Wu Chong-shi
Wu Chong-shi
Wu Chong-shi
2 a0 a1 a2
a3a4a5 a6a7a8a6a9a10a7
1. a11a12a13a14a15a16a17a18a19a20a21a19a20a22a23a24a25a26
(1) 1 + i√3; (2) eisinx, xa27a18a16;
(3) eiz; (4) ez;
(5) eiφ(x), φ(x)a28a18a29a16xa17a18a30a16; (6) 1?cosα+ isinα, 0 ≤α<2pi.
2. a31a13a14a32a33a34a35a36a37a38a39a40a12a41a26
(1) |z|< 2; (2) |z| = 2;
(3) |z|> 2; (4) Rez> 12;
(5) 1 <Imz <2; (6) 0 <arg(1?z) < pi4;
(7) |z?a| = |z?b|, a,ba27a42a16;
(8) |z?a|+|z?b| =c, a,b, ca43a27a42a16, c>|a?b|.
3. a44a13a14a45a14{zn}a17a46a47a23a48a49a50a51a52a28a18a16a45a14a50a53a54a55a44a12a56a48a49a23a13a48a49a26
(1) zn = (?)n n2n+ 1; (2) zn = (?)n 12n+ 1;
(3) zn =n+ (?)n(2n+ 1)i; (4) zn = (2n+ 1)+ (?)nni;
(5) zn =
parenleftbigg
1 + in
parenrightbigg
sin npi6 ; (6) zn =
parenleftbigg
1 + 12n
parenrightbigg
cos npi3 .
a3a57a5 a58a59a10a7
1. a60a61a13a14a30a16a62a36a63a64a65(a66a44a12a67a65a16)a20a62a36a63a68a69a26
(1) |z|; (2) z?;
(3) zm, m = 0, 1, 2, ···; (4) zRez;
(5) parenleftbigx2 + 2yparenrightbig+ iparenleftbigx2 +y2parenrightbig; (6) (x?y)2 + 2i(x+y).
2. a70a71a72a73a48a74a75a33(r,θ)a13a17Cauchy–Riemanna76a77a26
?u
?r =
1
r
?v
?θ,
?v
?r = ?
1
r
?u
?θ,
u(r, θ)a23v(r, θ)a78a79a27a15a29a30a16a17a18a19a23a21a19a80
3. a81a34
a72a73
a48a74a75a33(r,θ)a13a17Cauchy–Riemanna76a77a70a71a26
fprime(z) = rz
parenleftbigg?u
?r + i
?v
?r
parenrightbigg
= 1z
parenleftbigg?v
?θ ?i
?u
?θ
parenrightbigg
.
4. a82z = x+ iya50a83a84a68a69a30a16f(z) = u(x,y) + iv(x,y)a17a18a19 u(x,y)a51a13a50a85a44a12a68
a69a30a16f(z)a26
Wu Chong-shi
a86 a87 3
(1) x2 ?y2 +x; (2) xx2 +y2;
(3) ey cosx; (4) cosxcoshy.
5. a82z =x+iya50a83
a84
a68a69a30a16f(z) =u(x,y)+iv(x,y)a17a18a19a88a21a19a51a13a50a85a44fprime(z)a26
(1) u =x+y; (2) u = sinxcoshy.
6. a89f(z) = u(x,y)+ iv(x,y)a68a69a50a90
u?v = (x?y)(x2 + 4xy+y2),
a85a44f(z)a80
7. a68a13a14a76a77a26
(1) sinz = 34 + i4; (2) cosz = 4;
(3) sin2z? 32 sinz?1 = 0; (4) tanz = i;
(5) sinhz = 0; (6) 2cosh2z?3coshz+ 1 = 0.
8. a60a61a13a14a30a16a28a91a92a17a93a28a94a92a17a26
(1) √z2 ?1; (2) z+√z?1;
(3) sin√z; (4) cos√z;
(7)sin
√z
√z ; (8) cos
√z
√z ;
(9) lnsinz; (10) sinparenleftbigilnzparenrightbig.
9. a95a12a13a14a94a92a30a16a17a96a47a50a66a97a98za99a100a101a96a47a102a103a100a104a105
a106a107
a63a108a30a16a92a17a29a109a80
a51a52a54a55a99a110a101a20a111a101a20a112a113a114a94a101a96a47a100a104a50a30a16a92a115a51a36a29a109a116
(1) radicalbig(z?a)(z?b), anegationslash= b; (2)
radicalbiggz?a
z?b, anegationslash= b;
(3) 3radicalbig(z?a)(z?b), anegationslash= b; (4) 3radicalbig(z?a)2;
(5) √1?z3; (6) 3√1?z3;
(7) ln(z2 + 1); (8) lncosz.
10. a44a13a14a30a16a62a117a118a47a17a119a19a64a120a121a92a26
(1) lnz,z = 1,i,?1,1+ i;
(2) zi, z = 2,i,?1,(1+ i).
11. a122a118a30a16w = z 3√z?2a62a372.1 a123
a124a125
a56a126a17
a24a25a270a50a85a44a127a30a16a62
a124a125
a13a126z = 3a63a17a16a92a80
a115a128a26
a129
a101a30a16a130a35a101a91a92a78a96a116a44a12a62a67a131a78a96
a123
a124a125
a13a126z = 3a63a17a30a16a92a80 a1322.1
Wu Chong-shi
4 a0 a133 a2
a1322.2
12. a83
a84
a30a16w = ln(1 ?z2) a50a122a118w(0) = 0 a50a85a97a98a134z a49a135a62a37 2.2(a) a23 (b) a123a17
w(3)a92a80a89a136
a124a125
a51a372.2(c)a50a53a62
a124a125
a56a20a13a126z = 3a63wa115a121a36a92a116
a1322.3
13. a137a138a139a30a16arctanza17a118a140a27
arctanz ≡ 12i ln 1 + iz1?iz.
a89a136
a124a125
a51a372.3a50a66a122a118
arctanzvextendsinglevextendsinglez=0 = pi,
a44a30a16a62z = 2a63a17a65a16a92a80
14. a83
a84
a30a16f(z) =z?p(1?z)p, ?1 <p<2a80a89a62a18a141a56a1420a1061a136
a124a125
a50a122a118
a124a125
a56a126argz = arg(1?z) = 0a50a85a44f(±i)a23f(∞)a80
15. a89a30a16f(z) a62a143a144G a145a68a69a50a90a67a22a27a42a16a50a70a71f(z) a146a147a148
a149
a27a42a16a80
a3a150a5 a6a9a151a152
1. a85a153a154a118a17a155a156a157a158a13a14a159a78a26
(1)
integraldisplay 2+i
0
Rezdza50a159a78a155a156a27a26
(i)a125a160[0, 2]a23[2, 2 + i]a161a162a17a163
a125
a50 (ii)a125a160z = (2 + i)t, 0 ≤t≤ 1;
(2)
integraldisplay
C
dz√
z a80a122a118
√zvextendsinglevextendsingle
z=1 = 1a50a159a78a155a156a27a164z = 1a12a165a17a26
(i)a91a166a167a17a56a168a104a50 (ii)a91a166a167a17a13a168a104a80
2. a157a158a13a14a159a78a26
(1)
contintegraldisplay
|z|=1
dz
z ; (2)
contintegraldisplay
|z|=1
|dz|
z ;
(3)
contintegraldisplay
|z|=1
dz
|z|; (4)
contintegraldisplay
|z|=1
vextendsinglevextendsingle
vextendsinglevextendsingledz
z
vextendsinglevextendsingle
vextendsinglevextendsingle.
3. a157a158a13a14a159a78a26
(1)
contintegraldisplay
C
1
z2 ?1 sin
piz
4 dza50Ca78a79a27a26
Wu Chong-shi
a86 a87 5
(i) |z| = 12, (ii) |z?1| = 1,
(iii) |z| = 3, (iv) |z| = R,R→∞;
(2)
contintegraldisplay
C
1
z2 + 1e
izdz
a50Ca78a79a27a26
(i) |z?i| = 1, (ii) |z| = 2,
(iii) |z+ i|+|z?i| = 2√2, (iv)a169a170a171
a125r = 3?sin2θ
4.
4. a157a158a13a14a159a78a26
(1)
contintegraldisplay
|z|=2
cosz
z dz; (2)
contintegraldisplay
|z|=2
z2 ?1
z2 + 1dz;
(3)
contintegraldisplay
|z|=2
sin(ez)
z dz; (4)
contintegraldisplay
|z|=2
ez
coshzdz.
