Wu Chong-shi 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 a34a59 a60 ζ k a29a61a62a63a64a34a65a66a55a56a57a67a44a35a25 f(z) a68 a33a30 C a29a22 a23a34a69a44 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 a90 a58 a36 a34a65 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 Wu Chong-shi §3.1 a3 a4 a5 a6 a72a8 a102a103a34a20a21a22a23a29a25a67a104a105a106 ? 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 a300 → 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). Wu Chong-shi 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 a92a41a197a47a71 a23 a45a81a82 a29a198a77a199a200 C( a201a2023.3)a37 contintegraldisplay C f(z)dz = 0, a203a204a29C a205a206a207 a24Ga29a208a99a31 a703.3 a183a184a185a186a187a209Cauchya210a211 Wu Chong-shi §3.2 a212a213a214a215a216a217 Cauchya218a219 a74 a8 a220 a44a221 a193a222a201 a34a113a27 a36a223a224 a29a225a226a113a227a228a203 a71 a38a140a31a229a230a29a225a226a24 fprime(z) a36G a92a83a84a231a31 a36 a55a225a226a113 a206a207a232a233 Greena234a235contintegraldisplay 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, a236a237a28a27a29a198a77a199a200a22a23a238a44a27a22a23 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. a239a240Cauchy-Riemann a241a242 a34a243 a108 a74 a71 a22a23a92a29 a107 a22a35a25a244a44 0a34a245a37 contintegraldisplay C f(z)dz = 0. square a126a106Green a234a235 a29a246 a124 a34a203a204a247a248a29 a193 a83a194a148a149a34a249a250a24 a47a71 a37a99a148a149a34a111a120a250a24a251a252 ∞a46 a36a253 a29(a63a99)a148a149a31 a254a255f(z)a157∞a172a0a1a34a2a3∞a172a4a5a171a6a7a8a9a10a11a12a13 0a31 Cauchy a38a140a14a47a71a15 a27a16a17a18a195a196a35a25a29 a47a71 a85a86a19a87a89 a0a1a20a21a157a2a171a0a1a158a159a170a34a22 a172a171a20a21a23a24a25a26a27a28a171a31 ? Cauchy-Riemanna241a242 a24a203a151a64a29a29a30a23a31 a235 a34 ? Cauchya38a140a65a24a32a29a22a23a31a235a31 a126Cauchya38a140a33a111 a206a207 a53a34a113a27a29a35a142a89 a36a37 a50f(z) a36a193 a83a194a148a149G a92a195a196a34a65a20a21a22a23 integraldisplay C f(z)dz a60a115a116a63a64a31 (a131a38a39a40a190)a41a191a42a43 a44 a103 a36a193 a83a194a148a149a92a195a196a35a25a29a22a23a60a115a116a63a64a34a78a55a34a79a80a119a38 a222 a46z 0 a34a236a45a46a46z a44a21a46a34a65a48a44a22a23a28a57a29a35a25a34 integraldisplay z z0 f(z)dz = F(z) a24 a193 a83a194a148a149G a253 a29 a193 a67a35a25a34a66a44 f(z)a29a120a38a22a23a31 a191a192 3.1 a79a80a35a25f(z)a36a193 a83a194a148a149G a253 a195a196a34a65 F(z) = integraldisplay z z0 f(z)dz a231 a47a48f(z) a49Ga50a51a52a53 a54fprime(z) a55a49a53 a56fprimeprime(z) a57a55a49a53z ∈ Ga53 a58a59fprime(z) a184a60a53 a54a61a62a63a64a65?u/?x,?u/?y,?v/?x a66?v/?y a184a60a67 a68 a7 a61a69 Wu Chong-shi a0a1a2 a3 a4 a5 a6 a75 a8 a205a36G a253 a195a196a34a70a59 Fprime(z) = ddz integraldisplay z z0 f(z)dz = f(z). a220 a249a246 a130a71a124a72 F(z) a29a73a25a111 a206 a31 a703.5 a44a55a34a32z a24G a253a47 a46a34z +?z a24a32a29a74a46a34a79 a202 3.5a34a65 F(z) = integraldisplay z z0 f(ζ)dζ, F(z +?z) = integraldisplay z+?z z0 