a0 a1
star a2a3a4a5a6a7a8a9a10 6
star §5.8 a11a12a8a13a14a15a16a17a18a6a7
a0a1a2 a3a4a5a6a7a8a9a10a11a12 a131
a14
a15a16a17 a18a19a20a21a22a23a24a25a26a27
§5.1 a28a29a30a31a32 Taylor a33a34
a35a36a37a38a39a40a41a42a43a44a45a46a47a48a35a36a49a50a38a39a51
a52a53a54
a35a36a49a50a38a39a48a55a56a37a57a39a58
a59a60 5.1 (Taylor)
a61a62a63f(z)a64a65aa66a67a68a69a67C a70a71C a72a73a74a75a76a77a78a67a70a69a79a80z
a81
a75f(z)a82a83a84a85a63a86a87a66(a88a89a90a75f(z)a82a64a
a81
a86a87a66a84a85a63)
f(z) =
∞summationdisplay
n=0
an(z ?a)n,
a91a92
an = 12pii
contintegraldisplay
C
f(ζ)
(ζ ?a)n+1dζ =
f(n)(a)
n! ,
C a93a94a95a96a97a98a99a51
a100 a101a102Cauchy
a103a104a105a106a75a77a78a67C a70a79a107a108
a81z
a75a109
f(z) = 12pii
contintegraldisplay
C
f(ζ)
ζ ?zdζ.
a110a111
a75
1
ζ ?z =
1
(ζ ?a)?(z ?a) =
1
ζ ?a
∞summationdisplay
n=0
parenleftbiggz ?a
ζ ?a
parenrightbiggn
.
a112
a85a63a64
vextendsinglevextendsingle
vextendsinglevextendsinglez ?a
ζ ?a
vextendsinglevextendsingle
vextendsinglevextendsingle≤ r < 1
a69a113a114
a92
a108a115a116a117a75a118
a112
a82a65a119a120a103a104a75
f(z) = 12pii
contintegraldisplay
C
bracketleftBigg ∞summationdisplay
n=0
(z ?a)n
(ζ ?a)n+1
bracketrightBigg
f(ζ)dζ
=
∞summationdisplay
n=0
bracketleftbigg 1
2pii
contintegraldisplay
C
f(ζ)
(ζ ?a)n+1 dζ
bracketrightbigg
(z ?a)n
=
∞summationdisplay
n=0
an(z ?a)n,
an = 12pii
contintegraldisplay
C
f(ζ)
(ζ ?a)n+1 dζ =
f(n)(a)
n! . square
a90a121a122
1. a123a124a69a125a126a82a65a127a128a75a129a130f(z)a64C a70a73a74a131a82
a51
a99a132a133a134a135a136a137a138a139a140a141a142a143a144a134a132a145a139a146a147a148a149a150a151a152a153
§5.1 a3a4a5a6a7Taylora11a12 a132a14
a154a155a156a157a158a159
a42z
a75
a160a161a162a162a
a163
a45a164a165a35a45Cprime
a75
a54z
a166a167
a40a45a46a51 f(z)a40Cprime a46a168
Cprimea169a170a49a50a42a51
2. a171a172Taylora86a87a69a173a106a174a175a176a62a63
a92
a69 Taylora105a106a177a178a75
a110a111
a125a126a179a178
a51
star a64a175a176a62a63
a92
a75f(x)a69a79a80a180a181a63a182a64a75a183a179a184a65a185a186Taylora105a106a182a64(a88Taylora105a106a116
a117)a51
star a64a187a176a62a63
a92
a75a73a74a69a130a188 (a108a180a181a63a182a64)a189a184a65a185a186Taylora85a63a116a117
a51
3. a190a191a192a193 a62a63f(z)a69a194
a81a195a196a197
a123a198 Taylora85a63a69a116a117a199a200
a51
a61 ba111f(z)a69a201 aa81a202
a203
a69a194
a81
a75a76a108a204a90a205a75a116a117a199a200 R = |b?a|a51
f(z)a40a45|z ?a| < |b?a| a46a206a206a49a50a75f(z)a161a162a40a45a46a207a208a163Taylora57a39(a209a210a211a75
Taylora57a39a40a45|z ?a| < |b?a| a46a43a44)a51a154a212a170a211a75f(z)a42Taylora57a39a43a44a213a214a215a216
a157|b?a|
a51
a43a44a213a214a35a217a218a215a219a220
a157|b?a|
a51
a221a222
a75ba223
a212
a166a224
a40a43a44a45a46
a75a225a226
a37a57a39a40a43a44
a45a46a206a206a49a50
a75a227ba223a163a228a223
a42a229a230a231a232 (
a233a234ba223
a170
a161a235
a228a223a75a2365.5 a237)a51
1
1 + z2 =
∞summationdisplay
n=0
(?)nz2n, |z| < 1.
a62a63a69a194
a81z = ±i
a189
a197
a123a198Taylora85a63a69a116a117a199a200R = |±i| = 1a51
a238
a64a175a63a239a240a70a75Taylora85a63a69a116a117a199a200a241a62a63a242a243a244a245a69a246a247a189a248a65a249a250
a51
1
1 + x2 =
∞summationdisplay
n=0
(?)nx2n, ?1 < x < 1,
a189a248a65a124a73a116a117a199a200a66a80
a1111
a75a118a66a62a631/(1+x2)a64a251a252a175a253a72a254
a111a255a0
a82a181a1
a2a3
a79a80a180a181a63
a254
a111
a182a64a69a4
4. Taylor a5a6a7a8a9a10 a11a123a108a252a64a67C a70a73a74a69a62a63a75a76a12a69 Taylora86a87
a111a13
a108a69a75a131
a86a87a247a63an a111a195a196a14a123a69
a51
a100 a15
a123a109a16a252Taylora85a63a64a67C a70a254a116a117a17a178a108a252a73a74a62a63f(z)a75
f(z) = a0 + a1(z ?a) + a2(z ?a)2 +···+ an(z ?a)n +···
= aprime0 + aprime1(z ?a) + aprime2(z ?a)2 +···+ aprimen(z ?a)n +···.
a93a18a19z→aa75a76a20a78a85a63a64C a70a69a79a108a21a113a114
a92
a108a115a116a117a75a22a109
a0 = aprime0.
a119a120a23a24a75a25a93a18a19 z → aa75a26a27
a1 = aprime1.
a0a1a2 a3a4a5a6a7a8a9a10a11a12 a133
a14
a28a112a29a0
a75a131a82a186a27
an = aprimen, n = 0,1,2,···. square
Taylora86a87a69a13a108a242a30a31a32a33a122
star a179a250a83a34a35a97a36a75a27a17a69f(z)a64a178a108a252a67a70a69Taylora86a87
a111a13
a108a69
a51
a118
a112
a75a179a108a123a130a83a188
a181a63a69a37a36a123a86a87a247a63
a51
star a28a38a64a178a108
a81
a86a87a69a16a252Taylora85a63a177a39a75a76a82a65a119a120a40a41a247a63
a51
? a42a43
a111
a64a178a108
a81
a86a87a69a16a252 Taylora85a63a177a39a75a44a82a65a119a120a40a41a247a63
a51
? a178a108a252a62a63a64a179a178
a81
a86a87a27a17a69a16a252Taylora85a63a75a131a45a109a105a46a69a116a117a113a114a75
a47
a179a48a49
a50
a40a41a86a87a247a63
a51
§5.2 Taylora51
a6a52a53a54a55 a134
a14
§5.2 Taylor a56a31a57a58a59a60
a188Taylora85a63a69a97a36a61a248a108a108a62a63
a51
a171a172a129a64a65a108a66a67a68a69a70a69a97a36
a51
star a71a72a105a106a122
ez = 1 + z + z
2
2! +···+
zn
n! +··· =
∞summationdisplay
n=0
zn
n!, |z| < ∞,
sinz = e
iz ?e?iz
2i =
∞summationdisplay
n=0
(?)n
(2n + 1)!z
2n+1, |z| < ∞,
cosz = e
iz + e?iz
2 =
∞summationdisplay
n=0
(?)n
(2n)!z
2n, |z| < ∞,
1
1?z =
∞summationdisplay
n=0
zn, |z| < 1.
star a77a78
a91a73
a62a63a75a74
a111a75a76a77
a83a171a66a71a72a105a106
a51
1
1 + z2 =
∞summationdisplay
n=0
parenleftbig?z2parenrightbign = ∞summationdisplay
n=0
(?)nz2n, |z| < 1.
a109a124a62a63a74a82a65a83a78a104a104a106a69a97a36a79a66a80a81a82a69a173a106a75
1
1?3z + 2z2 = ?