5. a157a158a13a14a159a78a26
(1)
contintegraldisplay
|z|=2
sinz
z2 dz; (2)
contintegraldisplay
|z|=2
|z|ez
z2 dz;
(3)
contintegraldisplay
|z|=2
sinz
z4 dz; (4)
contintegraldisplay
|z|=2
dz
z2(z2 + 16).
6. (1)a157a158a159a78
contintegraldisplay
|z|=1
ez
z3dza172
(2) aa121a36a92a55a50a30a16F(z) =
integraldisplay z
z0
ez
parenleftBig1
z +
a
z3
parenrightBig
dza28a91a92a17a116
7. a44|sinz|a62a169a143a1440 ≤ Rez ≤ 2pi, 0 ≤ Imz ≤ 2pi a123a17a173a174a92a80
a3a175a5 a176a177a178a7
1. a60a61a13a14a179a16a17a180a181a182a183a184a185a180a181a182a26
(1)
∞summationdisplay
n=2
in
lnn; (2)
∞summationdisplay
n=1
in
n.
2. a70a71a179a16
∞summationdisplay
n=1
zn?1
(1?zn)(1?zn+1), |z|negationslash= 1
a180a181a50a66a44a67a23a80
3. a85a186a118a13a14a179a16a17a180a181a143a144a26
(1)
∞summationdisplay
n=1
zn!; (2)
∞summationdisplay
n=1
parenleftbigg z
1 +z
parenrightbiggn
;
(3)
∞summationdisplay
n=1
(?)n(z2 + 2z+ 2)n; (4)
∞summationdisplay
n=1
2n sin z3n.
Wu Chong-shi
6 a0 a187 a2
4. a70a71a179a16
∞summationdisplay
n=0
bracketleftBigzn+1
n+ 1 ?
2z2n+3
2n+ 3
bracketrightBig
a17a23a30a16a62z = 1a47a188a189a190a80
5. a70a71a26
ln(1?z) = ?z? z
2
2 ?
z3
3 ?
z4
4 ?···, |z|< 1,
a66a164
a191
a65a12
rcosθ?r2cos2θ2 +r3cos3θ3 ?+··· = 12 lnparenleftbig1 + 2rcosθ+r2parenrightbig,
rsinθ?r2sin2θ2 +r3sin3θ3 ?+··· = arctan rsinθ1 +rcosθ,
a67a123?1 <r<1a80
6. a44a13a14a179a16a192a23a26
(1) cosθ+ cos2θ2 + cos3θ3 + cos4θ4 +···, 0 <θ< 2pi,
sinθ+ sin2θ2 + sin3θ3 + sin4θ4 +···, 0 <θ< 2pi;
(2) cosθ+ cos3θ3 + cos5θ5 + cos7θ7 +···, 0 <θ<pi,
sinθ+ sin3θ3 + sin5θ5 + sin7θ7 +···, ?pi2 ≤θ≤ pi2;
(3) sinθ? sin3θ32 + sin5θ52 ? sin7θ72 +?···, ?pi2 ≤θ ≤ pi2;
(4) cosθ? cos5θ5 + cos7θ7 ? cos11θ11 +?···, ?pi3 <θ< pi3.
a193a194
a26a195a196a197a198a199a200a201
a202Abel
a203
a204a205a206
a80
7. a85a44a13a14a207a179a16a17a180a181a168a156a26
(1)
∞summationdisplay
n=1
1
nnz
n; (2)
∞summationdisplay
n=1
1
2nnnz
n;
(3)
∞summationdisplay
n=1
n!
nnz
n; (4)
∞summationdisplay
n=1
(?)n
22n(n!)2z
n;
(5)
∞summationdisplay
n=1
nlnnzn; (6)
∞summationdisplay
n=1
1
22nz
2n;
(7)
∞summationdisplay
n=1
lnnn
n! z
n; (8)
∞summationdisplay
n=1
parenleftbigg
1? 1n
parenrightbiggn
zn.
a3a208a5 Taylor
a209a210
a8 Laurent
a209a210
1. a211a13a14a30a16a62a117a118a47a212a213a27Taylora179a16a50a66a154a12a67a180a181a168a156a26
Wu Chong-shi
a86 a87 7
(1) 1?z2,a62z = 1a212a213a172 (2) sinz,a62z = npia212a213a172
(3) 11 +z+z2,a62z = 0a212a213a172 (4) sinz1?z,a62z = 0a212a213a172
(5) exp
braceleftbigg 1
1?z
bracerightbigg
,a62z = 0a212a213(a64a214a44a215a216a217)a80
2. a211a13a14a30a16a62a117a118a47a212a213a27Taylora179a16a50a66a154a12a67a180a181a168a156a26
(1) lnz,a62z = ia212a213a50a122a1180 ≤ argz< 2pi;
(2) lnz,a62z = ia212a213a50a122a118lnzvextendsinglevextendsinglez=i = ?32pi;
(3) arctanza17a218a92a50a62z = 0a212a213a172
(4) ln 1 +z1?z,a62z = ∞a212a213a50a122a118 ln 1 +z1?z
vextendsinglevextendsingle
vextendsingle
z=∞
= (2k+ 1)pia80
3. a44a13a14a219a220a179a16a192a23a26
(1)
∞summationdisplay
n=0
1
2n+ 1z
2n+1, |z|< 1; (2)
∞summationdisplay
n=0
1
(2n)!z
2n, |z|<∞.
4. a44a13a14a30a16a17Laurenta212a213a26
(1) 1z2(z?1),a62z = 1a221a222a212a213; (2) 1z2(z?1),a212a213a143a144a271 <|z|<∞;
(3) 1z2 ?3z+ 2,a212a213a143a144a271 <|z|< 2; (4) 1z2 ?3z+ 2,a212a213a143a144a272 <|z|<∞;
(5) (z?1)(z?2)(z?3)(z?4),a212a213a143a144a273<|z|<4; (6) (z?1)(z?2)(z?3)(z?4),a212a213a143a144a274<|z|<∞.
5. a34a179a16a223a224a17a76a225a44a13a14a30a16(a121a218a92a78a96)a62z = 0a47a221a222a17a179a16a212a213a26
(1) ?ln(1?z)ln(1 +z); (2) ln(1 +z2)arctanz.
6. a60a61a13a14a30a16a226a47a17a182a227a50a51a52a28a48a47a50a186a118a67a228a16a26
(1) 1z2 +a2, anegationslash= 0; (2) cosazz2 ;
(3) cosaz?cosbzz2 , anegationslash= b; (4) sinzz2 ? 1z;
(5) cos 1√z; (6)
√z
sin√z;
(7) 1(z?1)lnz; (8)
integraldisplay z
0
sinh√ζ√
ζ dζ.
7. a60a61a13a14a30a16a62∞a47a17a182a227a26
(1) z2; (2) 1z;
(3) coszz ; (4) zcosz;
(5) z
2 + 1
ez ; (6) exp
braceleftbigg
? 1z2
bracerightbigg
;
(7) 1cosh√z; (8) radicalbig(z?1)(z?2).
Wu Chong-shi
8 a0 a229 a2
a3a230a5 a231a232a152a233a234a235a236a178a7a58a237
1. a44a238a228
a125
a182a42a239a78a76a77a50a240a67a68a27a26
(1) w1(z) =z,w2(z) = ez; (2) w1(z) = exp
braceleftbigg1
z
bracerightbigg
, w2(z) = exp
braceleftbigg
?2z
bracerightbigg
;
(3) w1(z) = cos az, w2(z) = sin az; (4) w1(z) = z
2
z2 ?1,w2(z) =
z
z2 ?1.
2. a44a13a14a76a77a62z = 0a241a144a145a17a110a101a179a16a68a26
(1) wprimeprime ?z2w = 0; (2) wprimeprime ?zw = 0;
(3) (z2 ?1)wprimeprime +zwprime ?w = 0; (4) (1 +z+z2)wprimeprime + 2(1 + 2z)wprime + 2w = 0;
3. a44a13a14a76a77a62z = 0a241a144a145a17a110a101a179a16a68a26
(1) z2(1?z)wprimeprime+z(1?3z)wprime?(1+z)w = 0; (2) 9z2wprimeprime ?15zwprime + (36z4 + 7)w = 0.
