, ( ) 2 (312), 3 (402) Cardano

214 9 21, 215 4 21 ( ) 2 (312), 3 (42) 5.1.......... 5.2......................................... 6.2.1 Cardano................................... 6.2.2 Bombelli................................... 6.2.3 de Moivre.................................. 7.2.4 Euler..................................... 7.2.5 Gauss.................................... 7.2.6 Cauchy................................... 8.2.7 Weierstrass, Riemann............................ 8.2.8................................... 8.3................................... 8 1 9 1.1.................................. 9 1.2........................................ 11 1.3......................................... 12 1.4..................................... 12 1.4.1.............................. 14 1.5........................................ 14 1.6....................................... 15 1.7............................... 16 1.8,..................................... 17 1.9................................ 2 1

1.1 n........................................ 22 1.11 C............................... 26 1.12 C............................. 27 2 28 2.1............................... 28 2.2................................ 28 2.3.................................... 29 2.4......................................... 31 2.5 Cauchy-Riemann............................. 33 2.5.1...................... 35 2.5.2............................ 36 3 37 3........................................ 38 3.1 C..................................... 39 3.2................................... 4 3.3................................... 43 3.4.................................. 49 3.5............................... 54 3.6........................ 59 3.7 Abel, Abel...................... 61 4 64 4.1 log z.................................... 64 4.2 z α...................................... 68 5 69 5.1.................................... 69 5.2.............................. 75 5.3 R 2 ( )................... 78 6 Cauchy (1) 78 6.1....................................... 79 6.2.......................... 8 6.3................ 83 6.4 Cauchy....................... 85 7 91 7.1 Cauchy......................... 91 7.2........................... 92 2

8 94 8.1.................................. 94 8.2 Cauchy..................... 95 8.3............................... 97 8.4 (, )....... 99 9 11 9.1............................. 11 9.2..................................... 12 9.3............................. 15 9.4 Liouville................................... 17 9.5....................................... 19 9.6 Schwarz................................... 113 1, Laurent, 113 1.1 ()............ 113 1.2 Laurent.................... 116 1.3,,................ 122 11 : 132 11.1 1..................................... 133 11.2............................... 133 11.3.............................. 136 12 (residue theorem) 138 12.1..................................... 138 12.2....................................... 14 12.3.............................. 141 12.4..................................... 144 13 144 13.1 ()....... 145 13.2 13.3 e iax f(x) dx............................ 146 13.4 f(x)e iax dx...................... 152 2π r(cos θ, sin θ)dθ............... 156 13.5........................................ 159 13.5.1 Cauchy............................. 159 13.5.2 13.5............................. 163 14, 165 3

A 166 B 166 C 168 C.1 Cauchy-Hadamard............................. 168 C.1.1............................... 168 C.1.2 Cauchy-Hadamard............... 17 C.1.3 Cauchy-Hadamard................ 171 C.1.4 lim sup n a n..................... 171 C.2............................... 172 C.3...................... 172 C.4 Abel................................. 175 C.5.......................... 179 D 18 E 18 E.1 x α E.2 E.3 (log x) n E.4 (log x) n x α f(x)dx........................ 181 f(x) dx...................... 184 g(x)(log x) n dx................... 186 f(x)(log x) n dx.................. 187 E.5............................. 187 E.6................................. 188 F 193 F.1 [1] (1957) VI 4........................ 193 G 194 N = {1, 2, 3, } 4

1 ( ) ( ).1 Cauchy f(z) = 1 f(ζ) 2πi ζ z dζ ( ) C c f f(z) = a n (z c) n, n= a n = f (n) (c) n! = 1 2πi C f(ζ) (ζ z) n+1 dζ Taylor ( ) dx x 2 + 1 ( ) 1 = 2πi Res z 2 + 1 ; i = 2πi lim(z i) z i 1 z 2 + 1 = 2πi 1 z + i = 2πi 1 z=i 2i = π. (tan 1 dx x 4 + 1 Mathematica ) Cauchy 2 1 4 1 C C 2 1 () ( ) Cauchy 5

.2.2.1 Cardano 2 3 Cardano (Gerolamo Cardano, 151 1576 ) Ars magna de Rebus Algebraicis (1545) 3 3 4 ( ) 3 x 3 + px + q = (q 2 /4 + p 3 /27 < ) Cardano 3 q 2 + q 2 4 + p3 27 + 3 q 2 q 2 4 + p3 27 q 2 /4 + p 3 /27 Ars Magna () 1, 4 5 ± 15 ( PDF 67 ) 1: 1, 4 2 5 ± 1 (3 3 Cardano ).2.2 Bombelli Rafael Bombelli (1526 1572) Algebra (1572 ) 3 x 3 = 15x + 4 ( x = 4, 2 ± 3 ) Cardano x = 3 2 + 121 + 2 3 121 4 3 3 http://www.filosofia.unimi.it/cardano/testi/operaomnia/vol_4_s_4.pdf 6

