11

11.1 11.2 11.3 f(z) 11.4 f(z) 11.5 f(z) 11.6 11.7 f(z) 11.8 11 533

11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 -11 533

11.17 11.18 11.19 11.20 11.21 11.22 11.23 11.24

11 533

11.25 11.26 11.27 11.28 11.29 11 533

11.1 z (x, y) z = x + iy x y z z P(x, y)(r, ) z (modulus)| z | z (argument) arg. zz = r (cos + i sin ) = rei11 534

11.1 11 534

11.1 z = x + iyxiyz

11 534

11.2 x y z = x + iy(complex variable)R z (= x + iy)w (= u + iv) w z w = f(z) = u + ivw = z2 z = x + iy w = f(z) = u + iv u + iv = (x + iy)2 = (x2y2) + i(2xy) u = x2y2 v = 2xyw u v x y w = f(z) = u(x, y) + iv(x, y)11 534

11.2 z w w (single-valued function)z w w (multi-valued function)w = f(z) z w z w

11 534-535

11.3 f(z) 0 < | z z0 | < | f(x) l | < z0 - < z < z0 + , zz0l - < f(z) < l + f(z)zz0l

11 535

11.3 f(z) 11 535

11.3 f(z) x x0x x0 zz0z z0 z z011 535

11.4 f(z) f(z) f(z)z = z0f(z)z R f(z)R

11 535

11.5 f(z) w = f(z)z (= x + iy)

z0

11 536

11.6 f(z)R f(z)(analytic function regular function holomorphic function)(singular point)

11 536

11.7 f(z) w = f(z) = u(x, y) + iv(x, y)R

(i) R x y

(ii)

(ii) (Cauchy-Riemann equations) C-R (C-R equations)11 536

11.7 f(z) (a) w = f(z) = u(x, y) + iv(x, y)R

11 536

11.7 f(z) xyx y u, vzu, vz

(1)

w = f(z)R z0xy0 (1) 11 536

11.7 f(z) z0x y = 0 z = x [ z = x + iy, z + z = (x + x) + i(y + y) z = x + iy ](1)

(2)11 536-537

11.7 f(z) z0y x = 0z = iy (1)

11 537

11.7 f(z)

(2) (3)

C-R f(z)

11 537

11.7 f(z) (b) f(z) = u + iv

R C-R

f(z)f(z)R11 537

11.7 f(z) xy

11 537

11.7 f(z) 11 537

11.7 f(z) f(z)f(z)

1(conjugate functions)f(z) = u(x, y) + iv(x, y)u(x, y)v(x, y)C-R2f(z)z f(z) = z2 f(z) = 2z f(z) = sin z f(z) = cos z 11 538

11.8 (r, )(x, y)x = r cos , y = r sin ,z = x + iy = r(cos + i sin ) = rei u + iv = f(z) = f(rei) (1)

11 538

11.8 (1) r

(2)

(1)

| (2)

11 538

11.8

C-R

11 538

11.9

(harmonic function)f(z) = u + ivz u v

(1)

(2)11 538

11.9 (1) x (2) y

(3)

(4)

11 539

11.9

(3) (4)

(5)

(1) y (2) x

(6)

(7)11 539

11.9

(6) (7)

(8)

(5) (8) u v u v

11 539

11.10 f(z) = u + ivu(x, y) = c1v(x, y) = c2 u(x, y) = c1 (1) v(x, y) = c2 (2)

11 539

11.10 11 540

11.10 (1) x

(m1)

11 540

11.10

(2) (m2)

(3)11 540

11.10 f(z)u v C-R

(3) 11 540

11.10 (1) (2) 1

11 540

11.11 xy

11 540

11.11 V (1)(x, y)

(2)11 541

11.11

(1) (2) (3)

(x, y)(velocity potential function) (velocity potential)

11 541

11.11 div V = 0

(4)

vxvy(3) (4)

11 541

11.11 w = f(z) = (x, y) + i(x, y)(x, y)(x, y) = c

[C-R]

[(3)]11 541

11.11 (x, y) = c (stream line)(x, y)(stream function) (x, y) = c(equipotential lines)(x, y)(x, y)w = f(z)(x, y) = c(x, y) = c 11 541

11.11

[C-R]

[(3)]

w = f(z) (complex potential)11 541-542

11.11 (x, y) = c(x, y) = c(lines of force)(x, y) = c(x, y) = c(isothermals)(heat flow lines)