(3) zwprimeprime ?zwprime +w = 0; (4) zwprimeprime + (z?1)wprime +w = 0;
4. a44a76a77 d
2u
dz2 +
2
z
du
dz +m
2u = 0
a62z = 0a221a222a17a110a101a242a243a68a80
5. a44a76a77
d2w
dz2 +
1
z
dw
dz ?m
2w = 0
a62z = 0a221a222a17a110a101a242a243a68a80
a3a244a5 a58a59a245a246
1.a205a247a248a249a250a251a252a253a254a255a0a1a2a3a201
a4a5a6a7a8a9
a80a136a27a100a101a10a11a50a70a71
f1(z) = 1 +az+a2z2 +a3z3 +···
a183
f2(z) = 11?z ? (1?a)z(1?z)2 + (1?a)
2z2
(1?z)3 ?+···
a12
a27a68a69a13a14a80
2. a219a220a179a16a62a188a54a143a144a145a64a15a180a181
a106
a188a54a17a23a30a16a80a16
a255a0a17a18a2a19a20(
a248a249a250a251
a252a253)
a21a22
a250
a23a24a254a1a2a25a26a24
a50a27a28
a249a4a5a6a7a8a9
a80a136a27a100a101a10a11a50a70a71a179a16
∞summationdisplay
n=1
parenleftbigg 1
1?zn+1 ?
1
1?zn
parenrightbigg
a62a143a144|z|<1a183|z|>1 a145a78a79a29a39a110a101a68a69a30a16a50a30a188
a12
a27a68a69a13a14a80
3. a83
a84
a26
f(z) =
∞summationdisplay
n=0
z2n = z+z2 +z4 +z8 +z16 +···, |z|< 1.
(1)a70a71a26 z = 1a28f(z)a17a226a47a172
(2)a70a71a26 f(z) =z+f(z2)a50a31
a191
a50z2 = 1a17a32a148a33a28f(z)a17a226a47a172
Wu Chong-shi
a86 a87 9
(3)a34a35a36a70a71a26 z2k = 1a172k a101a32a148a28f(z)a17a226a47a50ka27a37a38a138a39a16a172
(4) a164
a191
a70a71a26a188a64a120a211f(z)a13a14
a106
a91a166a167
a40
a80
a3a41a5 a42a7a43a44a45a46a47a48
1. a44a13a14a30a16a62a117a118a47z0 a63a17a49a16a26
(1) 1z?1 expparenleftbigz2parenrightbig, z0 = 1; (2) 1(z?1)2 expparenleftbigz2parenrightbig,z0 = 1;
(3)
parenleftbigg z
1?cosz
parenrightbigg2
, z0 = 0; (4) z
2
z4 ?1, z0 = i;
(5) 1z2 sinz, z0 = 0; (6) 1+ e
z
z4 ,z0 = 0;
(7) e
z
(z2 ?1)2, z0 = 1;
(8) 1cosh√z, z0 = ?
parenleftbigg2n+ 1
2 pi
parenrightbigg2
, n = 0,1,2,···.
2. a44a13a14a30a16a62a226a47a63a17a49a16a26
(1) 1z3 ?z5; (2) 1(1 +z2)m+1,ma27a138a39a16;
(3) z1?cosz; (4)
√z
sinh√z;
(5) exp
bracketleftbigg1
2
parenleftbigg
z? 1z
parenrightbiggbracketrightbigg
; (6) cos 1√z;
(7) 1(z?1)lnz;
(8) 1z
bracketleftbigg
1 + 1z+ 1 + 1(z+ 1)2 +···+ 1(z+ 1)n
bracketrightbigg
.
3. a44a13a14a30a16a62∞a47a63a17a49a16a26
(1) 1z; (2) coszz ;
(3) zcosz; (4) (z2 + 1)ez;
(5) exp
parenleftbigg
? 1z2
parenrightbigg
; (6) radicalbig(z?1)(z?2).
4. a157a158a13a14a159a78a92a26
(1)
contintegraldisplay
|z?1|=1
1
1 +z4dz; (2)
contintegraldisplay
|z?1|=2
1
1 +z4dz;
(3)
contintegraldisplay
|z?1|=1
1
z2 ?1 sin
piz
4 dz; (4)
contintegraldisplay
|z|=3
1
z2 ?1 sin
piz
4 dz;
Wu Chong-shi
10 a0 a50 a2
(5)
contintegraldisplay
|z|=n
tanpizdz,na27a138a39a16; (6)
contintegraldisplay
|z|=2
1
z3(z10 ?2)dz;
(7)
contintegraldisplay
|z|=1
ez
z3dz;
(8)
contintegraldisplay
|z|=R
z2
e2piiz3 ?1dz,n<R
3 <n+ 1,n
a27a138a39a16a80
5. a157a158a13a14a159a78a26
(1)
integraldisplay 2pi
0
cos2nθdθ, na27a138a39a16; (2)
integraldisplay 2pi
0
dx
(a+bcosx)2,a>b>0;
(3)
integraldisplay pi
0
dθ
1 + sin2θ; (4)
integraldisplay pi
0
dθ
(1 + sin2θ)2.
6. a157a158a13a14a159a78a26
(1)
integraldisplay ∞
?∞
x2
1 +x4dx;
(2)
integraldisplay ∞
?∞
x2m
1 +x2ndx, n,ma43a27a138a39a16a50a90n>m;
(3)
integraldisplay ∞
?∞
1
(1 +x2)n+1dx, na27a138a39a16; (4)
integraldisplay ∞
?∞
dx
(1 +x2)cosh pix2
.
7. a157a158a13a14a159a78a26
(1)
integraldisplay ∞
0
cosx
1 +x4dx; (2)
integraldisplay ∞
0
cosx
(1 +x2)3dx;
(3)
integraldisplay ∞
?∞
xsinx
x2 ?2x+ 2dx;
(4)
integraldisplay ∞
0
sin(a+ 2n)x?sinax
(1 +x2)sinx dx, a>?1,na27a138a39a16.
8. a157a158a13a14a159a78a26
(1) v.p.
integraldisplay ∞
?∞
dx
x(x?1)(x?2); (2)
integraldisplay ∞
0
sin(x+a)sin(x?a)
x2 ?a2 dx, a> 0;
(3)
integraldisplay ∞
0
x?sinx
x3(1 +x2)dx; (4)
integraldisplay ∞
?∞
epx ?eqx
1?ex dx, 0 <p<1, 0 <q< 1.
9. a157a158a13a14a159a78a26
(1)
integraldisplay ∞
0
xα?1
1?xdx, 0 <s< 1; (2)
integraldisplay ∞
0
x?α(cospx?cosqx)dx, 0<α<2,p,q>0;
(3)
integraldisplay ∞
0
xs
(1 +x2)2dx, ?1 <s<3; (4)
integraldisplay ∞
0
xα?1 lnx
1+x dx, 0 <α<1;
(5)
integraldisplay ∞
0
lnx
x2 +a2dx, a>0; (6)
integraldisplay ∞
0
lnx
(x+a)(x+b)dx, b>a>0.
Wu Chong-shi
a86 a87 11
a3a51a5 Γ a10a7
1. a211a13a14a189a224a159a34Γa30a16a39a40a12a41a26
(1) (2n)!!; (2) (2n?1)!!;
(3) (1 +ν)(2 +ν)(3 +ν)···(n+ν);
(4) bracketleftbign(n+ 1)?ν(ν + 1)bracketrightbigbracketleftbig(n?1)n?ν(ν + 1)bracketrightbig···bracketleftbig0?ν(ν + 1)bracketrightbig.
2. a157a158a13a14a159a78a26
(1)
integraldisplay ∞
0
x?α sinxdx, 0 <α<2,
integraldisplay ∞
0
x?α cosxdx, 0 <α<1;
(2)
integraldisplay ∞
0
xα?1e?xcosθ cos(xsinθ)dx, α>0, ?pi2 <θ< pi2,
integraldisplay ∞
0
xα?1e?xcosθ sin(xsinθ)dx, α>0, ?pi2 <θ< pi2.
3. a82ψ(z) = ddz lnΓ(z) = Γ
prime(z)
Γ(z) a50a70a71a26
(1) ψ(z+ 1) = 1z +ψ(z); (2) ψ(z+n)?ψ(z) = 1z+ 1z+1+···+ 1z+n?1;
(3) ψ(1?z)?ψ(z) = picotpiz; (4) 2ψ(2z)?ψ(z)?ψ
parenleftbigg
z+ 12
parenrightbigg
= 2ln2.