.2.3 de Moivre de Moivre (Abraham de Moivre, 1667 1754) 173 cos nx = cos n n(n 1) x sin 2 x cos n 2 n(n 1)(n 2)(n 3) x + sin 4 x cos n 4 x +, 1 2 1 2 3 4 sin nx = n sin x cos n 1 n(n 1)(n 2) x sin 3 x cos n 3 x + 1 2 3 cos nθ + i sin nθ = (cos θ + i sin θ) n.2.4 Euler Euler Euler Euler Euler (Leonhard Euler, 177 Basel 1783 ) Euler e iθ = cos θ + i sin θ ( ) e iθ = cos θ i sin θ cos θ = eiθ + e iθ 2, sin θ = eiθ e iθ. 2i de Moivre Euler ( ).2.5 Gauss (Carolus Fridericus Gauss, 1777 4 3 1855 2 23 ) Gauss 4 Gauss 1 1 2 ( 1 ( n ) ) 4 d Alembert Gauss Gauss 7

Cauchy ( [2] ) (,, ) Gauss Gauss (1811 ) Wessel (Caspar Wessel, 1797 ), Argand (Jean-Robert Argand, 186 ) ( the Argand plane, an Argand diagram ).2.6 Cauchy Cauchy (Augustin Louis Cauchy, 1789 8 21-1857 5 23 ) Cauchy Cauchy Cauchy-Riemann Cauchy [3].2.7 Weierstrass, Riemann (: [4] IV. Abel ) (Karl Theodor Wilhelm Weierstrass, 1815 1897 (Georg Friedrich Bernhard Riemann, 1826 9 17-1866 7 2 ) Cauchy- Riemann Riemann..2.8 (a) ( ) (b).3 () I, II 8

1 n 1.1 i 2 = 1 (, the imaginary unit) i a + ib (a b ) (a complex nuber) a + i a, + ib ib, a + i1 a + i + i, + i1 i (a 1 + ib 1 ) + (a 2 + ib 2 ) = (a 1 + a 2 ) + (b 1 + b 2 )i, (a 1 + ib 1 ) (a 2 + ib 2 ) = (a 1 a 2 b 1 b 2 ) + (a 1 b 2 + b 1 a 2 )i () i i 2 1 () a + i a R C 1.1 () i i j JIS () i Mathematica I MATLAB i, j i j 1i 1j a + ib (a, b R) C (an imaginary number) a + ib (a, b R, b ) a =, b ( + i = ( ) ) z, w, ζ (ζ ) 9

z = x + iy (x, y R) x, y z Re z, Im z x = Re z, y = Im z. (Rz, Iz ) = + i, 1 = 1 + i z = x + iy (x, y R) w z zw = 1 w = u + iv (u, v R) (x + iy)(u + iv) = 1 u = x x 2 + y 2, v = y x 2 + y 2 z w w = x x 2 + y i y 2 x 2 + y 2 { xu yv = 1 xv + yu = 1. 1 x + iy (x, y R) 1 x + iy = x iy x 2 + y. 2 C C z, n z n 5 2. 3. i n (n Z) (1 + i) 2 5 n z n n z z = 1 ( a n z n z z z = z = 1 ), n z z n = 1 z n. 1 n=

1.2 c C z 2 = c z C c (a square root of c) 1.1 c C z 2 = c z C c = z = c 1 c = α + iβ (α, β R) z = ( α+ α ± 2 +β 2 + i β α+ 2 β α 2 +β 2 2 ) (β ) ± α (β = α ) ±i α (β = α < ). ( ) (x + iy) 2 = α + iβ (x + iy) 2 = α + iβ 4. 1 5. i c R x 2 = c x R c c c = c =. c > c > c c c. c1, c 2 c1 c2 = c 1 c 2 c 2 c c1 c2 = c 1 c 2 ( ) c ( ) (a) c c () (b) c c ( ) (c) c c 11

6 3 3i 3i ± 3i 2 a ± ib ( ) 1.2 2 az 2 + bz + c = (a, b, c C, a ) 2 z = b ± b 2 4ac. 2a 1.3 ( a > b a + c > b + c a > b c > ac > bc ) () ( ) (d(z, w) = z w ) 1: (?) 1.4 z = x + iy (x, y R) x iy z (the complex conjugate of z) z z, w C z = z. z + w = z + w, z w = z w, zw = z w, ( z w ) = z/w = z w. 6 c c ( c) c < c = ci 1 = i, 2 = 2i. c c c c = ci 12

6. z = x + iy (x, y R) z = x iy (1) x = z + z 2, y = z z. 2i (2) Re z = z + z 2, Im z = z z. 2i ( z C) z = z z R. 1.3 (2) z = e iθ = cos θ + i sin θ (θ R) cos θ = Re e iθ = eiθ + e iθ, sin θ = Im e iθ = eiθ e iθ. 2 2i z C cos z = eiz + e iz, sin z = eiz e iz 2 2i (z C) ( ) 7. cos 5 θ cos kθ, sin kθ (k = 1, 2, 3, 4, 5) (cos nθ, sin nθ cos θ, sin θ ) (1) x y z z 8. a C \ {}, β R az + az + β = 9. a C, β R, a 2 β < zz + az + az + β = (a c ) 13