f(ζ)dζ. a78a44a22a23a60a115a116a63a64a34a247 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)a24a83a84a29a34a245a117a106 a41 a118a29 ε > 0a34a58 a36δ > 0a34a52a51|ζ?z| < δa54 a34|f(ζ)?f(z)| < εa34a247 a207vextendsingle vextendsinglevextendsingle vextendsingle ?F ?z ?f(z) vextendsinglevextendsingle vextendsinglevextendsingle ≤ 1 |?z| ·ε·|?z| = ε, a111a53 Fprime(z) = lim ?z→0 ?F ?z = f(z). a203a75a227a228a18F(z) a36G a253a206 a73a34a70a59Fprime(z) = f(z)a31 square a76a39a40 a79a80a35a25Φ(z)a29a73a25Φprime(z) = f(z)a34a65Φ(z)a66a44f(z)a29a77a35a25a31a28a27a38a39a29f(z) a29a120a38a22a23a75a24f(z)a29 a47a71 a77a35a25a31a117a106a118a38a29 a47a71 a35a25f(z) a78 a248a34a77a35a25a120a24a79 a47 a29a31 a41a42 a74 a71 a77a35a25a143a144a249a80a81 a47a71 a98a25a31a203a24a78a44a34a79a80 Φ 1(z) a60Φ 2(z)a90 a24f(z)a29a77a35a25a34a65 Φprime1(z) = f(z), Φprime2(z) = f(z). Wu Chong-shi §3.2 a212a213a214a215a216a217 Cauchya218a219 a76 a8 a247 a207 a34bracketleftbigΦ 1(z)?Φ2(z) bracketrightbigprime = 0a34 Φ1(z)?Φ2(z) = C. a82a200a18 a107 a22a35a25a29a77a35a25a34 a206 a52a20a21a22a23a29a83a84a85a44a221a238a31a32 Φ(z)a44f(z)a29 a47a71 a77a35a25a34a65 f(z)a29a120a38a22a23 F(z) = integraldisplay z z0 f(z)dz = Φ(z) + C. a86a24a34a102a103a37 F(z0) = Φ(z0) + C = 0, C = ?Φ(z0). a247 a207 integraldisplay z z0 f(z)dz = Φ(z)?Φ(z0). a123 3.2 a83a84a22a23 integraldisplay b a zndz a34na44a87a25a31 a131 a51na44a88a103a25 a54 a34zn a36a89 a26a27a195a196a34 1 n + 1z n+1 a24a32a29 a47a71 a77a35a25a31a78a55a34a117a106 z a26a27 a28a29 a41a42a47 a225a22a23a115a30a34a244a37 integraldisplay b a zndz = 1n + 1bracketleftbigbn+1 ?an+1bracketrightbig. a51n = ?2,?3,?4,··· a54 a34zn a36 a120a251a252z = 0a46 a36a253 a29 a41a42a47a71a193 a83a194a148a149 a253 a195a196a34a91a77a35a25 a90 a206 a62a44 1 n + 1z n+1 a31a78a55a34a90a37 integraldisplay b a zndz = 1n + 1bracketleftbigbn+1 ?an+1bracketrightbig. a86a126a106a113 a47a91a92 3a29a77a78a34a55a93a80a117a106a120a251a252 z = 0a46a36a253 a29 a41a42 a148a149a244a94a33a31 a51n = ?1 a54 a34z?1 a205 a24 a36 a120a251a252z = 0 a36a253 a29 a41a47 a148a149 a253 a195a196a34a86a91a77a35a25 a232 a44 lnza31a78a55a34 a36 a120a251a252z = 0a29 a41a47a193 a83a194a148a149 a253 a34 integraldisplay b a dz z = lnb?lna. a95a246a19a150a96 a42 a34 a36a47a71a193 a83a194a148a149 a253 a34a28a27a29a22a23a51a103a60a115a116a63a64a31a86a24 a97a175a12a98a171a165a181a182 a158a159a34a98a99a171a100a172a101a102a172a8a103a104a105a106a12a98a171a6a7a23a31a14a83a84a29a107 a242a108 a34a203a204a29a77a35a25a24a109a67a35a25a34 a78a55a22a23a67a60a126aa21a238a34ba29 a241a235 a37a64a31a51a57a110 a36 a120a252z = 0a29 a47a71a193 a83a194a148a149 a253a54 a34a75a24a40 lnz a57a110 a36a111a47a71a193 a67a23a112 a253 a34a245a22a23a67 lnb?lnaa24a79 a47a113 a38a29a31a236a117a106a120a121a29 a193 a83a194a148a149a34a75 a206 a250 a117 a232 a106lnz a29a120a121 a193 a67a23a112a34a78a236a22a23a67 a205 a75 a206 a250a120a121a31 Wu Chong-shi a114a115a116 a117 a118 a119 a120 a1217 a122 §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 