4. a157a158a13a14a159a78a26
(1)
integraldisplay 1
?1
(1?x)p(1 +x)qdx, Rep>?1, Req>?1;
(2)
integraldisplay pi/2
0
tanαθdθ,
integraldisplay pi/2
0
cotαθdθ, ?1 <α< 1;
(3)
integraldisplay ∞
?∞
dx
(r?ix)a(s?ix)b, r> 0,s> 0, 0 <a< 1, 0 <b< 1, a+b> 1;
(4)
integraldisplay pi/2
0
cosa+b?2θ cos(b?a)θdθ, 0 <a<1, 0 <b< 1,a+b> 1;
5. a44a13a14a219a220a179a16a192a23a26
(1)
∞summationdisplay
n=1
1
n(4n2 ?1); (3)
∞summationdisplay
n=?∞
1parenleftbig
n2 + 1parenrightbig2
.
a3a52a5 Laplace a9a53
(a54a55a56a198a57
a254a58a18a2f(t)
a50a59a60
a206a6a5a61
a21η(t))
1. a44a13a14a30a16a17Laplacea62a63a26
Wu Chong-shi
12 a0 a64 a2
(1) tn, n = 0,1,2,···; (2) tα, Reα>?1;
(3) eλt sinωt, λ> 0,ω> 0; (4) sinωtt , ω> 0;
(5) 1?cosωtt2 , ω> 0; (6)
integraldisplay ∞
t
cosτ
τ dτ.
2. a89f(t)a27a104a65a30a16a50a104a65a27αa50a66f(t+α) = f(t), t>0a80a51a52f(t)a17Laplacea29a62
a67
a62a50a70a71a26a68a30a16a28
F(p) = 11?e?αp
integraldisplay α
0
e?ptf(t)dt.
3. a44a13a14a30a16a17Laplacea62a63a26
(1) |sinωt|, ω> 0; (2) t?a
bracketleftbiggt
a
bracketrightbigg
, a> 0.
4. a44a13a14Laplacea62a63a17
a107
a30a16a26
(1) a
3
p(p+a)3; (2)
ω
pparenleftbigp2 +ω2parenrightbig, ω> 0;
(3) 4p?1(p2 +p)(4p2 ?1); (4) p
2 +ω2
(p2 ?ω2)2, ω> 0;
(5) e
?pτ
p2 , τ > 0; (6)
1
p
e?αp
1?e?αp, α> 0.
5. a81a34Laplacea29a62a44a68a13a14a239a78a76a77(a161)a88a159a78a76a77a26
(1)a51a379.1a50a83
a84i(0) = 0, q(0) = 0,
a44i(t);
a132 9.1 a132 9.2
(2)a51a379.2a50a83
a84i(0) = 0, q(0) = 0,
a44i(t);
(3) y(t) = asint?2
integraldisplay t
0
y(τ)cos(t?τ)dτ; (4) f(t) + 2
integraldisplay t
0
f(τ)cos(t?τ)dτ = 9e2t.
6. a81a34Laplacea29a62a157a158a13a14a159a78a26
(1)
integraldisplay ∞
0
e?ax ?e?bx
x coscxdx, a> 0,b> 0,c> 0;
(2)
integraldisplay ∞
0
1?cosbx
x2 dx, b> 0; (3)
integraldisplay ∞
0
sinxt
x(x2 + 1)dx.
7. a34a69a70a137a71a72a63a44a13a14Laplacea62a63a17a107a30a16a26
Wu Chong-shi
a86 a87 13
(1) pp2 ?ω2, ω> 0; (2) e
?pτ
p4 + 4ω4, τ >0,ω> 0;
(3) 1pe?αp,α> 0; (4) 1pcosh(l?x)
√p
coshl√p , 0 <x<l.
8. a44a13a14a219a220a179a16a192a23a26
(1)
∞summationdisplay
n=0
(?1)n
3n+ 1; (2)
∞summationdisplay
n=0
(?1)n
4n+ 1;
(3)
∞summationdisplay
n=0
(?1)n
(3n+1)(3n+2)(3n+3); (4)
∞summationdisplay
n=0
1
(3n+1)(3n+2)(3n+3).
a3a52a4a5 δ a10a7
1. a70a71δa30a16a17a13a14a182a227a26
(1) δ(x) = δ(?x); (2) xδ(x) = 0;
(3) f(x)δ(x) =f(0)δ(x); (4) xδprime(x) = ?δ(x);
(5) δ(ax) = 1aδ(x), a> 0;
(6) δ(x2 ?a2) = 12abracketleftbigδ(x?a)+δ(x+a)bracketrightbig, a> 0.
2. a44a13a14a42a239a78a76a77a73a92a128a74a17a68a26
(1)
bracketleftBig d2
dx2 ?k
2
bracketrightBig
g(x;t) = δ(x?t), x, t> 0,k> 0,
g(0;t) = 0, dg(x;t)dx
vextendsinglevextendsingle
vextendsingle
x=0
= 0;
(2)
bracketleftBig d2
dx2 ?x
2
bracketrightBig
g(x;t) =δ(x?t), x, t> 0,
g(0;t) = 0, dg(x;t)dx
vextendsinglevextendsingle
vextendsingle
x=0
= 0;
(3)
bracketleftBigparenleftbig
1 +x+x2parenrightbig d
2
dx2 + 2(1 + 2x)
d
dx + 2
bracketrightBig
g(x;t) = δ(x?t), x, t>0,
g(0;t) = 0, dg(x;t)dx
vextendsinglevextendsingle
vextendsingle
x=0
= 0.
a193a194
a26a201a197a56a75a76
a2a77a78a79a80a81a254a6a82
a203
a83a84a85
a198a80
3. a34Greena30a16a76a225a44a68a13a14a42a239a78a76a77a73a92a128a74a26
(1) d
2y(x)
dx2 +k
2y(x) = f(x), x>0,k> 0,
y(0) =A, dy(x)dx
vextendsinglevextendsingle
vextendsingle
x=0
=B;
(2) d
2y(x)
dx2 ?k
2y(x) = f(x), x>0,k> 0,
Wu Chong-shi
14 a0 a64 a86 a2
y(0) =A, dy(x)dx
vextendsinglevextendsingle
vextendsingle
x=0
=B;
(3) d
2y(x)
dx2 ?x
2y(x) = f(x), x>0,
y(0) =A, dy(x)dx
vextendsinglevextendsingle
vextendsingle
x=0
=B.
4. a44a13a14a42a239a78a76a77a87a92a128a74a17a68a26
(1)
bracketleftBig d2
dx2 ?k
2
bracketrightBig
g(x;t) = δ(x?t), 0 <x, t< 1,k> 0,
g(0;t) = 0, g(1;t) = 0;
(2)
bracketleftBig d2
dx2 ?x
2
bracketrightBig
g(x;t) =δ(x?t), 0 <x, t< 1,
g(0;t) = 0, g(1;t) = 0;
(3)
bracketleftBigparenleftbig
1 +x+x2parenrightbig d
2
dx2 + 2(1 + 2x)
d
dx + 2
bracketrightBig
g(x;t) = δ(x?t), 0 <x, t<l< 1,
g(0;t) = 0, g(l;t) = 0.
5. a34Greena30a16a76a225a44a68a13a14a42a239a78a76a77a87a92a128a74a26
(1) d
2y(x)
dx2 +k
2y(x) = f(x), 0 <x<1,
y(0) =A, y(1) = B;
(2) d
2y(x)
dx2 ?k
2y(x) = f(x), 0 <x<1, k>0,
y(0) =A, y(1) = B;
(3) d
2y(x)
dx2 ?x
2y(x) = f(x), 0 <x<1,
y(0) =A, y(1) = B.
Wu Chong-shi
Wu Chong-shi
16 a0 a64 a133 a2
a3a52a57a5 a7a88a89a44a233a234a8a43a58a90a91
1. a100a92a27 l a20a93a94
a73
a159a27 S a17a43a95a96a182a97a50a83
a84
a100a98 (x = 0) a99a118a50a100a100a98 (x = l)
a132 12.1
a62a97a141a76a101a56a102a103a104F a136a34a105a106
a106a72a107 (
a108a37 12.1)a80a62t = 0
a55a50a109a110
a40
a104F a80a85a14a12a97a17a111a112a103a113a114a115a17a76a77a20a87a116a117a118
a23a73a119a117a118a80
2. a62a120a121a123a50
a122a123
a123a11a17a124a125a126a103
a40
a50a93
a67
a62a123a11a17a127a180
a23a128a129a130a77a80a82a62a91a166a55a131a145a20a91a166a132a159a123a127a180a23a128a129a17a123a11
a16a43a138a133
a134
a127a55a135a20a127a63a17a123a11a136a137 u(r,t)a50a31a105a138a128a123a11a16a64a39
a27αu(r,t)a50αa27a133a10a42a16a80a85a65a12u(r,t)a113a114a115a17a139a239a78a76a77a80
3. a130a92a27la17a43a95a140a97a50a141a142a130a67a110a98a20a62a91a166a55a131a145a20a143
a91a166
a73
a159a78a79a144a154a145a146q1 a183q2 a80a85a11a12a223a147a17a87a116a117a118a80
4. a130a100a168a156a27aa20a39
a73a148a149
a17a150a151a152a50a153a154
a134a155a156
a13 (a108a37
12.2)a50a62a157a158
a134a156a125
a17a91a166
a73
a159a56a50a91a166a55a131a145a127a180a145a146M a80a54
a55a50a152
a73
a153Newtona159a160a118a161a125a145(a188a162a121a104a163
a164
a227a17a165a137a270)a80
a85a62a166a134a17a74a75a33a123a11a12a87a116a117a118a80 a132 12.2
a3a52a150a5 a167a168a169a232a152a233a234a235a170a58
1. a44a13a14
a125
a182a171a172a139a239a78a76a77a17a142a68a26
(1) ?