1.4.1 n = 2 2 n = 3 ax 3 + bx 2 + cx + d = (a, b, c, d ) x = α + iβ (α, β ) x = α iβ 1.4 n N, f(z) = a z n + a 1 z n 1 + + a n R[z], c C, f(c) = f(c) =. a j a j = a j z C f(z) = a z n + a 1 z n 1 + + a n 1 z + a n = a z n + a 1 z n 1 + + a n 1 z + a n = a z n + a 1 z n 1 + + a n 1 z + a n = a z n + a 1 z n 1 + + a n 1 z + a n = f (z) f(c) = f(c). f(c) = f (c) = f(z) = Q(z) (P (z), Q(z) R[z]) f(z) = f (z) P (z) 1. c f(z) m c m (c f(z) m ) 1.5 z = x + iy (x, y R) z (the absolute value of z, the modulus of z, the magnitute of z) z z = x + iy = x 2 + y 2 R 2 (x, y) () zz = z 2 1.2 absolute value the maximum modulus principle modulus modulus magnitude ( 14

) MATLAB ( ) magnitude ( ) 1.3 zz = z 2 z z z z 2 = 1. 1 z = z z 2 1 z z (i) z. z =. (ii) zw = z w. ( z = z w w.) (iii) z + w z + w. a + ib C (a, b) R 2 () (i) (iii) (ii) z = a + ib, w = c + id zw = (ac bd) + (ad + bc)i zw = (ac bd) + (ad + bc)i = (ac bd) 2 + (ad + bc) 2 = a 2 c 2 + b 2 d 2 + a 2 d 2 + b 2 c 2 = (a 2 + b 2 ) (c 2 + d 2 ) = a 2 + b 2 c 2 + d 2 = z w. () zz = z 2 zw 2 = zwzw = zwz w = zzww = z 2 w 2 zw = z w. z = z 1.6 ( C, R 2 ) P O x y R 2 = {(x, y) x R, y R} (a, b) x a, y b ψ : R 2 P P R 2 1 1 P C = {x + yi x R, y R} R 2 = {(x, y) x R, y R} φ: C x + yi (x, y) R 2 15

ψ φ: C P P C 1 1 P (the complex plane) (the Gauss plane), (the Argand plane, an Argand diagram) R 2 x, y (real axis), (imaginary axis) 1.7 z = x + iy (x, y R) e z (exp z ) e z = e x+iy := e x (cos y + i sin y). z = x R e z = e x (cos + i sin ) = e x (1 + i) = e x = e z. e z 1+z 2 = e z 1 e z 2 e z 1 e z 2 = e x 1 (cos y 1 + i sin y 1 ) e x 2 (cos y 2 + i sin y 2 ) = e x 1 e x 2 [(cos y 1 cos y 2 sin y 1 sin y 2 ) + i (cos y 1 sin y 2 + sin y 1 cos y 2 )] = e x 1+x 2 (cos(y 1 + y 2 ) + i sin(y 1 + y 2 )) = e z 1+z 2. 11. e 1 = 1, e = z e z, e z = e z, x, y R e x+iy = e x e 2πi = 1. k Z e 2πki = 1 e z = 1 z 12. (1) z C e z = 1 ( k Z) z = 2kπi. (2) z, w C e z = e w ( k Z) w = z + 2kπi. z = iθ (θ R) e iθ = cos θ + i sin θ (θ R). e iθ 16

13. θ =, π 6, π 4, π 3, π 2, 2π 3, 3π 4, 5π 6, π, 3π 2, 2π eiθ θ R e iθ = 1, e iθ = e iθ, 1 e iθ = e iθ. 14. : e iθ cos, sin 1.8, ( ) ( ) z = x + iy (x, y R) (x, y) r, θ z = x + iy (x, y R) (3) z = r (cos θ + i sin θ) = re iθ, r, θ R r, θ (3) z (the polar form of z) 1.4 1 + 3i 1 + ( 3i = 2 cos 2π 3 + i sin π ) 3 z = re iθ z z = r(cos θ i sin θ) (r(cos θ + i sin θ) ) re iθ (r(cos θ + i sin θ) ) z = r =, θ = (θ ) ( θ ) z r = z (3) z = r e iθ = r 1 = r. 17

z r > (3) r (3) cos θ + i sin θ = z ( = x ) r r + iy r cos θ = x r, sin θ = y r. (x/r) 2 + (y/r) 2 = (x 2 + y 2 )/r 2 = r 2 /r 2 = 1 (x/r, y/r) θ () 1.5 z = 1 i z = x + iy = r iθ (x, y R, r, θ R) x = 1, y = 1 r = z = 1 2 + ( 1) 2 = 2. cos θ + i sin θ = 1 2 i 1 2. cos θ = 1 2, sin θ = 1 2 [, 2π) θ = 5 π. ( π, π] 4 θ = 3 π. z : 4 θ z = 2e i 5 4 π = 2e i 3 4 π. θ = 5 4 π + 2kπ (k Z). 15. z = 1 + 3i, i, 1 16. z = re iθ z 1 z ( z ) z (3) θ z arg z z = re iθ, r = z, θ R θ z (an argument of z) ( ) z θ z = {θ + 2kπ k Z}. ( ) 1 θ arg z π < θ π Arg z z (θ [, 2π) ) 1.6 Arg ( 1 i) = 3π. 5 π = arg( 1 i). 4 4 18