1
1?z +
2
1?2z = ?
∞summationdisplay
n=0
zn + 2
∞summationdisplay
n=0
(2z)n
=
∞summationdisplay
n=0
parenleftbig2n+1 ?1parenrightbigzn, |z| < 1
2.
a109a66a62a63a82a65a83a84a85a80a81a82a69a62a63a69a181a63a88a103a104a75a86
a238
a82a65a87a88a89a188a90
a91 Taylor
a85a63
a51
1
(1?z)2 =
d
dz
1
1?z =
d
dz
∞summationdisplay
n=0
zn
=
∞summationdisplay
n=1
nzn?1 =
∞summationdisplay
n=0
(n + 1)zn, |z| < 1.
star a28a38a62a63a82a65a83a84a85a16a252(a88a91a252)a62a63a69a92a103a75
a238a93
a108a78a104a69Taylora86a87a40a41a87a88a188a90a95a75
a76a82a94a83a85a63a177a92a69a97a36
a51
1
1?3z + 2z2 =
1
1?z ·
1
1?2z =
∞summationdisplay
k=0
zk ·
∞summationdisplay
l=0
2lzl
=
∞summationdisplay
k=0
∞summationdisplay
l=0
2lzk+l =
∞summationdisplay
n=0
parenleftBigg nsummationdisplay
l=0
2l
parenrightBigg
zn
=
∞summationdisplay
n=0
parenleftbig2n+1 ?1parenrightbigzn, |z| < 1
2.
a0a1a2 a3a4a5a6a7a8a9a10a11a12 a135
a14
a20a78a84a85a63a64a116a117a67a70a95a77a116a117a75a22a85a63a177a92
a111a96
a36a69a75a92a103a64a16a116a117a67a69a105a46a113a114a70a97a95a77a116
a117
a51
star a98a123a247a63a36
a51
a99 5.1
a188tanza64z = 0a69Taylora86a87
a51
a100
a20a78tanza111a194a62a63a75a22
a91
a64z = 0a69Taylora86a87a101a129a109a194a102a84a75
tanz =
∞summationdisplay
k=0
a2k+1z2k+1 = sinzcosz,
sinz = cosz ·
∞summationdisplay
k=0
a2k+1z2k+1,
∞summationdisplay
n=0
(?)n
(2n + 1)!z
2n+1 =
∞summationdisplay
l=0
(?)l
(2l)!z
2l ·
∞summationdisplay
k=0
a2k+1z2k+1
=
∞summationdisplay
n=0
parenleftBigg nsummationdisplay
k=0
(?)n?k
(2n?2k)!a2k+1
parenrightBigg
z2n+1.
a40a41a247a63a75a131a27
nsummationdisplay
k=0
(?)k
(2n?2k)!a2k+1 =
1
(2k + 1)!.
a103
a65
n = 0 : a1 = 1;
n = 1 : 12a1 ?a3 = 16, a3 = 13;
n = 2 : 124a1 ? 12a3 + a5 = 1120, a5 = 215;
...
a118
a112
a75a109
tanz = z + 13z3 + 215z5 + 17315z7 +···.
a86tanza69a194
a81
a82a65a104a105a75a85a63a69a116a117a199a200a101a66 pi/2a51
a106a107a108a159a109
a39a110
a75
a219a111a112
a109
a39a113a114a42a115a116a117
a109
a75
a118a222
a169
a161a162a119
a36a120a121a207a208
a109
a39
a75
a122
a35a217
a215a123a124a120a121a57a39a42a125a126a127a128 (
a129
a207a208
a109
a39an a42a49a50a48a130a128)a51
a52a131a132a133a134
a120a121a57a39a135a42a136a35a126
a209
a136a137a126
a109
a39
a75
a218
a161a162a138a107a108a159a109
a39a110a51
star a139a140a141a142a7Taylor a5a6 a77a78a143a144a62a63a75a64a145a146a147a123a198a82a144a104a148a149a75a131a82a150a82a144a62a63a151a152
a153Taylor
a86a87
a51
a99 5.2
a188a143a144a62a63(1 + z)α a64z = 0a69Taylora86a87a75a147a123z = 0a95 (1 + z)α = 1a51
§5.2 Taylora51
a6a52a53a54a55 a136
a14
a100
a82a49
a50
a188a90a62a63(1 + z)α a64z = 0a81a69a154a180a181a63a144a75
f(0) =1,
fprime(0) =α (1 + z)α?1vextendsinglevextendsinglez=0 = α,
fprimeprime(0) =α(α?1) (1 + z)α?2vextendsinglevextendsinglez=0 = α(α?1),
...
f(n)(0) =α(α?1)(α?2)···(α?n + 1) (1 + z)α?nvextendsinglevextendsinglez=0
=α(α?1)···(α?n + 1),
...
a118
a112
(1 + z)α =1 + αz + α(α?1)2 z2 +···
+ α(α?1)···(α?n + 1)n! zn +···
=
∞summationdisplay
n=0
parenleftbiggα
n
parenrightbigg
zn,
a91a92
parenleftbiggα
0
parenrightbigg
= 1 a174
parenleftbiggα
n
parenrightbigg
= α(α?1)···(α?n + 1)n!
a155
a66a67a156a69a157a120a106a86a87a247a63
a51
a85a63a69a116a117a113a114a75a183a130a158a159a160a69
a153
a36
a238
a123
a51
a116a117a199a200a39a78 z = 0a17a159a160a69
a202a161a162
a201a75
a103
a65a75
a202a163
a82a48a69a116a117a113a114
a111|z| < 1, R = 1
a51
a99 5.3
a188a143a144a62a63ln(1 + z)a64z = 0a69Taylora86a87a75a147a123 ln(1 + z)vextendsinglevextendsinglez=0 = 0a51
a100
a64a72a164a147a123a165a75a62a63ln(1 + z)a82a83a84a66a123a103a104a75a118
a112
ln(1 + z) =
integraldisplay z
0
1
1 + zdz =
integraldisplay z
0
∞summationdisplay
n=0
(?)nzndz
=
∞summationdisplay
n=0
(?)n
integraldisplay z
0
zndz =
∞summationdisplay
n=0
(?)n
n + 1z
n+1
=
∞summationdisplay
n=1
(?)n?1
n z
n.
a116a117a113a114
a47
a130a166a159a160a167a35
a153
a51
a116a117a199a200a39a78 z = 0 a17a159a160a69
a202a161a162
a201a75
a202a163
a82a48a69a116a117a113a114
a111
|z| < 1a75R = 1a51
star a168a169a170a171a172a7 Taylor a5a6
a28a38
a62a63f(z)a64z = ∞a81a73a74a75a76
a47
a82a65a64z = ∞a81a86a87a85
Taylora85a63
a51
a0a1a2 a3a4a5a6a7a8a9a10a11a12 a137
a14
a173a174 f(z)
a40∞
a223
a207a208a56Taylora57a39
a75
a212
a170a165a175a176 z = 1/t
a75a226a177 f(1/t)a40t = 0a223
a207
a208a56Taylora57a39a51
a225a163f(1/t)a40t = 0a223
a49a50
a75a178
f
parenleftbigg1
t
parenrightbigg
= a0 + a1t + a2t2 +···+ antn +···, |t| < r;
f(z) = a0 + a1z + a2z2 +···+ anzn +···, |z| > 1r.