2u
?x2 ?2
?2u
?x?y ?3
?2u
?y2 = 0; (2)
?2u
?x2 ?2
?2u
?x?y + 2
?2u
?y2 = 0;
(3) ?
2u
?x2 ?2
?2u
?x?y = 0; (4)
?2u
?t2 =
c2
r2
?
?r
parenleftbigg
r2?u?r
parenrightbigg
, cnegationslash= 0;
(5) parenleftbiga2 ?b2parenrightbig?
2u
?x2 + 2a
?2u
?x?t +
?2u
?t2 = 0, bnegationslash= 0;
(6) ?
4u
?x4 ?
?4u
?y4 = 0.
2. a44a13a14
a125
a182a173a171a172a139a239a78a76a77a17a142a68a26
(1) ?
2u
?x2 +
?2u
?y2 = x
2 +xy; (2) ?
2u
?x2 ?
?2u
?y2 = xy?x;
(3) ?
2u
?x2 ?2
?2u
?x?y +
?2u
?y2 = x
2 +y.
3. a44a68a139a239a78a76a77a26
(1) x2?
2u
?x2 ?2xy
?2u
?x?y +y
2?
2u
?y2 +x
?u
?x +y
?y
?y = 0;
Wu Chong-shi
a86 a87 17
(2) ?
2u
?x2 ?
?2u
?y2 =
parenleftbigx2 ?y2parenrightbigsinxy.
4. a44a139a239a78a76a77
?
?x
bracketleftbiggparenleftBig
1? xl
parenrightBig2 ?u
?x
bracketrightbigg
? 1a2
parenleftBig
1? xl
parenrightBig2 ?2u
?t2 = 0
a17a142a68a50a66a174a100a175a44a12a131a62a73a119a117a118
uvextendsinglevextendsinglet=0 = φ(x), ?u?t
vextendsinglevextendsingle
vextendsingle
t=0
=ψ(x)
a13a17a68a80
a3a52a175a5 a152a176a9a177a237
1. a92a27la20a110a98a99a118a17a43a95a178a50a73a119a55a50a178a179a103a213a51a37
14.1a50a180
a106a72a107
a108a181a182a183a184a80a44a68
a191
a128a74a80
2. a92a272la17a43a95a97a50a110a98a102a104a136a34a105a78a79a185a186
a123αl
a80
t = 0a55a109a110
a40
a104a80a44a68
a191
a97a17a111a112a103a128a74a80
3. a44a68a140a97a17a65a145a128a74a80a97a92la50a110a98(x = 0,l)a43a187
a188
a27a189a137a50a73a119a165a137a78a190a27u
vextendsinglevextendsingle
t=0 = b
x(l?x)
l2 a80
a132 14.1
4. a100a43a95a191a101a54a182a17a96a182a192a193a50 0 ≤x ≤ l, 0 ≤y ≤ la50a216a104a194a195a80a73a119a166a102a27Axy(l?
x)(l?y)a50a73a119a196a137a270a80a44a68a193a17a93a112a103a80
5. a44a68a26
?2u
?x2 +
?2u
?y2 = 0,
uvextendsinglevextendsinglex=0 =u0, uvextendsinglevextendsinglex=a =u0y,
?u
?y
vextendsinglevextendsingle
vextendsingle
y=0
= 0, ?u?y
vextendsinglevextendsingle
vextendsingle
y=b
= 0,
a67a123a27a83
a84
a42a16a80
6. a44a68a26
?2u
?t2 ?a
2?2u
?x2 =bx(l?x),
uvextendsinglevextendsinglex=0 = 0, uvextendsinglevextendsinglex=l = 0,
uvextendsinglevextendsinglet=0 = 0, ?u?t
vextendsinglevextendsingle
vextendsingle
t=0
= 0.
7. a44a68a197a198a238a199a1971a74a80
8. a100a140a92a97a50x = 0a98a99a118a50x =la98a102a104a65a104Asinωta136a34a80a44a68
a191
a97a17a111a112a103a128a74a80
a82a73a166a102a23a73a196a137a43a270a80
Wu Chong-shi
18 a0 a64 a200 a2
9. a62a201a38a143a1440 ≤x≤a, ?b/2 ≤y ≤b/2 a123a44a68a26
(1) ?2u = ?2, (2) ?2u = ?x2y,
ua62a87a116a56a17a16a92a43a270a80
10. a85a44a13a14a118a68a128a74a192a68a26
?2u
?t2 ?a
2?2u
?x2 = 0,
uvextendsinglevextendsinglex=0 = cos pilat, ?u?x
vextendsinglevextendsingle
vextendsingle
x=l
= 0,
uvextendsinglevextendsinglet=0 = cos pilx, ?u?t
vextendsinglevextendsingle
vextendsingle
t=0
= sin pi2lx.
11. a44a68a13a14a118a68a128a74a26
?u
?t ?κ
?2u
?x2 = 0,
uvextendsinglevextendsinglex=0 = Aexpbraceleftbig?α2κtbracerightbig, uvextendsinglevextendsinglex=l = Bexpbraceleftbig?β2κtbracerightbig,
uvextendsinglevextendsinglet=0 = 0.
12. a134a202a203a120a121a17a204a137a205a130a100a118a206a116a92a55a50a123a11a136a137a211a207a55a131a105a128a208a50a15a209a210a211a120a121a212
a213
a80
a129a214
a28
a107
a11a96a212
a213
a17a215a146a130a77a80a85a216a157a202a203a120a121a17a206a116a204a137a80a123a11a136a137a114a115a17a139a239a78a76a77
a108a197a198a238a199a1972a74a50a217a118a87a116a117a118a27a171a172a17a197a100a34a87a116a117a118a80
a3a52a208a5 a218a219a220a221a222a223a224
1. a100a101a168a156a27aa17a219a220a92a225a226a65a132a167
a227
a50a78a162a110a168a50
a12
a223a184a228a80a100a168a229
a230
a27 V a50a100a100
a168a229
a230
a27?V a80a44
a227
a145a17a229
a230
a78a190a80
2. a168a156a27aa20a39
a73a231a149
a17a43a95a150a151a167
a227
a50
a72
a183a62a36a56a50a102
a106a232a156a233a234
a50a62a157a158
a134a156a125
a17
a91a166
a73
a159a56a91a166a235a236a145a127a180a145a146a27M a50a54a55a50
a227a73
a153 Newtona159a160a118a161a101
a40
a125a145a80a85a44
a227
a145
a17a237a118a165a137a78a190a80a121
a40
a116a165a137a27 0a50a66a82a167
a227
a27a219a220a92a80
3. a44a62a238a38a143a144a≤r ≤b a145a114a115a87a116a117a118
uvextendsinglevextendsingler=a = f(φ), uvextendsinglevextendsingler=b =g(φ)
a17a239a23a30a16a80
4. a62a167a1440 ≤x2 +y2 ≤a2 a56a44a68a26
(1)
??
?
?2u = ?4,
uvextendsinglevextendsinglex2+y2=a2 = 0;
(2)
??
?
?2u = ?4y,
uvextendsinglevextendsinglex2+y2=a2 = 0;
(3)
?
?
?
?2u = ?4xy,
uvextendsinglevextendsinglex2+y2=a2 = 0;
(4)
?
?
?
?2u = ?4(x+y),
uvextendsinglevextendsinglex2+y2=a2 = 0.