z z = re iθ r, θ r = z, θ = arg z. z = z e i arg z, z = z e i Arg z. 1.7 ( ) r 1 (cos θ 1 + i sin θ 1 )r 2 (cos θ 2 + i sin θ 2 ) = r 1 r 2 (cos(θ 1 + θ 2 ) + i sin(θ 1 + θ 2 )). r 1 e iθ 1 r 2 e iθ 2 = r 1 r 2 e i(θ 1+θ 2 ). ( ) 1.8 (de Moivre ) θ R, n N (cos θ + i sin θ) n = cos nθ + i sin nθ. ( n Z ( e iθ) n = e inθ ) 17. cos 3θ, sin 3θ cos θ, sin θ 18. z C, n Z (e z ) n = e nz ( (e x ) y = e xy ( a (a x ) y = a xy ) ) (a) n N n = 1 n = k a k+1 = a k a 1.7 (cos θ + i sin θ) k+1 = (cos θ + i sin θ) k (cos θ + i sin θ) = (cos kθ + i sin kθ) (cos θ + i sin θ) = cos (kθ + θ) + i sin (kθ + θ) = cos ((k + 1)θ) + i sin ((k + 1)θ). n = k + 1 n N 19

(b) n = (cos θ + i sin θ) n = 1 = cos( θ) + i sin( θ) = cos(nθ) + i sin(nθ). (c) n Z, n < m := n m N (cos θ + i sin θ) n = (cos θ + i sin θ) m = ( (cos θ + i sin θ) 1) m = (cos( θ) + i sin( θ)) m = cos( mθ) + i sin( mθ) = cos(nθ) + i sin(nθ). z 1, z 2 z 1 z 2 = z 1 e i arg z 1 z 2 e i arg z 2 = z 1 z 2 e i(arg z 1+arg z 2 ). arg z 1 + arg z 2 z 1 z 2 (Q) z 1 = i = e iπ/2, z 2 = 1+i 2 arg(z 1 z 2 ) = arg z 1 + arg z 2 = e i3π/4 z 1 z 2 = 1 i 2 = e i 5 4 π = e i 5 4 π. arg(z 1 z 2 ) 5 4 π arg(z 1z 2 ) = arg z 1 + arg z 2 3 4 π 2π arg(z 1 z 2 ) arg z 1 + arg z 2 (mod 2π) (θ 1 θ 2 (mod 2π) ( k Z) θ 2 θ 1 = 2kπ) 1.9 z = x + iy (x, y R) (R 2 (x, y) ) ( z z ) (arg z 2π ) z = re iθ ( ), e x+iy e z = e Re z z z z = re iθ z = re iθ ( 1 ) (R 2 ) ( ) z 1 z 2 = r 1 e iθ1 r 2 e iθ 2 = r 1 r 2 e i(θ 1+θ 2 ) z 1 z 2 = z 1 z 2, arg(z 1 z 2 ) arg z 1 + arg z 2 (mod 2π). 2

2: TikZ 21

e iθ (e iθ z = e iθ re iθ = re i(θ +θ) ) i = e i π 2 π/2 1 = e iπ π cos, sin e iθ z = (cos θ + i sin θ) (x + iy) = (x cos θ y sin θ) + i (x sin θ + y cos θ). w = e iθ z, z = x + iy (x, y R), w = u + iv (u, v R) ( ) ( ) ( ) u cos θ sin θ x =. v sin θ cos θ y 1.1 n 2 n c z n = c z c n z n = c 2 (binomial equation) ( ) c 2 c c 3 c n 1.9 ( n ) n 2 c = ρe iφ (ρ >, θ R) n n ρe i( φ n + 2π n k) (k =, 1,, n 1) ( n ) n ρ n ( n, n 3 ) ( n ρ r n = ρ, r r ( ) ) z = r (cos θ + i sin θ) (r, θ R) z n = c r n (cos nθ + i sin nθ) = ρ (cos φ + i sin φ). r n = ρ. r = n ρ. cos nθ + i sin nθ = cos φ + i sin φ. nθ φ (mod 2π). ( k Z) nθ φ = k 2π. ( k Z) θ = φ n + k 2π n. c n n ρe i( φ n +k 2π n ) 22 (k Z).