a179a180
a122f(z)a40∞a223
a42Taylora57a39a135
a132a181a182
a39a126a168a183a37a126
a75
a184a181a185
a37a126
a75a226
a43a44a186
a167a163
|z| > 1/ra75
a218
a212
a170
a211a75
a57a39a40
a162∞
a163
a45a164a42a136a36a45a46a43a44a51
§5.3 a3a4a5a6a7a187a188a189a190a10a191a3a4a5a6a7a192a193a10 a138a14
§5.3 a28a29a30a31a32a194a195a196a197a198a199a28a29a30a31a32a200a201a198
star a64a65a73a74a62a63a69a16a252a202a130a242a243a75a12a33a203a109a204a69a202a130a69a124a250a205a144
a51
a59a206 a28a38f(z)
a64aa81a71a91a207a114a70a73a74a75f(a) = 0a75a76
a155z = a
a66 f(z)a69a208
a81
a51
a61f(z)a64z = a
a81
a71
a91a207
a114a70a73a74a75a76a146|z ?a|a209a104a210a95a75
f(z) =
∞summationdisplay
n=0
an(z ?a)n,
a22a211z = aa66a208
a81
a75a76a42a109
a0 = a1 = ··· = am?1 = 0, am negationslash= 0.
a112
a95a75
a155 z = aa81
a66f(z)a69ma180a208
a81
a75a177a101a89a75
f(a) = fprime(a) = ··· = f(m?1)(a) = 0, f(m)(a) negationslash= 0.
a208
a81
a69a180a63a254
a111a14
a123a69a212a251a63 a64a62a63a69a73a74a113a114a70a75a179a82a48a109a104a63a102a69a208
a81
a51
a49a50a38a39a213
a223
a42a35a36a214
a134a215a216
a170a41a42a217a218
a215
a59a60 5.2
a211 f(z)a179a219a39a78a208a75
a3
a64a220a221 z = aa64a70a69a113a114a70a73a74a75a76a42a48a222a17a67|z ?a| =
ρ(ρ > 0)a75a45a64a67a70a223a198z = aa82a48a66a208
a81a224
a75f(z)a225
a91a73
a208
a81
a51
a154
a36
a159a226a227
a163
a49a50a38a39a42a213
a223
a217a218
a215a159a226
a51a228a229
a154
a36
a159a226
a75
a161a162
a116a121a49a50a38a39a213
a223
a42a230a231a232a36a214
a134a215a216
a122
a233a234 1
a61f(z)a64G : |z ?a| < R a70a73a74
a51
a211a64G a70a182a64f(z)a69a225a235a143a252a208
a81{z
n}a75
a3
limn→∞zn = a,
a110z
n negationslash= aa75a76f(z)a64G a70a219a660a51
a116a2361 a135a42a237a238 lim
n→∞zn = a
a161a162a239a240
a163a241a242{zn}a42a35a36a243a244a223a163aa51
a233a234 2
a61f(z)a64G : |z ?a| < R a70a73a74
a51
a211a64G a70a182a64a245aa81a69a108a246a247 la88a221a109aa81a69a108
a252a248a113a114ga75a64la72a88g a70f(z) ≡ 0a75a76a64a251a252a113a114 G a70f(z) ≡ 0
a51
a116a2362a42a56a218a186
a167
a170
a162z = a
a223a163
a45a164a42a45
a249
a75
a122
a170a250a123a124a116a251a112a35a217a252a253a42a254
a249
a51
a233a234 3
a61f(z)a64G a70a73a74
a51
a211a64G a70a182a64a108
a81z = a
a71a245aa81a69a108a246a247 la88a221a109aa81a69a108
a252a248a113a114ga75a64la72a88g a70f(z) ≡ 0a75a76a64a251a252a113a114G a70f(z) ≡ 0a51
a0a1a2 a3a4a5a6a7a8a9a10a11a12 a139
a14
a2555.1
a250a123a124
a54
a116a236 1
a0a1
a56a49a50a38a39a42a2a35
a215a159a226
a51
a59a60 5.3
a61a64a113a114 G a70a109a16a252a73a74a62a63 f1(z) a174 f2(z) a75
a3
a64 G a70a182a64a108a252a3a63 {zn}a75
f1(zn) = f2(zn)a51a211{zn}a69a108a252a18a19
a81z = a(negationslash= z
n)
a47a4
a64G a70a75a76a64G a70a109f1(z) ≡ f2(z)a51
a5a6
a75
a161a162a54
a116a236 3
a0a1a163
a116a236 4a51
a233a234 4
a61f1(z)a174f2(z)a254a64a113a114G a70a73a74a75
a3
a64G a70a69a108a246a247a88a108a252a248a113a114a70a177a39a75a76a64
G a70f1(z) ≡ f2(z)a51
a165
a163
a41a42a7a8a9a252
a75a10
a181
a122
a233a234 5
a64a175a253a72a85a11a69a219a39a106a75a64 z a12a13a72a97a14a85a11a75a129a130a171a252a219a39a106a16a15a69a62a63a64 z a12
a13a72a254
a111
a73a74a69
a51
§5.4 a16a17a18a19a20Laurenta21a22 a2310a24
§5.4 a28a29a30a31a32 Laurent a33a34
a108a252a62a63a223a198a82a64a73a74
a81a153 Taylor
a86a87
a224
a75a109a95a183a25a130a26a12a64a194
a81a27a203a28a29
a85a30a31a32a33
a34
a35a36a37a38 Laurenta28a29
a33
a39a40 5.4 (Laurent)
a41a42a32f(z)a43a44ba45a46a47a48a49a50a51a52R1 ≤ |z ?b| ≤ R2
a53a54a55a56a57a58
a59a60a61
a49a52a62a48a63a64 z a65
a58
f(z)a66a44a67a30a31a32
a28a29
a45
f(z) =
∞summationdisplay
n=?∞
an(z ?b)n, R1 < |z ?b| < R2,
a68a69
an = 12pii
contintegraldisplay
C
f(ζ)
(ζ ?b)n+1dζ,
C a70a49a52a62a71a62a46a72a73a48a63a74a72a75a76a77a78a79 (a80a815.2)a33
a825.2 Laurent
a83a84
a85 a86
a49a52a48a62a87a88a89a90a91a92a45 C1
a93
C2
a58
a59a94a95a96a97a98
a51a52a48 Cauchya99a90a100a101
a58a102
f(z) = 12pii
contintegraldisplay
C2
f(ζ)
ζ ?zdζ ?
1
2pii
contintegraldisplay
C1
f(ζ)
ζ ?zdζ.
a60a61C
2 a53
a48a99a90
a58
a66a44a103a104a105a67a44a106a48a107a108
a58
1
2pii
contintegraldisplay
C2
f(ζ)
ζ ?zdζ =
∞summationdisplay
n=0
an(z ?b)n, |z ?b| < R2,
an = 12pii
contintegraldisplay
C2
f(ζ)
(ζ ?b)n+1dζ.
a60a61C
1 a53
a48a99a90
? 12pii
contintegraldisplay
C1
f(ζ)
ζ ?zdζ =
1
2pii
contintegraldisplay
C1
f(ζ)
(z ?b)?(ζ ?b)dζ
= 12pii
contintegraldisplay
C1
f(ζ)
z ?b
∞summationdisplay
k=0
parenleftbiggζ ?b
z ?b
parenrightbiggk
dζ
a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12211
a123
=
∞summationdisplay
k=0
(z ?b)?k?1 · 12pii
contintegraldisplay
C1
f(ζ)(ζ ?b)kdζ.
a124?k?1 = n, k = ?(n + 1)
a58
a59
? 12pii
contintegraldisplay
C1
f(ζ)
ζ ?zdζ =
?∞summationdisplay
n=?1
(z ?b)n · 12pii
contintegraldisplay
C1
f(ζ)
(ζ ?b)n+1dζ
=
?∞summationdisplay
n=?1
an(z ?b)n, |z ?b| > R1,
a68a69
an = 12pii
contintegraldisplay
C1
f(ζ)
(ζ ?b)n+1dζ.
a125a126a127
a90a77a128a129a130
a58
a36
a102
f(z) =
∞summationdisplay
n=?∞
an(z ?b)n, R1 < |z ?b| < R2,
an = 12pii
contintegraldisplay
C
f(ζ)
(ζ ?b)n+1 dζ.
a34a131
a107a108a132a45a42a32 f(z) a43a49a52 R1 < |z ?b| < R2 a62a48 Laurent
a28a29
a58
a68a69
a48a31a32a132a45 Laurenta31
a32a33 square
a133a134a135a136a137 a
n a138a139a140a141a142a143a144a145a146a147a148a149a150C a33 (a151a152a153a154
a133a155a156a157)
a158a159a160
star Laurenta28a29a48a75a161a162a66a44a163a164a45 f(z)a43R1 < |z ?b| < R2 a62
a54a55a56a57
a33
star a60a61 Laurenta28a29a130a165
a58a166
a32 (a167a168a70a169a30a170a48
a166
a32)
an negationslash= 1n!f(n)(b).
