a224a225 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. a126a106f(ζ) a36C a28a83a84a34a14a36C a28a37|f(ζ)| ≤ M a34a32za59C a29a226a227a228a229a44δa34la44C a29a100a101a34a65a37vextendsingle vextendsinglevextendsingle vextendsingle 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, a78a55a34a22a23a222a113 a124 a56a57a77a223a34 fprime(z) = 12pii contintegraldisplay C f(ζ) (ζ ?z)2dζ. a121a200a34 a206a207a124a74 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. a79a55a230a84a34a111 a206a124a74 f(n)(z)a31 square star a203a71 a104a80a17a220a34 a47a71 a20a21a35a25a34a18a15 a36a47a71 a148a149a92 a47a216 a75a25a206a206a58 a36 (a231a232 a49a158a159a170a171a25a26 a45a46) a34a65a57a29 a41a197a216 a75a25 a90 a58 a36 a34a72a59 a90 a24a203 a71 a148a149 a22 a29a195a196a35a25a31 §3.5 Cauchya5a6a198a199 §3.6 a233a234a235a236a2a237a238a239a236 a714 a8 star a36 a72a21a35a25a92a72a240a79a55a31a241a242a72a120a19a126fprime(x) a29a58 a36 a60a214 a74fprimeprime(x) a29a58a36 a31 star a20a21a35a25a92f(z) a36a47 a148a149a92a206a206 a206 a75 (a111a195a196) a24 a47a71a243a244 a29a15 a124 a31a72a21a35a25a92fprime(x) a29a58 a36 a18a20a21a51 x a36 a25a125a28 ( a47 a38a148a144 a22) a21a7a54 a117f(x) a29a15 a124 a34a5a20a21a35a25a92fprime(z) a29a58 a36 a65 a20a21a43 a36a245a246 a26a27a148a149a28a117 f(z)a29a15 a124 a31 a0a1a2 a3 a4 a5 a6 a715 a8 §3.7 Cauchy a247a18a19a248a249a250a251a18a19a137a207a208a252 a36 a28 a47a102 a64a106a195a196a35a25 a244a216 a75a25 a226a227 a29a219a220a118 a11 a92a34f(z)a29a195a196a87a18a24a204a194 a36 a89 (1) f(z) a206 a225 Cauchya22a23a226a227 a93a94 a97 (2)f(z) a36C a28a83a84a31a78a55a34a215 a20a28a27a29a253a254a34a86 a206a207 a219a220a89 a36a47 a217a23 a45a81a82 a29 (a198a77a142a120a198a77) a33a30C a28a83a84a29a35a25φ(ζ) a16a125a105a29a22a23 f(z) = 12pii integraldisplay C φ(ζ) ζ ?zdζ (a66a44Cauchya255a67a68) a24a33a30a188 a46z a29a195a196a35a25a34fprime(z) a206 a194a118a22a23a222a113 a124 a75a5a53a59a34 f(p)(z) = p!2pii integraldisplay C φ(ζ) (ζ ?z)p+1dζ. a123 3.4 a94a95a22a23 f(z) = 12pii contintegraldisplay |ζ|=1 ζ? ζ ?zdζ, |z| negationslash= 1. a131 a203a24 a47a71 Cauchya0 a22a23a31a78a44 a36|ζ| = 1a28ζ? = 1/ζ a34a14 f(z) = 12pii contintegraldisplay |ζ|=1 1 ζ(ζ ?z)dζ. a51|z| > 1 a54 a34a55a22a23 a206a207a225 Cauchya22a23a226a227 a94a95a34 f(z) = 12pii contintegraldisplay |ζ|=1 1 ζ bracketleftbigg 1 ζ ?z bracketrightbigg dζ = ?1z. a510 < |z| < 1 a54 a34 f(z) = 12pii contintegraldisplay |ζ|=1 1 z bracketleftbigg 1 ζ ?z ? 1 ζ bracketrightbigg dζ = 0. a1a2 a119a74 a34a55a104a80a117a106 z = 0a101a105a58a31a3a77 a207 a28a104a80a34a86a37 f(z) = 12pii contintegraldisplay |ζ|=1 ζ? ζ ?zdζ = ? ? ? ?1z, |z| > 1, 0, |z| < 1. a126a55 a206a201 a34f(z) a36|z| negationslash= 1a206a195a196a34a4a5ζ? a36a100 a26a27a120a195a196a31 a6 a225Cauchya0 a22a23a34a86 a206a207 a60 a74a7a8a9a67a68a190 a131a63a10a31 a191a192 3.2 a32 1. f(t,z)a49ta11z a171a181a12a45a46a34t ∈ [a,b]a34z ∈ Ga34 2. a108a175[a,b]a13 a171a162a163ta48a34f(t,z)a49G a13 a171a165a48a25a26a45a46a34 a65F(z) = integraldisplay b a f(t,z)dta36G a22 a24a195a196a29a34a59 Fprime(z) = integraldisplay b a ?f(t,z) ?z dt. a212 a78a44f(t,z) a36Ga28a195a196a34a14a117a106G a22 a29 a41a197a47 a46z a34Cauchya22a23 a226a227 a105a58a34 f(t,z) = 12pii contintegraldisplay C f(t,ζ) ζ ?z dζ. §3.7 Cauchya14 a5a6a15a16a17a18a19a20 a2a233a234a21§3.6 a233a234a235a236a2a237a238a239a236 a2216a23 a24a25F(z) a26a27a28a29a30a31a32a33a34a35a36 (a37a38f(t,z)a39a40) a29a41 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. a42a43a44a45 Cauchy a46a33a34a29 integraldisplay b a f(t,z)dta39a40a29a47F(z)a38G a48a26a49a50a51a52a29a53 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 a54a55 a29 a42a45a56a57a58a59a60a61 integraldisplay C f(t,z)dta62 a42a63a64a65a66a67C a43 a34a68a69a70a71a72a29 a65 t a73C a74a75a76a29 z ∈ Ga63a29f(t,z)a43 ta77z a26a39a40a51a52a62a78a79a26a80a81a82a74a83a84a85a62