k ±n ( k n) k =, 1,..., n 1 z n = c (4) z = n ρe i( φ n +k 2π n ) (k =, 1,..., n 1). 1.1 (1 n ) n 2 1 n e i 2π n k (k =, 1,, n 1) ( n ) ω = e i 2π n ρ = 1, φ = ω k (k =, 1,..., n 1) 19. z n 1 = (z 1)(z ω)(z ω 2 ) (z ω n 1 ) 1.11 (1, 1 n ) n 1 n 1 n n = 2 z 2 = 1 e i k 2π 2 = e ikπ (k =, 1) e = 1, e iπ = 1. z 2 1 = (z + 1)(z 1). z 2 = 1 e i( π 2 +k 2π 2 ) = e i (2k+1)π 2 (k =, 1) e i π 2 = i, e i 3π 2 = i. z 2 + 1 = (z + i)(z i).n n = 3 z 3 = 1 e i k 2π 3 (k =, 1, 2) e = 1, e i 2π 3 = 1+i 3 e i 4π 3 = 1 i 3. 2 ( ) ( ) z 3 1 = (z 1)(z 2 + z + 1) = (z 1) z 1 + i 3 z 1 i 3. 2 2 z 3 = 1 e i( π 3 +k 2π 3 ) = e i (2k+1)π 3 e i π 3 = 1+i 3 2, e i 3π 3 = 1, e i 5π 3 = 1 i 3 2. z 3 + 1 = (z + 1)(z 2 z + 1) 1 ± i 3. 2 2, 23

n = 4 z 4 = 1 e i k 2π 4 = e ik π 4 (k =, 1, 2, 3) e = 1, e i π 2 = i, e iπ = 1, e i 3π 4 = i. z 4 1 = (z 2 + 1)(z 2 1) = (z + i)(z i)(z + 1)(z 1).N z 4 = 1 e i( π 4 +k 2π 4 ) = e i (2k+1)π 4 e i π 4 = 1+i 2, e i 3π 4 e i 7π 4 = 1 i 2. z 4 + 1 = (z 2 + i)(z 2 i). = 1+i 2, e i 5π 4 = 1 i 2, z 2 = i, z 2 = i z 4 + 1 = z 4 + 2z 2 + 1 2z 2 = ( z 2 + 1 ) 2 ( ) 2 2z = (z 2 + ) ( 2z + 1 z 2 ) 2z + 1 2 2 2 ± i 2, 2 2 ± i 2. 2 n = 5 z 5 = 1 e ik 2π 5 (k =, 1,, 4) e = 1, e i 2π 5, e i 4π 5, e i 6π 5, e i 8π 5. z 5 1 = (z 1)(z 4 + z 3 + z 2 + z + 1) z 4 + z 3 + z 2 + z + 1 = z 2 + z + 1 + 1 z + 1 z = ( 2 z + 1 ) 2 + z + 1 z z 1 =. X = z + 1 z X2 + X 1 = X = 1 ± 5 2 z + 1 z = 1 + 5, z + 1 2 z = 1 5 2 2z 2 + (1 5)z + 2 =, 2z 2 (1 + 5)z + 2 =. z = (1 5) ± i 1 + 2 5 4, ( 5 + 1) ± i 1 2 5. 4 z 5 = 1 e i( π 5 +k 2π 5 ) = e i (2k+1)π 5 (k =, 1,, 4) e i π 5, e i 3π 5, e i 5π 5 = 1, e i 7π 5, e i 9π 5. e i 8π 5. z 5 + 1 = (z + 1)(z 4 z 3 + z 2 z + 1) 24

z 4 z 3 + z 2 z + 1 = z 2 z + 1 1 z + 1 z = ( 2 z + 1 ) 2 ( z + 1 ) 1 =. z z X = z + 1 z X2 X 1 = X = 1 ± 5 2 z + 1 z = 1 + 5, z + 1 2 z = 1 5 2 2z 2 (1 + 5)z + 2 =, 2z 2 (1 5)z + 2 =. z = (1 + 5) ± i 1 + 2 5 4, (1 5) ± i 1 2 5. 4 n = 6 n = 7 z 7 = 1 e ik 2π 7 (k =, 1,, 6) e = 1, e i 2π 7, e i 4π 7, e i 6π 7, e i 8π 7, e i 1π 7, e i 12π 7. z 7 = 1 e i( π 7 +k 2π 7 ) = e i (2k+1)π 7 (k =, 1,, 6) e i π 7, e i 3π 7, e i 5π 7, e i 7π 7 = 1, e i 9π 7, e i 11π 7, e i 13π 7. (1, 1 ) ( Gauss ) n Gauss n n n = 2 k F m n = 17 = F 2 n n = 2, 3, 4, 5, 6, 8, 1, 12, 15, 16, 17, 2,. F m = 2 2m +1 F = 3, F 1 = 5, F 2 = 17, F 3 = 257, F 4 = 65537 F 5 (F 5 = 4294967297 = 641 67417 () Euler ) 2. c C c = ρe iφ 3 ρe i φ 3, 3 ρe i( φ 3 + 2π 3 ), 3 ρe i( φ 3 + 4π 3 ) c = α + iβ, z = x + iy (x + iy) 3 = α + iβ ( ) 25

21. c 4 c ( ) 1, i 4 1.11 C Hamilton ( ) Hamilton C R 2 (5) (a, b)(c, d) = (ac bd, ad + bc). 1.12 R 2 (a, b) + (c, d) = (a + c, b + d) (5) R 2 (, ), (1, ) (x, y) (, ) (x, y) 1 (x, y) 1 = ( x x 2 + y 2, y x 2 + y 2 ). ( ) a b b a (a, b R) 1 i ( ) ( ) 1 1 I =, J = 1 1 ( J 2 = I e iθ cos θ sin θ ) sin θ cos θ R[x] (x 2 + 1) C = R[x]/(x 2 + 1) ( ) () 26