11 542

11.12 y = f(x)xy z = f(x, y)w = f(z)u + iv = f(x + iy)x, y u, v 11 548

11.12 z w z z = x + iyw w = u + ivw = f(z)z z C(x, y) (u, v)w w Cw = f(z)z w 11 548

11.12 w = z + (1i)z x = 0, y = 0, x = 1y = 2D w Dw = z + (1i) u + iv = (x + iy) + (1i) = (x + 1) + i(y1)u = x + 1v = y1z x = 0, y = 0, x = 1y = 2w u = 1, v =1, u = 2v = 1DD11 548

11.12 11 549

11.13 C1, C2Pw = f(z)w C1, C2PP Pf (conformal transformation)

11 549

11.13 11 549

11.13

11 549

11.14 f(z)z R f(z) 0w = f(z)R

11 550

11.14 --ProofP(z)z R w P(w) RP C PCQ(z + z)C P Q(w + w)C

11 550

11.14 --Proofzr PQ PQ x z =reiw = reirw

11 550

11.14 --Proof11 550

11.14 --ProofC P x CPu z0

(1)11 550

11.14 --Prooff(z) 0f(z) = ei = | f(z) | = f(z)

(1)

(2)

(3)

11 550

11.14 --ProofC1z P C1w P' C1P x C1P'u '(3) = (4)

11 550

11.14 --Proof(3) (4) = = = w = f(z)f(z) 0

11 551

11.14 --Proof11 551

11.14 --Proof1f(z) = 0(critical point)2(2) P = | f(z) |z w | f(z) |z | f(z) |211 551

11.14 --Proof3(3) = + C P = amp [f(z)]4

11 551

11.15 1. w = z + cc z = x + iyc = a + ibw = u + ivu + iv = (x + iy) + (a + ib) = (x + a) + i(y + b) u = x + a v = y + bz OMPN w = z + (1 + 2i)w OMPN11 552

11.15 11 552

11.15 2. w = czc c = pei, z = reiw = Rei Rei = rei(+) R = r = +z (r, )w P(r , +)P = | c | = amp (c)z w 11 552

11.15 z OMPN w = (1 + i)zw OMPN

11 552-553

11.15 11 553

11.15 3. z = rei w = Rei

=

z P(r, )

11 553

11.15 11 553

11.15 z w P (r, )

P1

P1P [P O k OP OP . OQ = k2 Q]11 553

11.15 P1x P

z | z | = 1

x

| z | = 1

| w | = 1| z | = 1| w | = 111 553-554

11.15 z = 0w = (point at infinity)z w

11 554

11.15 z x2 + y2 + 2gx + 2fy + c =0 (1)

11 554

11.15 x y (1)

(2)11 554

11.15 c0((1)) (2) w c = 0 ((1)) (2) 2gu2fv + 1 = 0w

11 554

11.15 4. w

a, b, c, dadbc0w (bilinear or Mobius transformation)

11 554

11.15

(1)

(2)

11 554-555

11.15 (1) cwz + wdazb = 0w z 11 555

11.15 (1) z w (2) w z 11 555

11.15 (i) w = z + c(ii) w = cz(iii) w 11 555

11.15 z w1w1w2w2w3w3w 11 555

11.15 1w1, w2, w3w4z1, z2, z3z4w

2

11 555

11.16 -w z

11 560-561

11.16 -

(1)

(2)

1, 2,, nw1, w2,, wnx1, x2,, xnz A B 11 561

11.16 -11 561

11.16 - --Proof(1)

(3)11 561

11.16 - --Proofz x1w wnw1w1z x1w wnw1amp (w) z x11 = amp (zx1)0(3) 111 561

11.16 - --Proofw w1w2z x22 = amp (zx2)0 2w w2w3w3z x w 11 561

11.17 f(z)z = x + iyC A B C

C n

11 563

11.17 11 563

11.17 zi = zizi-1Pi-1Pi

nzi0f(z)C (line integral)

11 564

11.17 P0PnC (contour integral)11 564

11.17 f(z) = u(x, y) + iv(x, y)dz = dx + idy

11 564

11.17

C a b

11 564

11.18 (simple closed curve1)(multiple curve2)

11 569

11.18 11 569

11.18 (simply connected) (multiply connected) 3 C1C2()(AB) 11 569

11.19 f(z) f '(z) C

11 569

11.19 --ProofR C

11 569

11.19 --Prooff(z) = u(x, y) + iv(x, y)

(1)

11 570

11.19 --Proof

R

(2)11 570

11.19 f(z)R

(2)