star f(z)a43a62a46C1 a69a171
a56a57
a33a72a172a165a130
a58
a43C1
a53
a70
a102a173
a65a48a33
a174a61b
a65
a58
a66a175a70f(z)a48
a173
a65
a58
a162a66a175a70f(z)a48
a56a57
a65a33
? a176a108ba65a70C2 a62a48a177a72
a173
a65
a58
a59C
1 a66a44a178a179a180a181a58
a182a183a184a185a36a186a1870 < |z?b| < R
a33
a34a35a36a37a38f(z)
a43a188a189
a173
a65 ba48a190a52a62a48 Laurenta28a29a33
? a87a46C2 a48a191a192a162a66a44a45∞
a58a193
a174
a43∞a65a162
a182a183
a33
star Laurenta28a29a194
a102
a169a30a170
a58a195a102a196
a30a170a33
§5.4 a112a113a114a115a116Laurenta120a121 a12212a123
? a169a30a170a43a46C2 a62(|z?b| < R2)a197
a60a182a183
a58
a43C2 a62a48a63a74a72
a131
a76a51a52
a69
a72a198
a182a183
a58
a132
a45Laurenta31a32a48a169
a59a127
a90a199
?
a196
a30a170a43a46C1 a87(|z?b| > R1)a197
a60a182a183
a58
a43C1 a87a48a63a74a72
a131
a76a51a52
a69
a72a198
a182a183
a58
a132
a45Laurenta31a32a48a200a201
a127
a90a33
? a126a127a90a77a129a130
a58
a36a202a187 Laurent
a31a32
a58
a43a49a52
R1 < |z ?b| < R2
a62a197
a60a182a183
a58
a43a49a52a62a48a63a74a72
a131
a76a51a52
a69
a72a198
a182a183
a33
? a203R1 = 0a35
a58
Laurenta31a32a48a200a201
a127
a90
a36a204a205a206a207a208 f(z)
a43z = ba65a48
a173a209a210
a33
star Laurent a211a212a213a214a215a216 a41f(z)a43a49a52R1 < |z ?b| < R2 a62
a102
a126a131 Laurent
a31a32
a58
f(z) =
∞summationdisplay
n=?∞
an(z ?b)n =
∞summationdisplay
n=?∞
aprimen(z ?b)n.
a126a217a218a219
a44 (z ?b)?k?1
a58a220
a49a52a62a71a62a46a72a73a48a63a72
a185a221 C
a99a90 (a34a126a131a31a32a43
a185a221
a53a222
a223
a72a198
a182a183
a58a224a225
a66a44a226a170a99a90)
a58
a59a227a61
contintegraldisplay
C
(z ?b)n?k?1dz = 2piiδnk,
a228
a102
ak = aprimek a33
a224
a45ka63a74
a58
a228
a102
ak = aprimek, k = 0,±1,±2,···.
a167a229
a37 Laurenta28a29
a48a177a72
a210
a33
a230a231a232a233 Laurent
a234a235a236a237a215a238a239a240a241a241a242a243
a58 a244a245a246a247a248
a235a242a243 (a249a250a251a252a253
a248
a235) a33
a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12213
a123
§5.5 Laurent a254a255a0a1a2a3
a4 Laurent
a5a6
a58a7a8a9a10a11a12a138a139
a4a136a137 (a133a13a14a15a16
a144a145a142a143a58a147a17a18a19a20a21a22
a141
)a33a23a24a25a26
a58a27a28
a4 Laurent
a5a6
a141a29a30a31a32
a33
a33a34a35a137a36a37a38a39a40a41
a141
Laurenta5a6
a18a42a147a141a58a43
a24
a58
a44a45
a12
a152a153
a31a32a58a46
a14a47a48
a150
a36a133a49a39a40a41a50a51a48 f(z)
a141a52a53
a137
a58a54a55a56a147
a38
a18
f(z)
a141
Laurenta5a6a33
Taylora5a6
a140a57a58a141a31a32a58a8a59a28a60a141a61a62a58a63a7a8a64a12a65
a4 Laurent
a5a6a33
a66 5.4
a67
1
z(z ?1) a430 < |z| < 1 a62a93|z| > 1 a62a48a68
a29
a101a33
a69 1
z(z ?1) a430 < |z| < 1 a62a48a68
a29
a50a101a72a70a70
∞summationtext
n=?∞
anzn a33a71a44
1
z(z ?1) = ?
1
z
1
1?z = ?
1
z
∞summationdisplay
n=0
zn
= ?
∞summationdisplay
n=0
zn?1 = ?
∞summationdisplay
n=?1
zn, 0 < |z| < 1.
a162a66a44a67
a127
a90a90a101a48a72a73
a160
1
z(z ?1) = ?
1
z ?
1
1?z = ?
1
z ?
∞summationdisplay
n=0
zn = ?
∞summationdisplay
n=?1
zn.
1
z(z ?1) a43|z| > 1 a62a48Laurenta68
a29
a50a101a162a70
∞summationtext
n=?∞
anzn
a58
1
z(z ?1) =
1
z2
1
1? 1z
= 1z2
∞summationdisplay
n=0
parenleftbigg1
z
parenrightbiggn
=
?∞summationdisplay
n=?2
zn, |z| > 1.
a133a134a74a75a76a48
a58a77a147
a49a35a137a36a44
a77a141a78
a40a41
a141
Laurenta5a6
a18a79
a44a80
a77a141
a33 1/z(z?1)
a360 < |z| < 1 a41
a141
Laurenta5a6
a46a28a147
a49a81
a52a82a58
a83a36 |z| > 1 a41
a141
Laurenta5a6
a28a84a85
a86a49a81
a52a82a58a87a88a27a28a89a52a82
a33
a66 5.5
a67a90a70
a166
a32a73a67 cotz a43z = 0a190a52a62a48 Laurenta68a91a33
a69
a90a70
a166
a32a73a92a175a67
a61
a102
a179
a131
a196
a30a170 (a169a30a170)a48a93a50a33
cotz =
∞summationdisplay
n=?1
b2n+1z2n+1.
(a151a152a153
a46a28a147
a49a81
a52a82a58
a133a49
a145a94
a36 5.6
a95a96
a45
a33)
cosz = sinz
∞summationdisplay
n=?1
b2n+1z2n+1,
∞summationdisplay
n=0
(?)n
(2n)!z
2n =
∞summationdisplay
k=0
(?)k
(2k + 1)!z
2k+1
∞summationdisplay
l=0
b2l?1z2l?1
§5.5 Laurenta97
a115a98a99a100a101 a12214
a123
=
∞summationdisplay
k=0
∞summationdisplay
l=0
(?)k
(2k + 1)!b2l?1z
2(k+l)
=
∞summationdisplay
n=0
bracketleftBigg nsummationdisplay
l=0
(?)n?l
(2n?2l + 1)!b2l?1
bracketrightBigg
z2n.
a227a102a37a38a103a104a105
a166 n
summationdisplay
l=0
(?)l
(2n?2l + 1)!b2l?1 =
1
(2n)!.
a226a106a67
a56a58
a167
a37
n = 0 : b?1 = 1;
n = 1 : 13!b?1 ? 11!b1 = 12!, b1 = ?13;
n = 2 : 15!b?1 ? 13!b1 + 11!b3 = 14!, b3 = ? 145;
n = 3 : 17!b?1 ? 15!b1 + 13!b3 ? 11!b5 = 16!, b5 = ? 2945;
...
a71a44
cotz = 1z ? 13z ? 145z3 ? 2945z5 ?···.
a94a95cotz
a48
a173
a65a90a107
a58
a66a108a109
a102
a31a32a48
a182a183a184a185
a45 0 < |z| < pia33
a110a111a112
a66a44a113a67a31a32a114a73a33
cotz = 1tanz = 1
z + 13z3 + 215z5 + 17315z7 +···
= 1z 1
1 + 13z2 + 215z4 + 17315z6 +···
= 1z
bracketleftBig
1?