Wu Chong-shi
a86 a87 19
5. a100a101a164
a240a241
a65a132a242a162a17a219a220a92a243a65a244a50a67a94
a73
a43a95a50a51a37 15.1 a113a40a80a244a145a27a245a225a80
a217a118a100a101
a72a73 (
a66a37a123a17a100a117a158a87) a17a229
a230
a27V a50a67a246
a73
a56
a17a229
a230
a43a270a80a85a44a243a65a244a145a17a229
a230
a78a190a80
6. a44a68a152a145a17a118a68a128a74a26
?u
?t ?
κ
r2
?
?r
parenleftBig
r2?u?r
parenrightBig
= 0,
uvextendsinglevextendsingler=0a130a116, uvextendsinglevextendsingler=1 = Aexp
braceleftBig
?(ppi)2κt
bracerightBig
,
uvextendsinglevextendsinglet=0 = 0.
a193a194
a26
1
r2
?
?r
parenleftBig
r2?u?r
parenrightBig
≡ 1r?
2(ru)
?r2 . a132 15.1
a3a52a230a5 a247a10a7
(a62a13a14a191a74a123a50k,la43a27a248a182a16)
1. a70a71a26 integraldisplay
1
x
Pk(x)Pl(x)dx = parenleftbig1?x2parenrightbig P
prime
k(x)Pl(x)?P
prime
l(x)Pk(x)
k(k+ 1)?l(l+ 1) , knegationslash=l.
2. a157a158a159a78 integraldisplay
1
?1
(1 +x)kPl(x)dx,
a249
a38a78a79a97a98k≥la23k,la110a250a251a38a80
3. a157a158a13a14a159a78a26
(1)
integraldisplay 1
?1
Pl(x)ln(1?x)dx; (2)
integraldisplay 1
?1
Pl(x)(1?x)?αdx, 0 <α<1.
4. a252Legendrea94a217a63a17a253a162a30a16a70a71a26
(1) Pl(?1/2) =
2lsummationdisplay
k=0
Pk(?1/2)P2l?k(1/2); (2) Pl(cos2θ) =
2lsummationdisplay
k=0
(?)kPk(cosθ)P2l?k(cosθ).
5. a157a158a13a14a159a78a26
(1)
integraldisplay 1
0
Pk(x)Pl(x)dx; (2)
integraldisplay 1
?1
xPl(x)Pl+1(x)dx;
(3)
integraldisplay 1
?1
x2Pl(x)Pl+2(x)dx; (4)
integraldisplay 1
?1
bracketleftBig
xPl(x)
bracketrightBig2
dx;
6. a211a13a14a118a140a62[?1, 1]a56a17a30a16a153Legendrea94a217a63a212a213a26
(1) f(x) =x2; (2) f(x) = √1?2xt+t2;
(3) f(x) = |x|; (4) f(x) = 12bracketleftbigx+|x|bracketrightbig.
Wu Chong-shi
20 a0 a64 a229 a2
7. a44a68a225a226a152a254a145a17a118a68a128a74a26
?2u = 0, a<r<b,
uvextendsinglevextendsingler=a = u0, uvextendsinglevextendsingler=b =u0cos2θ.
8. a44a68a152a145a17a118a68a128a74a26
?2u = 0, 0 <r<a, 0 <θ<pi,
uvextendsinglevextendsingler=0a130a116, uvextendsinglevextendsingler=a = u0η(α?θ).
9. a44a68a197a198a238a199a1974a74a80
10. a100a255a119a0a1a17a43a95a140
a125
a50 x = 0 a98a99a118a62a95a196a2a3a4a5a6a7a8a9a10a11 ω a7a12a13a14
(x = l) a15
a16
a17a18a19a20a21a22a23a24
a4a25a26a27a7
a16a28a29a30a31a32
a20
a4a33a34a7a35a36a37a4a38a39a40a41a11a42a38
a37
a17a43
a35a37a44a45
a28
a38a39a40a41a33a46a47a3a48a7a49a50a51a52a53a25a26a11
?2u
?t2 ?
ω2
2
?
?x
bracketleftBigparenleftbig
l2 ?x2parenrightbig?u?x
bracketrightBig
= 0,
uvextendsinglevextendsinglex=0 = 0, uvextendsinglevextendsinglex=l
a54a55
,
uvextendsinglevextendsinglet=0 = φ(x), ?u?t
vextendsinglevextendsingle
vextendsingle
t=0
= ψ(x).
a56a57
a53a35a52a53a58a59
a17
11. a60
a54
a13a61a62a11aa4a63a64a61a65a7a65a66a67a68a11a69a70u0 a7
a71
a66a67a68a110a17a57a61a65a72a4a73a52
a67a68a74a75
a17
12.
a54
a13a61a62a11ba4a76a77a63a64a65a78a7a72
a79a80
a54
a13a81a82
a83
a7
a83
a4a61a62a11aa7
a83a32a84
a65
a32
a19
a85
a7
a83
a6a86a87a88a89a7a90a89
a91
a11Qa17a57a65a72a4a89a92a74a75a17
13. a93a27a94a95a70a96a65a97a95a70Yml (θ,φ)a98a99a100
(1) sin2θcos2φ; (2) parenleftbig1 + cosθparenrightbigsinθcosφ.
14. a13a61a62a11a4a86a87a63a64a65a7a101a66a67a68a11a100
(1) uvextendsinglevextendsingler=a = P11(cosθ)cosφ; (2) uvextendsinglevextendsingler=a=P1(cosθ)sinθcosφ.
a56a57a102
a65a72a4a73a52a67a68a74a75
a17
15. a57a53a65a72a58a59a100
?2u = A+Br2 sin2θcosφ,
uvextendsinglevextendsingler=a = 0,
a103a104A, B
a11a105
a106
a69a70
a17
Wu Chong-shi
a107 a108 21
a109a110a111a112 a113a114a115
1. a116a117a100
(1) cosx = J0(x)?2J2(x)+2J4(x)?+···, (2) x = 2bracketleftbigJ1 + 3J3(x) + 5J5(x) +···bracketrightbig;
sinx = 2J1(x)?2J3(x)+2J5(x)?+···;
(3) x2 = 2
∞summationdisplay
n=1
(2n)2J2n(x); (4) J20(x) + 2
∞summationdisplay
n=1
J2n(x) = 1.
2. a93a95a70f(θ) = cos(xsinθ)a118g(θ) = sin(xsinθ)a98a99a11 Fouriera119a70a17
3. a120a121a27a94a122a74a100
(1)
integraldisplay x
0
x?nJn+1(x)dx; (2)
integraldisplay a
0
x3J0(x)dx;
(3)
integraldisplay t
0
J0parenleftbig
radicalbig
x(t?x)parenrightbigdx; (4)
integraldisplay t
0
bracketleftbigradicalbigx(t?x)bracketrightbignJ
n
parenleftbigradicalbigx(t?x)parenrightbigdx.
4. a61a62a11Ra4a82
a123a124
a7a125a126a127a52a7a128a129
a123a130a131a132a133a134a135
a66
uvextendsinglevextendsinglet=0 = A
parenleftbigg
1? r
2
R2
parenrightbigg
a123
a7a128a136a110a17a57a53a82
a124
a4a46a47a3a58a59
a17
5. a57a53a27a94a52a53a58a59a100
?u
?t ?κ
bracketleftBig1
r
?
?r
parenleftBig
r?u?r
parenrightBig
+ 1r2 ?
2u
?φ2
bracketrightBig
= 0,
uvextendsinglevextendsingler=0
a54a55
, uvextendsinglevextendsingler=a = 0,
uvextendsinglevextendsinglet=0 =u0 sin2φ.
6. a13a137a11pia138a61a62a111a4a82a139a123a63a64a7a139a64a4a140a66a118a6a27a71a4a67a68a86a141a142a11 0a7a128a129a48
a139
a64a72a4a67a68a74a75a11f(r)sinza7
a57a139
a64a72a67a68a4a74a75
a84a143a144
a17
7. a13a145
a32
a82
a139
a7a72a61a62a11aa7
a146
a61a62a11ba7
a147
a142a72
a146a139
a66a4a67a68a11 0a17a148a60a139a64a149ha7
a6a27
a71a150a151
a7a128a67a11u0 a7
a57a139
a64a72a67a68a4a74a75
a84a143a144
a17
8. a61a62a11Ra4a82
a123a152
a7a125a126a127a52a7
a18a153
a40a154
a91
a6a155a156a157
a20
(1) f(r,t) = Asinωt, (2) f(r,t) =A
parenleftbigg
1? r
2
R2
parenrightbigg
sinωt
a4a33a34a7
a57
a53a82
a124
a4a158a159a47a3a7a60a128a40a160
a84
a128a136a68a86a11 0a17
9. a120a121a27a94a122a74a100
(1)
integraldisplay 1
0
√1?xsinparenleftbiga√xparenrightbigdx, a> 0; (2) integraldisplay ∞
0
e?axJ0parenleftbig
√
bxparenrightbigdx, a> 0,b≥ 0;
(3)
integraldisplay ∞
0
e?axJν(bx)xν+1dx, ν >?1,a>0, b> 0;
Wu Chong-shi
22 a161 a162 a163 a164
(4)
integraldisplay ∞
0
expbraceleftbig?a2x2bracerightbigJν(bx)xν+1dx, ν >?1,a>0, b> 0.