1.12 C R 2 ( ) z C x := Re z, y := Im z, z := (x, y) z R 2. z = (x, y) R 2 z := x + iy z C (). z = x 2 + y 2 = z 2 z 1 = x 1 + iy 1, z 2 = x 2 + iy 2 (x 1, y 1, x 2, y 2 R), z 1 := (x 1, y 1 ), z 2 := (x 2, y 2 ) z 1 z 2 = z 1 z 2. C C C R 2 Bolzano- Weierstrass Weierstrass C {z n } n N n z n N n z n C {z n } c lim z n c = n ε-δ ( ε > )( N N)( n N : n N) z n c < ε z n, c x n := Re z n, y n := Im z n, z n := (x n, y n ), a := Re c, b := Im c, c := (a, b) lim z n c = n lim x n = a lim y n = b n n lim z n = c lim z n = c n n ( lim x n = a n ) lim y n = b. n 27

2 Ω C f : Ω C 2.1 f : Ω C (Ω C ) u(x, y) := Re f(x + iy), v(x, y) := Im f(x + iy) u, v u v f u, v Ω := { (x, y) R 2 x + iy Ω } u, v : Ω R (Ω Ω C R 2 Ω = Ω.) 2.1 (1) f : C C, f(z) = z 2 f(z) = (x + iy) 2 = x 2 y 2 + 2ixy u(x, y) = x 2 y 2, v(x, y) = 2xy. (2) Ω := C \ {}, f : Ω C, f(z) = 1 z 1 z = 1 x + iy = x iy x u(x, y) = x 2 + y2 v(x, y) = y x 2 + y. 2 x 2 + y 2, (3) f : C C, f(z) = e z. (e x+iy = e x (cos y+i sin y)) f(z) = e x+iy = e x (cos y + i sin y) u(x, y) = e x cos y, v(x, y) = e x sin y. 22. f : Ω C f u, v (1) f(z) = z 3 (z C) (2) f(z) = 1 z 2 + 1 (z C \ {±i}) (3) f(z) = 1 2 (eiz + e iz ) (z C) ( (3) f(z) = cos z ) 2.2 c C, r > c r D(c; r) : D(c; r) := {z C z c < r}. Ω C Ω Ω : Ω := {z C ( ε > ) D(z; ε) Ω }. Ω C Ω C (an open subset of C) ( z Ω)( ε > ) D(z; ε) Ω 28

R n a R n, r > a, r : B(a; r) := {x R n x a < r}. Ω R n Ω : Ω := {x R n ( ε > )B(x; ε) Ω }. Ω R n Ω R n : ( x Ω)( ε > ) B(x; ε) Ω. 2.3 () ( ) 2.2 ( ) Ω C, f : Ω C (1) c Ω, γ C z c f(z) γ ( ε > )( δ > )( z Ω : z c < δ) f(z) γ < ε lim z c f(z) = γ (2) c Ω f c lim f(z) = f(c) z c (3) f Ω Ω z f u v f f : Ω R 2 ( ) u(x, y) f(x, y) := v(x, y) 29

z = (x, y) ( ) ( ) u(z) u(x, y) f(z) = f(x, y) = =. v(z) v(x, y) ( ) ( ) a α c = a + ib (a, b R), γ = α + iβ (α, β R), c =, γ = b β lim f(z) = γ lim f(z) = γ z c z c lim u(x, y) = α lim v(x, y) = β. (x,y) (a,b) (x,y) (a,b) f c f c u v c = (a, b) T. f Ω f Ω u v Ω. f ( ) 2.3 () Ω C, c Ω. f : Ω C, g : Ω C z c lim z c lim z c lim z c (f(z) + g(z)) = lim f(z) + lim g(z), z c z c (f(z) g(z)) = lim f(z) lim g(z), z c z c (f(z)g(z)) = lim f(z) lim g(z), z c z c lim f(z) g(z) = z c lim f(z) lim z c g(z). z c ( ) ( [5]) ( 3.7) ε-δ 2 f = u 1 + iv 1, g = u 2 + iv 2, fg = u 3 + iv 3 u 3 = u 1 u 2 v 1 v 2, v 3 = u 1 v 2 + v 1 u 2 u 3, v 3 u 1, u 2, v 1, v 2 2.4 () ( ) 3

p(z) p(z) f : C C, f(z) = p(z) (z C) C f c (z) = c, f id (z) = z 2.4 f f p 23. f c, f id ε-δ r(z) r(z) = q(z) p(z), q(z) C[z] p(z) Ω := {z C p(z) } f(z) = r(z) (z Ω) f : Ω C r(z) Ω p(z), q(z) p, q Ω p(z) (z Ω) f = q p 2.5 f(z) = 1 z {z C z } 2.6 ( e z ) ( ) φ: R R R 2 (x, y) φ(x) R R 2 (x, y) φ(y) R f 1 (x, y) = e x, f 2 (x, y) = cos y, f 3 (x, y) = sin y f 1, f 2, f 3 : R 2 R f(z) = e z f 1 f 2, f 1 f 3 f C e z = n= z n n! (z C) e z 2.4 Ω C f : Ω C, c Ω f c f(c + h) f(c) lim h h f (c) Ω z f z f Ω (regular,, holomorphic) 2.1 () f 1 c c U ( Ω) f U : U C 31