parenleftbigg1
3z
2 + 2
15z
4 + 17
315z
6 +···
parenrightbigg
+
parenleftbigg1
3z
2 + 2
15z
4 + 17
315z
6 +···
parenrightbigg2
?
parenleftbigg1
3z
2 + 2
15z
4 + 17
315z
6 +···
parenrightbigg3
+
parenleftbigg1
3z
2 + 2
15z
4 + 17
315z
6 +···
parenrightbigg4
?+···
bracketrightBig
= 1z
bracketleftBig
1? 13z2 +
parenleftbigg
? 215 + 19
parenrightbigg
z4
a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12215
a123
+
parenleftbigg
? 17315 + 2× 13 × 215 ? 127
parenrightbigg
z6 +···
bracketrightBig
= 1z
bracketleftbigg
1? 13z2 ? 145z4 ? 2945z6 ?···
bracketrightbigg
= 1z ? 13z ? 145z3 ? 2945z5 +···.
a115a116a117
a235a213 Laurent a211a212
a66 5.6
a67a42a32ln z ?2z ?1 a431 < |z| < 2a118 2 < |z| < ∞ a62a48a30a31a32a68a91a33
a69 a110a111a69a119
a70a48a68a91a51a52a70a49a50a51a52
a58
a71a44
a58
a176a108a175a120a30a31a32a68a91a48a121
a58
a37a38
a48a72a70a70Laurent
a31a32a33
a42a32ln z ?2z ?1
a102
a126a131a122
a65
a160 z = 1
a93
z = 2
a58
a228
a43a49a52 1 < |z| < 2 a62
a171
a66a175a120Laurenta68a91a33
a43a49a522 < |z| < ∞ a62
a58
a42a32ln z ?2z ?1 a70
a54a55a56a57
a48
a58
a123a124a125a126a127a128a129
a70
a54a55
a90
a122a130
a58
a72a66a120
Laurenta68a91a33a131a176
a58a132
a129
a70a43a133a79
a53a134
arg(z ?2)?arg(z ?1) = pi
a58
a59
ln z ?2z ?1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=∞
= 0.
a61
a70
a102
ln z ?2z ?1 = ln 1?2/z1?1/z = ln
parenleftbigg
1? 2z
parenrightbigg
?ln
parenleftbigg
1? 1z
parenrightbigg
=
bracketleftBigg
?2z ? 12
parenleftbigg2
z
parenrightbigg2
? 13
parenleftbigg2
z
parenrightbigg3
?···
bracketrightBigg
?
bracketleftBigg
?1z ? 12
parenleftbigg1
z
parenrightbigg2
? 13
parenleftbigg1
z
parenrightbigg3
?···
bracketrightBigg
= ? 1z ? 32 1z2 ? 73 1z3 ?···? 2
n ?1
n
1
zn ?···.
a66 5.7
a67exp
braceleftbiggz
2
parenleftbigg
t? 1t
parenrightbiggbracerightbigg
a430<|t|<∞ a62a48Laurenta68a91a33
a69
a67a31a32
a219
a73a33
a224
a45
ezt/2 =
∞summationdisplay
k=0
parenleftBigz
2
parenrightBigk tk
k!, |t| < ∞,
e?z/2t =
∞summationdisplay
l=0
parenleftBigz
2
parenrightBigl (?)l
l!
parenleftbigg1
t
parenrightbiggl
,
vextendsinglevextendsingle
vextendsingle1t
vextendsinglevextendsingle
vextendsingle< ∞a167|t| > 0,
§5.5 Laurenta97
a115a98a99a100a101 a12216
a123
a71a44
exp
braceleftbiggz
2
parenleftbigg
t? 1t
parenrightbiggbracerightbigg
=
∞summationdisplay
k=0
parenleftBigz
2
parenrightBigk tk
k!
∞summationdisplay
l=0
parenleftBigz
2
parenrightBigl (?)l
l!
parenleftbigg1
t
parenrightbiggl
=
∞summationdisplay
k=0
∞summationdisplay
l=0
(?)l
k!l!
parenleftBigz
2
parenrightBigk+l
tk?l
=
∞summationdisplay
n=0
bracketleftBig ∞summationdisplay
l=0
(?)l
l!(l + n)!
parenleftBigz
2
parenrightBig2l+nbracketrightBig
tn
+
?∞summationdisplay
n=?1
bracketleftBig ∞summationdisplay
l=?n
(?)l
l!(l + n)!
parenleftBigz
2
parenrightBig2l+nbracketrightBig
tn
=
∞summationdisplay
n=?∞
Jn(z)tn,
a68a69
Jn(z) =
?
???
???
???
???
∞summationdisplay
l=0
(?)l
l!(l + n)!
parenleftBigz
2
parenrightBig2l+n
, n = 0,1,2,···;
∞summationdisplay
l=?n
(?)l
l!(l + n)!
parenleftBigz
2
parenrightBig2l+n
, n = ?1,?2,?3,···
a132a45na135Bessela42a32a33
a176a108a178a136a137a65a70a42a32 f(z) a48
a173
a65
a58a225
a43a178a136a137a65a48a190a52a62
a54a55a56a57
a48a121
a58
a59
a66
a86 f(z)
a43∞
a65a48a190a52a62a120 Laurenta68a91(
a102
a35a36a138
a54a139
a165
a187
a43 ∞a65a120 Laurenta68a91)a33
star a140a141f(z)a36∞a142
a141a143
a40a41(∞
a142a23a26)a144a145a146a147
a58a56a148a149a150a151a152a153
t = 1/z
a58
a35a137f(1/t)a36
t = 0a142
a141a143
a40a41(t = 0
a23a26)a144a145a146a147
a58a43
a83
f
parenleftbigg1
t
parenrightbigg
=
∞summationdisplay
n=?∞
antn, 0 < |t| < r,
f(z) =
∞summationdisplay
n=?∞
anz?n, 1r < |z| < ∞.
a133a134
a141
a50a51a154
a144a7a8
a76
a149a18a8
∞a142a151a155a156
a141a147
a49a39a40
a33
star f(1/t)
a141
Laurent
a53
a137
a140a89a52a82
(a157a158a159
a137
a82
)a160
a143a18a89a161
a160
a143a58
a81
a52a82a18a162
a14
a160
a143
a33
a43
a24
a58
a163a164a165
a64a166a58
a74a75a135 f(z)a36z = ∞
a142
a143
a40a41
a141
Laurent
a53
a137
a140a58
z
a141
a81
a52a82a167
a151
a89a161
a160
a143a58
a83
a89a52a82a167
a151
a162
a14
a160
a143
a33
a89a52a82
a163a164a168a169
a150
a35a137 f(z)a36∞
a142
a141a170a171a172
a33
a53a173
a48a131 4
a93
a1316a48a174a175a176a93a50a44a118a131 7
a58
a162a177a66a44a178
a187
a70a43∞a65a190a52a62 Laurenta68a91a33
a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12217
a123
§5.6 a179a180a181a255a182a183a184a185a186
a39a187
a41f(z)a45
a54a55
a42a32(a188a189
a55
a42a32a48a72
a131
a54a55
a90
a122)
a58
ba65a70a190a48
a173
a65a33a176a108a43ba65a191
a43a72
a131
a190a52
a58
a43a192a190a52a62(a114ba65a87)
a58
f(z)a193a193a66a194
a58
a59
a132 ba45f(z)a48a188a189
a173
a65a33
a195
a188a189
a173
a65a48a131a196a33
a60a61
a42a321/sin(1/z)
a58a222
a223
a58
1/z = npi
a58
a167z = 1/npi
a58
n = 0,±1,±2,···a70a190a48
a173
a65a33
a59z = 0
a70
a34a197
a173
a65a48a198a65(a199a179a65)a160a43z = 0a48a63a74a72
a131
a190a52
a69
a58
a200
a191a43a178a136a189
a131
a173
a65
a58
a228z = 0
a70
a195
a188
a189
a173
a65a33
a176a108z = ba70
a54a55
a42a32f(z)a48a188a189
a173
a65
a58
a59
a72a70a191a43a72
a131
a49a520 < |z?b| < R
a58
a43a192a49a52a62
a58
f(z)a66a44a68a91
a187 Laurent
a31a32
a58
f(z) =
∞summationdisplay
n=?∞
an(z ?b)n.