11 570

11.19 1f(z) R P Q R R P Q

11 570

11.19 11 570

11.19 PAQPBQP Q

11 570

11.19 2 f(z)C C1

11 570

11.19 CC1AB

11 571

11.19

()

11 571

11.19 C C1, C2,, Cn

11 571

11.20 f(z) C a C

11 571

11.20 C z = a

11 571

11.20 Proof:a C C1 C C111 571

11.20 11 572

11.20

(1)

C1| za | = za = ei

11 572

11.20 dz = i ei d

(1)

(2)11 572

11.20 C10C1a (2)

11 572

11.20

(1)

a C

11 572

11.20 (1) a

11 572-573

11.20

11 573

11.21 f(z)C| z | = C C z = rei

z = reiC (1)

z C C 11 577

11.21 11 577

11.21

(2)

11 578

11.21 (1) (2)

11 578

11.21 w = ei(3)

11 578

11.21 ei

11 578

11.21 (Poissons integral formula)

11 578

11.21

11 578-579

11.22 f(z)z a = + i

R C a = + iC a = + iC 11 579

11.22 11 579

11.22

(1)

(2)

11 580

11.22 (1) (2)

CRC 11 580

11.22 R

(3)

11 580

11.22

11 580

11.23 a1, b1, a2, b2,(a1 + ib1) + (a2 + ib2) ++ (an + ibn) +

(1)11 581-582

11.23 anbn(1) anbna b(1) (1) ()

11 582

11.23 | a1 + ib1 | + | a2 + ib2 | ++ | an + ibn | +(1) (absolutely convergent) | an | | an + ibn || bn | | an + ibn |11 582

11.23

(2)

S(z)Sn(z)n 11 582

11.23 z NR z n > N| S(z) Sn(z) | < (2) R (uniformly convergent)

11 582

11.23 | za | < Rzz = aR (3) (circle of convergence)R (radius of convergence)

11 582

11.23 MnR z | an(za)n | Mn(3) R Wierstrass M (Wierstrass M-test)

11 582

11.24 f(z)a C C z

11 582

11.24 z C C1C z w C1

11 582-583

11.24 11 583

11.24

(1)11 583

11.24 (1)

C1w

11 583

11.24 f(z)C1C1

11 583

11.24 (2)

...(3)f(z)z = a(Taylors series)

11 583

11.24 1 (3) z = a + h

11 583

11.24 2(3)

(Maclaurins series)11 584

11.25 f(z)ar1r2(r1 > r2)C1 C2R Rz f(z) = a0 + a1(za)+a2(za)2+ +a-1(za)-1 + a-2(za)-2 +

11 584

11.25 z R

(1)

11 584

11.25 11 584

11.25 (1) w C2

| za | < | wa |

11 584

11.25 C1w

11 584-585

11.25 (1) w C2

| wa | < | za |

11 585

11.25 C2w

11 585

11.25 (1) (2) (3)

11 585

11.25 1

f(z)C111 585

11.25 2f(z)C1 a-n = 0

11 585

11.25 3C R C1

11 585-586

11.25

11 586

11.25 11 586

11.25 4anf(z)

11 586

11.26 f(z)(singular point)z = 2 11 589

11.26 f(z)z = a(isolated singular point)z = 1, 1 z = 1, 2,11 589-590

11.26 z = af(z)f(z)z = a

(1)11 590

11.26 mam 0 ; am1 = am2 == 0z = am (pole)(simple pole)

z = af(z)(essential singular point)11 590

11.26 f(z)z = a(za)-1f(z)z = a(residue)(1) a1 a1 = Res{f(z), a}

11 590

11.27 f(z)C

(C )

11 590

11.27 11 590

11.27 a1, a2,, anz = a1, a2,, anC C1, C2,, Cn

11 591

11.27 f(z)C, C1, C2,, Cn

11 591

11.28 (1) z = af(z) Res{f(z), a} =

z = a

(za)

11 591

11.28 (2) f(z)z = am

z = af(z)m

11 591

11.28 (za)m

z (m1)za

11 591

11.29

11 595

11.29 (1) F(cos sin )cos sin

z = eidz = iei d 11 595-596

11.29 0 2z

C | z | = 1

11 596

11.29 (2) f(x)F(x)x F(x)x 11 599

11.29 11 599

11.29 C R R R C1

R C

11 599

11.29

(1)11 599

11.29 z = Reiz C1R0

11 599

11.29 R

11 599

11.29 (1) 11 599