(analytic) ( ) regular holomorphic () ( ) holomorphic (regular ) 2.7 f(z) = γ ( ) z C f(z + h) f(z) lim h h f (z) =. f C f(z) = z z C f(z + h) f(z) lim h h f (z) = 1. f C = lim h γ γ h = lim h (z + h) z h = lim =. h h = lim 1 = 1. h h 24. f(z) = z 2 C f (z) = 2z 2.8 Ω C c Ω. f : Ω C g : Ω C c f + g, f g, fg, f ( g(c) ) c g (f + g) (c) = f (c) + g (c), (f g) (c) = f (c) g (c), (fg) (c) = f (c)g(c) + f(c)g (c), ( ) f (c) = g(c)f (c) g (c)f(c). g g(c) 2 32

2.9 (1) k f(z) = z k C f (z) = kz k 1. (2) C ( n ) a k z k = k= n n 1 ka k z k 1 = (j + 1)a j+1 z j. k=1 j= (3) r(z) = q(z) (p(z), q(z) C[z]) Ω := p(z) {z C p(z) } z r(z) C 25. 2.9 (1) 26. 27. ( f f 1 f c f 1 f(c) (f 1 ) (f(c)) = 1 ) ( ) f (c) 2.5 Cauchy-Riemann 2.1 Ω C f : Ω C f f u v ( ) u x = v y, u y = v x ( ) Cauchy-Riemann (the Cauchy-Riemann equations, the Cauchy-Riemann relations) f u x = v y, u y = v x f c = a + ib (a, b R) ( ) f (c) = u x (a, b) + iv x (a, b) = 1 i (u y(a, b) + iv y (a, b)) u x (a, b) = v y (a, b), u y (a, b) = v x (a, b). (( ) f = f x = 1 i f y ) 33

f c f (c) = lim h f(c + h) f(c) h h (a) h = h x (h x R) () f(c + h x ) = u(a + h x, b) + iv(a + h x, b) f f(c + h x ) f(c) (c) = lim hx h hx R x = lim hx (u(a + h x, b) + iv(a + h x, b)) (u(a, b) + iv(a, b)) = lim hx h x ) ( u(a + hx, b) u(a, b) h x + i v(a + h x, b) v(a, b) h x = u x (a, b) + iv x (a, b). (b) h = ih y (h y R) ( ) f(c + ih y ) = u(a, b + h y ) + iv(a, b + h y ) f f(c + ih y ) f(c) (c) = lim hx ih hy R y = 1 i lim h y = 1 i lim (u(a, b + h y ) + iv(a, b + h y )) (u(a, b) + iv(a, b)) h y h y ) ( u(a, b + hy ) u(a, b) h y + i v(a, b + h y) v(a, b) h y = 1 i (u y(a, b) + iv y (a, b)). f c ( p, q R) lim f(c + h) f(c) h (p + iq) h = h = h x + ih y (h x, h y R) (p + iq)h = (p + iq)(h x + ih y ) = (ph x qh y ) + i(ph y + qh x ) f(c + h) f(c) (p + iq) h = [u(a + h x, b + h y ) + iv(a, b)] [u(a, b) + iv(a, b)] (p + iq) h x + ih y = [u(a + h x, b + h y ) u(a, b) (ph x qh y )] + i [v(a + h x, b + h y ) v(a, b) (qh x + ph y )] h x + ih y = [u(a + h x, b + h y ) u(a, b) (ph x qh y )] + i [v(a + h x, b + h y ) v(a, b) (qh x + ph y )] h 2 x + h 2 y u(a + h x, b + h y ) u(a, b) (ph x qh y ) = + i v(a + h x, b + h y ) v(a, b) (qh x + ph y ) h 2 x + h 2 y h 2 x + h 2. y 34

u(a + h x, b + h y ) u(a, b) (ph x qh y ) f c ( p, q R) lim (h x,h y) (,) h 2 x + h 2 y v(a + h x, b + h y ) v(a, b) (qh x + ph y ) lim = (h x,h y) (,) h 2 x + h 2 y ( p, q R) u v (a, b) u x (a, b) = p, u y (a, b) = q, v x (a, b) = q, v y (a, b) = p. u v (a, b) u x (a, b) = v y (a, b), u y (a, b) = v x (a, b). ( ) ( ) u f = f u x u y = v v x v y ( ) f c f (c) = p + iq f (a, b) f p q (a, b) =. q p ( ) det f (a, b) = u x (a, b) 2 + u y (a, b) 2 = v x (a, b) 2 + v y (a, b) 2 = f (c) 2. 2.2 () f : Ω C f c Ω, f (c) = p + iq f : Ω R 2 C 1 det f (a, b). f f(a, b) C 1 f 1 ( f 1 ) (f(x, y)) = (f (x, y)) 1 = 1 p 2 + q 2 ( p q ) q, p := u x (a, b), q := u y (a, b). p f f(c) f 1 f 1 Cauchy-Riemann = 2.5.1 I R, f : I R f = f No I = R \ {} f(x) = 1 (x > ), f(x) = (x < ) f f = I f = f Ω R l ( Ω C l ) Ω 2 Ω 35