10. a13a63a64a65a7a61a62a11aa7a128a67a11a69a70u0 a7a65a66a67a68a110a17a57a65a72a67a68a4a74a75a118
a143a144
a17
11. a120a121a122a74a100
(1)
integraldisplay ∞
0
e?ax/2 sinbxI0
parenleftBigax
2
parenrightBig
dx,
integraldisplay ∞
0
e?ax/2 cosbxI0
parenleftBigax
2
parenrightBig
dx, a103a104a>0,b> 0a165
(2)
integraldisplay ∞
0
J0(αx)K0(βx)xdx, α> 0, Reβ > 0.
12. a149a11ha138a61a62a11aa4a82a139a64a7a6a27a71a141a142a67a68a110a7a166a139a66a67a68a11u0 sin 2pih za7a57a139
a64a72a4a73a52a67a68a74a75
a17a167a168a169
a52a6a27
a71a170
a18
a4a38a66a74a171a11z = ha118z = 0a17
13. a93a27a94a95a70
a18t = 0
a4a172a173a72a33Taylora98a99a100
(1) 1z sin√z2 + 2zta7a174a52
√z2 + 2ztvextendsinglevextendsinglevextendsingle
t=0
= z;
(2) 1z cos√z2 ?2zta7a174a52√z2 ?2zt
vextendsinglevextendsingle
vextendsingle
t=0
=z;
(3) 1z sinh√z2 ?2izta7a174a52 √z2 ?2izt
vextendsinglevextendsingle
vextendsingle
t=0
=z;
(4) 1z cosh√z2 + 2izta7a174a52
√z2 + 2iztvextendsinglevextendsinglevextendsingle
t=0
= z.
14. a57a137a82a139a123a118a82a123a175a176a4a177
a55
a61a62
a17
a109a110a178a112 a179a180a181a182a183a184a185
1. a93a27a94a49a50
a144
a11Sturm–Liouvillea186a49a50a4a187a188
a123a189
a100
(1) xd
2y
dx2 + 2
dy
dx + (x+λ)y = 0; (2) x(1?x)
d2y
dx2 + (a?bx)
dy
dx ?λy = 0;
(3) xd
2y
dx2 + (1?x)
dy
dx +λy = 0; (4)
d2y
dx2 ?2x
dy
dx + 2λy = 0.
2. a57a53a190a191a192a58a59a100
1
r
d
dr
parenleftbigg
rdRdr
parenrightbigg
+ λr2R = 0,
R(a) = 0, R(b) = 0,
a103a104b>a> 0
a17
3. a60
a54
a190a191a192a58a59
d
dx
bracketleftBig
p(x)dydx
bracketrightBig
+bracketleftbigλρ(x)?q(x)bracketrightbigy = 0,
y(b) = α11y(a) +α12yprime(a),
yprime(b) =α21y(a)+α22yprime(a),
Wu Chong-shi
a107 a108 23
a103a104p(a) = p(b)
a17
a56
a116a117a7
a43 vextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
α11 α12
α21 α22
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
a48a7a45a193a194a195a190a191a192a4a190a191a95a70a196a197
a17
4. a60a190a191a192a58a59
?2Φ +λΦ = 0,
ΦvextendsinglevextendsingleΣ = 0
a4a53(a190a191a95a70)a11Φk a7a45a193a4a190a191a192a11λk a7
a167a168
a4ka198a190a191a192a4a199a200
a17
a56
a116a117a100
a43λ = 0
a194a198a190a191a192a48a7Poissona49a50a4a201a13a202a125a192a58a59
?2u = ?f,
uvextendsinglevextendsingleΣ = 0
a4a53a11
u =
summationdisplay
k
Ak
λk Φk,
Ak a198a203a204a205a206f a96{Φk}a98a99a4a207a70a7
f =
summationdisplay
k
AkΦk.
a167a168a208
a60Φk a105
a209
a13
a144
a17
5. a34a2014a59a4a49a210
a57
a53a211
a123a212
a1730 ≤x≤a, 0 ≤y ≤b a72Poissona49a50a4a52a53a58a59
?2u
?x2 +
?2u
?y2 = ?f(x, y),
uvextendsinglevextendsinglex=0 = 0, uvextendsinglevextendsinglex=a = 0,
uvextendsinglevextendsingley=0 = 0, uvextendsinglevextendsingley=b = 0.
a109a110a213a112 a214a179a181a215
1. a34Laplacea143a216a57a53a61a217
a55
a58a59a100
?u
?t ?κ
?2u
?x2 = 0, x> 0,t>0,
uvextendsinglevextendsinglex=0 = u0, uvextendsinglevextendsinglex→∞
a54a55
, t> 0,
uvextendsinglevextendsinglet=0 = 0, x> 0.
2. a60
a54a218
a81a61a217
a55a219
a7a67a68a74a171a11 0a118u0 a7
a18t = 0
a48a93
a218a219
a14a220a44a76a7
a57 t> 0
a48
a219
a104a221
a220a4a67a68a74a75
a17
Wu Chong-shi
24 a161 a222 a162 a164
3. a223a34Laplacea143a216a57a53a201a224a225a226a20111a59
a17
4. a34Fouriera143a216a49a210
a57
a53a13
a147
a217
a55a227
a6a4a158a159a47a228a58a59
?2u
?t2 ?a
2?2u
?x2 =f(x,t),
uvextendsinglevextendsinglet=0 = φ(x), ?u?t
vextendsinglevextendsingle
vextendsingle
t=0
= ψ(x).
5. a34Fouriera143a216a49a210a57a53a229a147a217
a55
a38a66a6a4a15
a16
a47a228a58a59
?2u
?t2 ?a
2
bracketleftbigg?2u
?x2 +
?2u
?y2
bracketrightbigg
= 0,
uvextendsinglevextendsinglet=0 = φ(x,y), ?u?t
vextendsinglevextendsingle
vextendsingle
t=0
= ψ(x,y).
6. a13a61a217
a55a227
x≥ 0a7
a230a231a28
a38a39
a130a232
a17
a60
a18t> 0
a48x = 0a14a33a233a234a47a228Asinωta17a56
a57
a227
a6
a221
a220a4a235a228
a17
7. a89
a236a237a238a104
a69a239a240a13a241a242
a153
a4a243a89
a244a245 a246
a62a247a248
a245
a7a249a4
a218a250
a198
a218
a81a217a251a76a252
a4
a246
a62(a60a11a)a195a5a137a82a248a7
a103
a89
a92
a74a171a11V0 a118?V0 a17a57a248a72a4a243a89
a92
a17
a253a254
a100a255a0a1a2a3a4u
vextendsinglevextendsingle
r=a = V0e
?k|z|sgnz
a5a6a7Fouriera8a9a10a11a7a12a13a14k → 0a17
a109a15a110a112 Green a114a115a16a183
1. (1)a34a89
a17
a210
a57a102
a65a72Laplacea49a50a201a13a202a125a192a58a59a4Greena95a70G(r; rprime)a165
(2)a57a102a125
a55
a66(a65a66r = a)a6
a221
a220a4a18a19a89
a20a21
a68σ(θ, φ)a165
(3)a116a117
a17
a89
a20
a118a18a19a89
a20
a18
a65a72
a22a23a246a24
a165
(4)a116a117a65a72Laplacea49a50a201a13a202a125a192a58a59
?2u = 0,
uvextendsinglevextendsingler=a =f(θ, φ)
a4a53a198
u(r,θ, φ) = a
parenleftbiga2 ?r2parenrightbig
4pi
integraldisplay 2pi
0
bracketleftbiggintegraldisplay pi
0
f(θprime, φprime)parenleftbig
a2 +r2 ?2arcosψparenrightbig3/2
sinθprimedθprime
bracketrightbigg
dφprime,
a103a104ψ
a198r(r, θ, φ)a84rprime(rprime, θprime,φprime)a4a25a8a7
cosψ = cosθcosθprime + sinθsinθprime cosparenleftbigφ?φprimeparenrightbig.