a34a35
a66a175a201a202a203a176a93a204
a160
star a31a32a68a91a101
a171a205
a196a206
a170
a160 b
a65a132a45f(z)a48a66a207
a173
a65a33
z = 0a36a70a42a32
sinz
z =
∞summationdisplay
n=0
(?)n
(2n + 1)!z
2n, |z| < ∞
a93 1
z ?cotz =
1
3z +
1
45z
3 + 2
945z
5 +···, |z| < pi
a48a66a207
a173
a65a33
star a31a32a68a91a101a92
a205
a102
a179
a131
a196a206
a170
a160 b
a65a132a45f(z)a48a199a65a33
star a31a32a68a91a101
a205
a102
a178a136a189
a131
a196a206
a170
a160 b
a65a132a45f(z)a48
a110
a210a173
a65a33
a208
a173
a90a91
a158a159
a42a32a43a203a176
a173
a65a193a48a209a45a33
a250a210a211a212
a227a61
a43a66a207
a173
a65a193
a58
a31a32a68a91a101
a69a171a205
a196a206
a170
a58
a228
a31a32
a171
a92a70a43a49a52a62
a182a183
a58a225
a213
a43a49a52a48
a69
a47
a58
a167a66a207
a173
a65 z = ba193a162a70
a182a183
a48a33arrowvertexdbl
arrowvertexdblarrowdblbt
a34a35
a48
a182a183
a51a52a70a72
a131
a46
a58
a46a47a43a66a207
a173
a65 z = b
a58
a31a32a43
a182a183
a46a62a48a63a72a76a51a52
a69
a72a198
a182a183
a58arrowvertexdbl
arrowvertexdblarrowdblbt
a93
a42a32
a97a214
a58
lim
z→b
f(z) = lim
z→b
∞summationdisplay
n=0
an(z ?b)n = a0.
§5.6 a215a216
a114a115a116a217a218a219a220 a12218
a123
a42a32a43a66a207
a173
a65a193a48a199a179
a55
a70
a102
a179a48a33 arrowvertexdbl
arrowvertexdblarrowdblbt
a67
a102
a199a179
a55
a120a45 f(z)a48a70a221
a58
f(z) =
??
?
f(z), z negationslash= b;
lim
z→b
f(z), z = b,
a34a222a37a38
a48 f(z)a43ba65a162
a36
a70
a56a57
a48
a208
a33
a34
a169a70a66a207
a173
a65
a34
a72a132a223a48
a227
a130a33
a206a224
a58
a176a108z = ba70a42a32f(z)a48a188a189
a173
a65
a58a225
a213f(z)
a43z = ba48a190a52a62
a102
a89
a58
a59z = b
a70f(z)
a48a66a207
a173
a65a33
a85 a86f(z)
a43z = ba48a190a52a62a120 Laurenta68a91
a58
f(z) =
∞summationdisplay
n=?∞
an(z ?b)n, 0 < |z ?b| ≤ ρ.
a224
a45a43a46C : |z ?b| = ρ
a53a58
|f(z)| < M
a58
a71a44
|an| =
vextendsinglevextendsingle
vextendsinglevextendsingle 1
2pii
contintegraldisplay
C
f(z)
(z ?b)n+1dz
vextendsinglevextendsingle
vextendsinglevextendsingle≤ 1
2pi
contintegraldisplay
C
|f(z)|
|z ?b|n+1|dz| <
M
ρn.
a124ρ → 0
a58
a167
a37
an = 0, n = ?1,?2,?3,···. square
a225
a212 a42a32a43a199a65a190a52a62a48 Laurenta68a91
a102a102
a179
a131
a196a206
a170
a58
f(z) =
∞summationdisplay
n=?m
an(z ?b)n
= a?m(z ?b)?m + a?m+1(z ?b)?m+1 +···
+a?1(z ?b)?1 + a0 + a1(z ?b) +···
= (z ?b)?mbracketleftbiga?m + a?m+1(z ?b) + a?m+2(z ?b)2 +···bracketrightbig
= (z ?b)?mφ(z),
φ(z)a43z = ba65a118
a68
a190a52a62a70
a56a57
a48
a58
a?m negationslash= 0a33 ba65
a36
a132a45f(z)a48 ma135a199a65a33
a92a201|z ?b|a226a227a181
a58
|f(z)|a66a44a228
a61
a63a64a169a32
a58
a167
lim
n→b
f(z) = ∞.
a117
a235a236
a225
a212a241a213
a225a229a116a230 ∞
a58a231a232a233a58
a117
a235a236
a225
a212a234a235
a230a236a237
a213a33
a206a224
a58
a176a108 ba70f(z)a48a188a189
a173
a65
a58
a213 lim
z→b
f(z) = ∞
a58
a59b
a70f(z)a48a199a65a33
a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12219
a123
a85
a224
a45 lim
z→b
f(z) = ∞
a58
a228
a203|z ?b| < δ a35
a58
|f(z)| > M, |f(z)|?1 < 1M = ε,
a167
lim
z→b
1
f(z) = 0.
a61
a70a66
a124
1
f(z) = (z ?b)
mg(z),
a68a69lim
z→b
g(z) negationslash= 0
a58
a213g(z)
a43z = ba118
a68
a190a52a62
a56a57
a33a71a44
f(z) = (z ?b)?m · 1g(z) = (z ?b)?mφ(z). square
a238
a62
z = b
a18
f(z)
a141
ma239a240a142
a58a161a241
a38
a18
1/f(z)
a141
ma239a242a142a33
a168
a25a243a244a33
a105a67
a34a131a105
a166a58
a66a44a245a246a247a248a249a250a199a65a33
? z = npia701/sinz a48a72a135a199a65a199
? z = 2kpii, k = 0,±1,±2,···a701/parenleftbigez ?1parenrightbiga48a72a135a199a65a199
? z = 1a701/(z ?1)2 a48a175a135a199a65a33
a251
a216a211a212 a42a32a43
a110
a210a173
a65a190a52a62a48 Laurenta68a91a252
a102
a178a136a189
a131
a196a206
a170a33
? a238
a62
z = b
a18
a35a137f(z)
a141a253a172a170
a142
a58a161a254
z → ba13
a58
f(z)
a141
a240a255
a44a0a36
a33
? a1a2a3
a166a4a58
z → b
a141a31a139
a44
a77a58
f(z)
a7a8a5a6
a44
a77a141
a137
a145a33
a131a176
a58
z = 0a70a42a32
e1/z =
∞summationdisplay
n=0
1
n!
parenleftbigg1
z
parenrightbiggn
, 0 < |z| < ∞
a48
a110
a210a173
a65a33a203 z a44
a171a218
a72a101a7
a61 0a35
a58
a36
a102
a171a218
a48a107a108
a160
?
a254
z a8
a89a9a10a11
a34 0a13
a58
e1/z → ∞a199
?
a254
z a8
a81
a9a10a11
a34 0a13
a58
e1/z → 0a199
?
a254
z a8a12
a10a11
a34 0a13
a58
e1/z a44
a11
a34
a147
a49
a3
a38
a141
a137
a33
a13
a91a70
a58
?
a254
z
a8a14a15
±i/2npi, n = 1,2,3,···
a11
a340a13
a58
e1/z a16a1511 (
a43
a83
a8
1a151a17a18a142) a199
?
a254
z
a8a14a15
±i/(2n+1)pi, n = 0,1,2,3,···
a11
a340a13
a58
e1/z a16a151?1 (
a43
a83
a8
?1a151a17a18a142)a199
?