a24a123a94a20a83a194a148a149Ga29a208a99a29a124 a71 a23 a45a81a82 a198a77a33a30a34C 1,C2,···,Cn a90 a251 a252 a36C0 a29a253a125 a34a236a59a247a37a29a22a23a115a116a126a96a80a121a31 a703.6 a189a184a185a186a187a209Cauchya210a211 a220 a79 a2023.6a34a120a127a62C0, C1, C2, ···, Cn a244a44a95a54a128a241 a96a31a48a129a51a29a43a30a40C 1, C2, ···, Cn a49 C0 a83a93a222a78 a34a14a236a53a34 a47a71a193 a83a194a148a149Gprime a34f(z) a36a193 a83a194a148a149Gprime a253 a24a195a196a29a34a78a236 a206a207a232a233a193 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 a253a193 a67a34a245 a68 a121 a47 a43a30a74a130a29a22a23a67a131a80a132a133a34 integraldisplay bi ai f(z)dz + integraldisplay ai bi f(z)dz = 0. a247 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) Wu Chong-shi §3.3 a3a213a214a215a216a217 Cauchya218a219 a78 a8 a123 3.3 a83a84 contintegraldisplay C zndz a67a34na44a87a25a34C a29a126a96a44a95a54a128a241a96a31 a131 a51na44a88a103a25 a54 a34a102a103a34a134a135 a193 a83a194a148a149a29 Cauchya38a140 contintegraldisplay C zndz = 0. a51na44a136a87a25 a54 a34a79a80 C a253 a120a252z = 0a34a65 a205 a37 contintegraldisplay C zndz = 0. a79a80C a253 a252a37z = 0a34a65a134a20a83a194a148a149a29 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,···. a137a93a28a27a29a93a80a34a75a37 contintegraldisplay C zndz = ?? ? 2pii, n = ?1, a59Ca253a252a37z = 0; 0, a91a138a139a31. a140a141a34 a223a47a122a142 a34 contintegraldisplay C (z ?a)ndz = ? ? ? 2pii, n = ?1, a59Ca253 a252a37z = a; 0, a91a138a139a31. Wu Chong-shi a0a1a2 a3 a4 a5 a6 a79 a8 §3.4 a143a144a145a146a137a147a139 a148 a192 3.1 a79a80a35a25f(z) a36z = aa46a29a74a149a253a83a84a34 a70a59a51 θ 1≤ arg(z ?a)≤ θ2, |z ?a| → 0a54 a34(z ?a)f(z) a47a149a142 a150a151a106 k a34a65 lim δ→0 integraldisplay Cδ f(z)dz = ik(θ2 ?θ1), a91a92C δ a24 a207 z = a a44a152a153a34 δ a44a154a116a34a155a156a44 θ2 ? θ1 a29a152 a157a34|z ?a| = δ, θ 1 ≤ arg(z ?a) ≤ θ2 a34 a201a2023.7a31 a220 a78a44 integraldisplay Cδ dz z ?a = i(θ2 ?θ1), a703.7 a247 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) a47a149a142 a150a151a106ka34a203 a42a158a159?ε > 0a34?(a60 arg(z ?a)a63a64a29) r(ε) > 0a34a52a51|(z ?a)| = δ < ra54 a34|(z ?a)f(z)?k| < εa31a247 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 Wu Chong-shi §3.4 a160a161a162a163a217a164a219 a710 a8 a148 a192 3.2 a32 f(z) a36 ∞ a46a29a74a149a253a83a84a34a51 θ1 ≤ argz ≤ θ2 a34z → ∞a54 a34zf(z) a47a149a142 a150a151a106K a34a65 lim R→∞ integraldisplay CR f(z)dz = iK(θ2 ?θ1), a91a92C R a24 a207 a77a46a44a152a153a34Ra44a154a116a165a155a156a44θ 2?θ1 a29a152a157a34 |z| = R, θ1 ≤ argz ≤ θ2 (a201a2023.8)a31 a220 a55a166a140a29a227a228a49 a148 a192 3.1 a29a227a228a80a167a31a78a44 integraldisplay CR dz z = i parenleftbigθ 2 ?θ1 parenrightbiga34a247 a207 a703.8 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) a47a149a142 a150a151a106K a34a203 a42a158a159?ε > 0 a34?(a60argz a63a64 a29)M(ε) > 0a34a52a51|z| = R > M a54 a34|zf(z)?K| < εa31a247 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