2. a13a217a26a137
a227
a7t = t0 a48
a18x = x0
a231
a155a240a27a48a4a28a29a7
a30a91
a11Ia17a56a57a53
a227
a4a46a47a228a7
a60a128a40a160a118a128a136a68a86a110a17
3. a34Greena95a70a49a210a53a217
a55a227
a4a46a47a228a58a59a7a52a53a58a59a11
?2u
?t2 ?a
2?2u
?x2 = 0,
Wu Chong-shi
a107 a108 25
uvextendsinglevextendsinglet=0 = φ(x), ?u?t
vextendsinglevextendsingle
vextendsingle
t=0
= ψ(x).
4.
a218
a14a127a52a4
a227
a7a137a11l
a17t = t0
a48a34a36a31a32a29
a227
a6x =x0 a220a7
a33a34a35a231a36a34a30a91I
a17
a57
a53
a227
a4a46a47a228a7a60a128a40a160a118a128a136a68a86a11 0a17
5. a34Greena95a70a49a210a53a201a224a225a226a2016a59
a17
6. a3420.5a37
a104a57
a53a38
a147
a217
a55
a145a39a40a228a49a50 Greena95a70a4a49a210a7
a57a151a41
a63a49a50a4 Greena95
a70
a17
a109a15a110a42a112 a181a179a183a43a44
1. a45
a102a33
a27a94a46a95
a169
a250
a192a4Euler–Lagrangea49a50a7a47
a57
a53a100
(1)
integraldisplay x1
x0
radicalBig
1 +y2yprime2dx; (2)
integraldisplay x1
x0
parenleftbigy2 +yprime2parenrightbigdx;
(3)
integraldisplay x1
x0
x
x+yprimedx; (4)
integraldisplay x1
x0
√1 +xradicalBig1 +yprime2dx.
a174a52
a250
a192a48a37a86a49a50a38a66a6a4a105
a106
a220 (x0, y0)a118(x1, y1)a17
2. a57a51a66x2 +y2 = z2 a6a4 a52
a53
a50a37a54(a188a55a56a7a57a11a58a77a37a7 Geodesic)a17
3. a57a82
a139
a66a6a59a58a77a37a7a60a82
a139
a59a60a37a38a61
a28z
a5
a17
4. a237a18a62a63a10a11na59a64a154
a104
a59
a41a65
a136a10a11v = dsdt = cn a7ca198a66a145
a104
a59a136a10a7
a28
a198
a237a16
Aa220(x0,y0)a41a65a240Ba220(x1,y1)a59a48a39a67a198
T =
integraldisplay (x1,y1)
(x0,y0)
ds
v =
1
c
integraldisplay (x1,y1)
(x0,y0)
nds.
Fermata230a68a56a7
a237
a37
a16A
a240Ba59a69a70a71a62a193
a43
a33T
a169
a250
a192
a17
a56a57a237
a18
a27a94a64a154
a104a41a65
a48a59
a69a70a72a73a100
(1) n = k(x+ 1); (2) n = k√y;
(3) n = k2x+ 3; (4) n = ky;
(5) n = key; (6) n = k√x+y;
(7) n = kr?1/2; (8) n = kr.
a103a104k
a86a11a105
a106
a69a70a7r2 = x2 +y2 a17
5. a56a45
a102
a190a191a192a58a59
?2u+λu = 0,
bracketleftbigg
αu+β?u?n
bracketrightbigg
Σ
= 0
a170
a45a193a59a46a95
a250
a192a58a59a7a60β negationslash= 0a17
Wu Chong-shi
26 a161a222a162a222a164
6. a34Rayleigh–Ritza49a210
a57a102
yprimeprime +λy = 0,
y(?1) = 0, y(1) = 0
a59a74a75a59
a218
a81a190a191a192a59a252a76a192a7
a169
a56a77
a95a70a11a100
(1) y =c1parenleftbig1?x2parenrightbig+c2xparenleftbig1?x2parenrightbiga165 (2) y = c1parenleftbig1?x2parenrightbig+c2x2parenleftbig1?x2parenrightbiga17
a109a15a110a15a112 a115a78a79a80a81a82a83a84
1. a85a86a27a94a49a50a59a202a186a7a47a93a249a87
a144
a11a187a188
a123a189
a100
(1) ?
2u
?x2 + 2
?2u
?x?y ?3
?2u
?y2 + 2
?u
?x + 6
?u
?y = 0;
(2) ?
2u
?x2 + 4
?2u
?x?y + 5
?2u
?y2 +
?u
?x + 2
?u
?y = 0;
(3) ?
2u
?x2 +y
?2u
?y2 +
1
2
?u
?y = 0;
(4) parenleftbig1 +x2parenrightbig?
2u
?x2 +
parenleftbig1 +y2parenrightbig?2u
?y2 +x
?u
?x +y
?u
?y = 0;
(5) tan2x?
2u
?x2 ?2ytanx
?2u
?x?y +y
2?
2u
?y2 +y
2?u
?y = 0.
(6) ?
2u
?x2 ?2sinx
?2u
?x?y ?cos
2x?
2u
?y2 ?cosx
?u
?y = 0.
2.
a54a88
a49a50a7a45a89
a106
a95a70a33a90
a43
a59
a143a216a91
a7
a21a22a92a93
a13a94a95a63a70a206
a17
(1)a116a117a100
a18
a143a216
u(x,y) = e?(ax+by)v(x,y)
a27a7a49a50
?2u+ 2a?u?x + 2b?u?y = 0
a144
a11Helmholtza49a50
?2v?parenleftbiga2 +b2parenrightbigv = 0,
a103a104a, b
a11a69a70a165
(2)a96
a57
a90
a43
a59
a143a216
a7
a33
a49a50
?2u
?x2 ?
?2u
?y2 + 2a
?u
?x + 2b
?u
?y = 0
a18
a143a216a91
a194a97a98
a54
a13a94a95a63a70a206a165
Wu Chong-shi
a107 a108 27
(3)a60
a54
a49a50
a?
2u
?x2 + 2b
?2u
?x?y +c
?2u
?y2 +d
?u
?x +e
?u
?y +fu =
?u
?t,
a103a104a, b,c, d, e,f
a11a69a70a7a99b2 ?acnegationslash= 0a17a116a117a100
a18
a143a216
u(x,y,t) = eαx+βy+γtv(x,y,t)
a27a7
a21
a33v(x,y,t)
a100a101a49a50
a?
2v
?x2 + 2b
?2v
?x?y +c
?2v
?y2 =
?v
?t.
3. a57a53
a227
a47a228a49a50a59Goursata58a59a100
?2u
?t2 ?a
2?2u
?x2 = 0,
uvextendsinglevextendsinglex?at=0 =φ(x), uvextendsinglevextendsinglex+at=0 = ψ(x),
a103a104φ(x), ψ(x)
a100a101φ(0) =ψ(0)a17
4. a18a40a228a49a50
?2u
?t2 ?a
2?2u
?x2 = 0
a104
a34iya102a103ata7a104a87a105a106
a34
a240Laplacea49a50a59 a52a128a192a54a58a59
?2u
?x2 +
?2u
?y2 = 0,
uvextendsinglevextendsingley=0 = φ(x), ?u?y
vextendsinglevextendsingle
vextendsingle
y=0
= ψ(x)
a59
a123a189
a53a11
u = 12bracketleftbigφ(x+ iy)+φ(x?iy)bracketrightbig+ 12i
integraldisplay x+iy
x?iy
ψ(ξ)dξ.
(1)a107
φ(x) =x, ψ(x) = e?x,
a108
a21
a34
u(x,y) = x+ e?x siny.
a109
a116
a167
a81a101a110
a189a231a231
a100a101Laplacea49a50a7a111a100a101y = 0a48a59 a52a128a129a54a25a26a165
(2)a112a113
φ(x) = 11 +x2, ψ(x) = 0,
a108a123a189
a53
a143
a11
u(x,y) = 1 +x
2 ?y2
parenleftbig1 +x2 ?y2parenrightbig2 + 4x2y2.
a116a117a100
a167
a81a95a70
a18(0, ±1)
a220a194a114a115a7a116a35a7a117a118
a18a167
a88
a220a6a7a47a194a100a101 Lpalacea49a50
a17
a167
a56a117a100
a18
a13a119a120a121a27a7Laplacea49a50a59 a52a128a192a54a58a59a217a53
a17