a254
z
a8a14a15
±i/(2n+ 1/2)pi, n = 0,1,2,···
a11
a34 0a13
a58
e1/z a16a151?i (
a43
a83
a8
?ia151a17a18a142) a33
a66a44a229
a127
a58
a60a61a110
a210a173
a65z = ba130a165
a58
a63a74a19a70a72
a131
a32A(
a102
a179a188∞)
a58
a200
a66a44a250
a38
a72
a131
a20a21z
n → ba58
a168
a37f(z
n) → A(
a171
a229)a33
§5.6 a215a216
a114a115a116a217a218a219a220 a12220
a123
a22a23a128
a139
a165
a58
a43
a110
a210a173
a65a48a63a74a72
a131
a181a190a52a62
a58
a42a32 f(z) a66a44a24 (a128
a213
a24a178a136a189a106)
a63a74a48
a102
a179a32
a55a58a25
a189a66a175
a102
a72
a131
a131a87a33
a236a26a27
a212
a98a28a186a29z = 1/t
a58
a125f(z)
a30
a187f(1/t)
a130
a158a159a31
star
a132
t = 0a65a70f(1/t)a32a66a207
a173
a65
a58
a59 z = ∞
a65a70f(z)a32a66a207
a173
a65a199
star
a132
t = 0a65a70f(1/t)a32a199a65
a58
a59z = ∞
a65a70f(z)a32a199a65a199
star
a132
t = 0a65a70f(1/t)a32
a110
a210a173
a65
a58
a59 z = ∞
a65a70f(z)a32
a110
a210a173
a65
a31
? z = ∞a701/(1 + z)a32a66a207
a173
a65a199
? z = ∞a701 + z2 a32a175a135a199a65a199
? z = ∞a70ez, sinz, cosz, ···a32
a110
a210a173
a65
a31
a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12221
a123
§5.7 a33 a34 a35 a36
a72
a131
a131a196
a31
a206a37a38 ∞summationdisplay
n=0
zn = 1 + z + z2 +···
? a39a40z = 0a65a41a42a43a32a44a45a42g1 : |z| < 1 a46a47a48a49a50a51a52a53a54a55a56
a38
a49a57a41 f1(z)a31
? a39a42a58a49
a37a38a59a60a61
a32
a31
a62a63a64
a53a65
a37a38
a51a66a67a49a68a40a69a70 f1(z)a39a44a45a42a46a71a72a52a73a74a56a75a76a77a78a79a80a75a76a81
a82a83
a49a84z = i/2a73a49a85
f1
parenleftbiggi
2
parenrightbigg
= 1 + i2 +
parenleftbiggi
2
parenrightbigg2
+···,
fprime1
parenleftbiggi
2
parenrightbigg
= 1 + 2· i2 + 3·
parenleftbiggi
2
parenrightbigg2
+···,
fprimeprime1
parenleftbiggi
2
parenrightbigg
= 2·1 + 3·2· i2 + 4·3·
parenleftbiggi
2
parenrightbigg2
+···,
...
f(n)1
parenleftbiggi
2
parenrightbigg
= n! + (n + 1)!1! i2 + (n + 2)!2!
parenleftbiggi
2
parenrightbigg2
+···,
...
a86a87
a49f1(z)a84z = i/2a73a74Taylora88a89
a59
∞summationdisplay
n=0
1
n!f
(n)
1
parenleftbiggi
2
parenrightbigg
·
parenleftbigg
z ? i2
parenrightbiggn
.
a64
a53a90a75a91a92a93a84a94a74a47a48a42g2 : |z ?i/2| < r a46a47a48a49a93a50a51a95a52a53a54a55a56a75a49a57a41 f2(z)a81
a965.3
a97a98a99a100
f1(z)a101f2(z)a102a103a104a105a106a107
§5.7 a108 a109 a110 a111 a11222a113
star a114g1
a115
g2
a116a117a118a119a120
g1intersectiontextg2
a121
a49f1(z) ≡ f2(z)a81
a122a123a124
a49
f1(z) = 11?z,
a125a126
f(n)1
parenleftbiggi
2
parenrightbigg
= n!parenleftbigg
1? i2
parenrightbiggn+1,
a127a128
a49a93a129a69a130
f2(z) =
∞summationdisplay
n=0
1parenleftbigg
1? i2
parenrightbiggn+1
parenleftbigg
z ? i2
parenrightbiggn
= 11?z,
vextendsinglevextendsingle
vextendsinglevextendsinglez ? i
2
vextendsinglevextendsingle
vextendsinglevextendsingle<
√5
2 .
star f1(z)a101f2(z)a131a132a133a134a135a136a137a138a139 (a1401/(1?z))a114a132a135
a119a120a121a116a141a142a143
a81
? a64a144a53a51a66a67a145a85a78a146a74a85a147a148a149a150
g1 : |z| < 1 a151 g2 : |z ?(i/2)| < √5/2.
? a84a152a153a154a155 g1intersectiontextg2 a49f1(z) ≡ f2(z)a81
star a64a156a157a158a159a160a84a154a155 g1 a46a74a65a90a75a70
a60
a49a130a161a95a84a162a52a154a155 g2 a46a74a162a52a65a90a75a51a66a67a81
star a84
a144
a53a154a155a74a152a153a163a164 g1intersectiontextg2 a46a165a166a167a168a81
star a169a170
a64
a53a171a172a49
a157
a68a129a173a70a174a175a74
a159a160
a148a149a49a176a177a68a129a178a88a161a179a53 z a180a181a81
a182a183 a184
a56a75f1(z) a84a154a155g1 a46a54a55a49a56a75 f2(z)a84a154a155 g2 a46a54a55a49a185a84 g1
a186
g2 a74a152a153a154
a155g1intersectiontextg2 a46a49f1(z) ≡ f2(z)a49a187a188f2(z)a189f1(z)a84g2 a46a74a54a55a190a191a192a193a194a49 f1(z)
a59
f2(z)a84g1
a46a74a54a55a190a191a81
? a195a196a197a198a199a200
a116a201a202
a49a203a204a205a206a138a139
a116
a182a183
a120
a101a197a198a207a208a81
a82a83
a49
a63
a90a75a209a210a164
a159a160
a74a56a75a49a211a175a145a85a52
a159
a74a212
a63
a148a149a49a213a214a54a55a190a191a49a68a129
a159a160
a70a84
a215a216
a148a149a46a74a54a55a56a75a81
a124
a181a74
a82a217a218
a49
a157
a130a161a95a84a154a155 g1
uniontextg
2 a46a54a55a74a56a75 f(z)a49
f(z) =
braceleftBigg
f1(z), z ∈ g1;
f2(z), z ∈ g2.
a219a220a221
a108a109a222a223a224a225a226a227a228a229 a11223a113
a162a52a53
a123a230a82a217
a49
a157
a59
Γa56a75a81
a231a232a63
a74a210a164
a159a160(
a233a84a234a235a180a181a54a55)a70
a60
a49
a236
a214a54a55a190a191a49
a158
a185a130a161a94a84a237a180a181a74
a159a160 (
a238a239 9.1a240)a81
? a241a136a242a243a196a244a197a198a199a200
a116a245a246
a134a247a248a249a250a251
a116a252
a197
a245a246
a81
a82a83
a49a253a254a165a79
a232a255
a164a0a1
d2w
dz2 + p(z)
dw
dz + q(z)w = 0,
a213
a232
a93a233a129a69a130a84a52
a159
a148a149a46a74a54a67a81a213a214a54a55a190a191a49a68
a128a158a64
a53a54a67a2a3a70a84a4a5a148a149a46a74a51
a66a67a81
a6 5.1 a184w
1 a59
a0a1
d2w
dz2 + p(z)
dw
dz + q(z)w = 0, (5.1)
a74a54a49a84a154a155 G1 a46a54a55a81a7 tildewidew1
a59
w1 a84a154a155G2 a46a74a54a55a190a191a49a8
w1 ≡ tildewidew1, z ∈ G1intersectiontextG2,
a187 tildewidew1
a9a59
a0a1 (5.1)a74a54a81
a10 a184
d2tildewidew1
dz2 + p(z)
dtildewidew1
dz + q(z)tildewidew1 = g(z),
g(z)a84G2 a46a54a55a81
a86
a189w1
a59
a0a1(5.1)a84a154a155G1 a46a74a54a49
a11
a84a4
a217
a154a155G1intersectiontextG2 a46a49
a9a12a13
a0a1
d2w1
dz2 + p(z)
dw1
dz + q(z)w1 = 0.
a185a84
a87a217
a154a155a46a49 w1(z) ≡ tildewidew1(z)a49
a11
d2tildewidew1
dz2 + p(z)
dtildewidew1
dz + q(z)tildewidew1 = 0, z ∈ G1
intersectiontextG
2,
a8g(z) ≡ 0, z ∈ G1intersectiontextG2 a81a14a15a54a55a56a75a74a16a52a17a49a18a8a19a130
g(z) ≡ 0, z ∈ G2,
a20
a8 tildewidew1 a84G2 a46
a12a13
a0a1
d2tildewidew1
dz2 + p(z)
dtildewidew1
dz + q(z)tildewidew1 = 0. square
a6 5.2 a184w
1 a151w2 a145a59
a0a1 (5.1)a74
a144
a53a21a17a22a23a54a49
a126a24
a84a154a155 G1 a46a54a55a81a7 tildewidew1 a151 tildewidew2
a164a25
a59
w1 a151w2 a84a154a155 G2 a46a74a54a55a190a191a49a8a84 z ∈ G1intersectiontextG2 a218
w1 ≡ tildewidew1, w2 ≡ tildewidew2.
a187 tildewidew1 a151 tildewidew2
a9
a21a17a22a23a81
§5.7 a108 a109 a110 a111 a11224a113
a10 a231a821
a26a49 tildewidew1 a151 tildewidew2
a9a59
a0a1 (a84G2 a46)a74a54a81
a231
a254w1 a151w2 a74a21a17a22a23a17a27a49
?[w1,w2] ≡
vextendsinglevextendsingle
vextendsinglevextendsinglew1 w2
wprime1 wprime2
vextendsinglevextendsingle
vextendsinglevextendsinglenegationslash= 0, z ∈ G1.
a184
?[tildewidew1, tildewidew2] ≡
vextendsinglevextendsingle
vextendsinglevextendsingle tildewidew1 tildewidew2
tildewidewprime1 tildewidewprime2
vextendsinglevextendsingle
vextendsinglevextendsingle= g(z),
g(z)a84G2 a46a54a55a81
a231
a254a84 z ∈ G1intersectiontextG2 a218a49
w1 ≡ tildewidew1, w2 ≡ tildewidew2,
a11g(z) negationslash= 0, z ∈ G
1
intersectiontextG
2 a81a9
a92a14a15a54a55a56a75a74a16a52a17a49
a157
a19a130
g(z) negationslash= 0, z ∈ G2.
a127a128
a49 tildewidew1 a151 tildewidew2(a84G2 a46)
a9
a21a17a22a23a81 square
a54a55a190a191
a59
a170a28a56a75a29a30
a218a31
a169a32a74a33a34a194a52a81
a197a198a199a200
a116a35a36a37
a243a38a39
a245a246
star a197a198a199a200a40a41a42a43a49a44a45a46a47a138a139
a116a48a49
a249a50a81
a83a51
a84a65a90a75f1(z)a74a47a48a52g1 a74a53a54
a124 a55a56
a12
a95a57
a58
a73a49a8a84a47a48a52a59
a124
a71a72a52a73a49a4a71
a72a60a74a61a155a46a145a85f1(z)a74
a58
a73a49
a62a63
a49a84g1 a46a169a64a65Taylora88a89a49a4a47a48a148a149a66a67a68a129a173
a70g1 a81
star a197a198a199a200
a116a68a69
a134a41
a115a70a71
a102a105a107
a72a73
a67a74a75a76a190a191 (a161a74a52a154a155) a74a77
a51
a59a78
a167a74a107
a213a214a54a55a190a191a130a161a74a56a75
a59a79
a76a74a49a80
a59a81
a76a74a107
a27 a82a83a84a85a86a87a88
?[w1,w2]a89G1
a90a91a92a93a94a95a96a97a98
a85a99a100a101
a97
?(z) ≡ ?[w1(z),w2(z)] = ?(z0) · exp
bracketleftBig
?integraltextzz0p(ζ)dζ
bracketrightBig
,
a102a103 ?[w
1(z),w2(z)]a89a104a93a94
z0
a97a98
a85
?[w1(z),w2(z)]a105
a97a98a106
a219a220a221
a108a109a222a223a224a225a226a227a228a229 a11225a113
§?5.8 Bernoulli a107a108 Euler a107
a109a110
a30a56a75
f(z) = zez ?1
a84z = 0a73a74a65a90a75a88a89a81
a231
a254
a87
a56a75a164a111a74a112a73a189 2npi, n = 0,±1,±2,···a49
a126a24
a189a52a79a112a73a192a74
a113z = 0
a114
a59
a56a75a164
a217
a74a52a79a112a73a49
a86a87 z = 0
a73
a59
f(z)a74a68a115
a58
a73a49
limz→0 zez ?1 = 1.
f(z)a68a84a52a155|z| < 2pi a116a65a117a118a88a89a49
z
ez ?1 =
∞summationdisplay
n=1
Bn
n! z
n,
Bn a188a189a119a120a62a75a81a121a181a157a175a69 Bn a81a86a189
z
ez ?1 =
z
2
parenleftbiggez + 1
ez ?1 ?1
parenrightbigg
= z2 e
z/2 + e?z/2
ez/2 ?e?z/2 ?
z
2,
a122
a52a123a189z a74a124a56a75a49a88a89
a113
a233a85 z a74a124a125a65a49
a127a128
B2n+1 = ?12δn0, n = 0,1,2,···.
a126a63a127a159a128
a75a129a69a70 B2n a81a189
a87
a49a130 (5.41)a67a131a132a133
ez ?1
z
bracketleftbigg
? z2 +
∞summationdisplay
n=0
B2n
(2n)!z
2n
bracketrightbigg
= ?12
∞summationdisplay
k=1
1
k!z
k +
∞summationdisplay
k=0
bracketleftbigg[k/2]summationdisplay
n=0
1
(k?2n + 1)!
B2n
(2n)!
bracketrightbigg
z2k = 1,
a157
a68
a128
a69a130 B0 = 0a128a77B2n a74a134a2a23
a128
[k/2]summationdisplay
n=0
k!
(k?2n + 1)!
B2n
(2n)! =
1
2.
a158
a185a2a130
B2 = 16, B4 = ? 130, B6 = 142, B8 = ? 130,
B10 = 566, B12 = ? 6912730, B14 = 76, B16 = ?3617510 , ···.
a67a135a136a70 B4n?2 > 0, B4n < 0, n = 1,2,3,···a81
a137a63a124
a181a74a77
a51
a80a68
a128
a130a161a138
a81
a85
a63
a74a88a89a67a49
a82a83
z
2 cot
z
2 =
iz
2
eiz/2 + e?iz/2
eiz/2 ?e?iz/2 =
∞summationdisplay
n=0
(?)n B2n(2n)!z2n, |z| < 2pi;
z
2 tan
z
2 =
z
2 cot
z
2 ?z cotz =
∞summationdisplay
n=1
(?)n?12
2n ?1
(2n)! B2nz
2n, |z| < pi;
z cscz = z2 cot z2 + z2 tan z2 =
∞summationdisplay
n=0
(?)n?12
parenleftbig22n?1 ?1parenrightbig
(2n)! B2nz
2n, |z| < pi;
§?5.8 Bernoullia223a139Eulera223 a11226a113
ln sinzz =
∞summationdisplay
n=1
(?)n 2
2n?1
n(2n)!B2nz
2n, |z| < pi;
lncosz =
∞summationdisplay
n=1
(?)n2
2n?1(22n ?1)
n(2n)! B2nz
2n, |z| < pi
2;
ln tanzz =
∞summationdisplay
n=1
(?)n?1 2
2n(22n?1 ?1)
n(2n)! B2nz
2n, |z| < pi
2.
a63
a74
a156
a74a140a129a80a68
a128
a130a161
2ez/2
ez + 1 = sech
z
2 =
∞summationdisplay
n=0
En
n!
parenleftBigz
2
parenrightBign
, |z| < pi,
a4
a218E
n a188a189a141a142a75a49
E2n+1 = 0, n = 0,1,2,···;
E0 = 1, E2 = ?1, E4 = 5, E6 = ?61, ···.
a94a143
a12a13
a134a2a23
a128
ksummationdisplay
l=0
(2k)!
(2l)!(2k?2l)!E2l = 0, k ≥ 1.
a124
a181a74a77
a51
a80a68
a128
a132a133
2ez/2
ez + 1 = sech
z
2 =
∞summationdisplay
n=0
E2n
(2n)!
parenleftBigz
2
parenrightBig2n
, |z| < pi.