Ω R 2 u: Ω R u = (u x = u y = ) u 2.11 Ω C ( ) f : Ω C (1) f f (2) f f (1) f u, v u = C ( ) u x = u y =. Cauchy-Riemann ( ) v x = u y =, v y = u x = v f = u + iv = u 28. 2.11 (2) ( ) 2.12 Ω C ( ) f : Ω C ( z Ω) f (z) = f 29. 2.12 2.5.2 2.13 ( ) Ω C f : Ω C f u, v u xx (x, y)+u yy (x, y) =, v xx (x, y)+v yy (x, y) = ((x, y) Ω := { (x, y) R 2 x + iy Ω } ) f Ω u v C Cauchy-Riemann u x = v y, u y = v x u xx + u yy = u x x + u y y = x v y + y ( v ) = 2 v x x y 2 v y x =. 36

v C 2 (v 2 ) v xx + v yy = n u(x 1,..., x n ) ( ) n j=1 2 u x 2 j = u (harmonic function) ( ) Laplace 2 R 2 Ω u ( ) v u 3. v u u v Ω u u () 31. () v u 2.3 ( 2? ) ( 3 ) Cauchy-Riemann u, v C 2 Cauchy-Riemann ( f = f x = 1 i f y ) u, v C 2 ( ) Cauchy-Riemann (2 ) ( ) 3 (a power series) 37

= () 3. n= a n(z c) n ( z c a n ) 3 (1 2 ) 1. ( ) e x, sin x, cos x, log x, (1 + x) α ( ) Taylor f (n) (a) (f(x) = (x a) n ) x z n! n= () 2. ( ( ) ) ( ) = ( ) ( ) Ω C f : Ω C (Ω ) c Ω D(c; ε) Ω ε > {a n } f(z) = a n (z c) n ( z c < ε) () 3. n= a n (z c) n c D(c; R) n= ( ) a n (z c) n = (n + 1)a n+1 (z c) n, n= a n (z c) n dz = C n= n= a n n= C (z c) n dz. ( C D(c; R) ) 38

3.1 C ( ) 3.1 (Cauchy ) C (R, R l, C l OK) {a n } Cauchy ( ε > )( N N)( n N : n N)( m N : m N) a n a m < ε 3.2 C (R, R l, C l OK) Cauchy {a n } a ε ε/2 >. ( N N) ( n N: n N) a n a < ε. n N, m N n, m N 2 a n a m = a n a + a a m a n a + a a m < ε 2 + ε 2 = ε. {a n } Cauchy C (R, R l, C l OK) Cauchy 3.3 R, C, R l, C l C R 2 C l R 2l R l {x n } R l Cauchy {x n } ( ) {x n } Bolzano-Weierstrass ( a R l ) ( {x nk } k N : {x n } ) lim x nk = a. k ε {x n } Cauchy ( N N) ( n N: n N) ( m N: m N) x n x m < ε. k N k N n k k N (m n k ) x n x nk < ε. k x n a ε. lim n x n = a {x n } 32. Cauchy 39

33. R l {x n }, a, c R l, r > lim x n = a, ( n N) x n c < r n a c r ( : x n c < r c r < x n < c + r {x n }, {y n } lim x n = a, lim y n = b, ( n N) x n y n a b n n c r a c + r ) 3.2 ( ) 2 9% 3.4 (Weierstrass M-test (M )) ( ) n a n (z) (z K) 2 {b n } n a n (z) K (i) ( n) ( z K) a n (z) b n. (ii) n b n 3.5 () n a n 2 {b n } n a n (i) ( n) a n b n. (ii) n b n 3.5 3.6 ( ) n a n n a n C (Cauchy ) 4

3.7 ( ) a n n s n := n a k, S n := k=1 n a k k=1 {S n } 3.2 Cauchy n, m N n > m n s n s m = a k k=1 m a k = k=1 n k=m+1 a k n k=m+1 a k = n a k n < m s n s m < S m S n k=1 m a k = S n S m. k=1 s n s m S n S m {s n } Cauchy 3.3 {s n } a n n ( 1) n 1 = 1 1 n 2 + 1 3 1 4 + n=1 34. n=1 1 n = 35. ( 1) n 1 {s n } {s 2n } n N {s 2n 1 } n N n n=1 ( 1) n 1 2 n n=1 Abel log 2 3.7 41

S n := n a k, T n := k=1 3.2 Cauchy n, m N n > m n b k {T n } k=1 n n S n S m = a k b k = T n T m. k=m+1 k=m+1 n < m S n S m T m T n S n S m T n T m. {S n } Cauchy 3.3 {S n } n a n n a n n b n {b n } b n = Mr n ( r < 1) b n = M n α ( α > 1) ( ) 1 ( n 7 ) α n=1 3.8 () {b n } n N 2 b n n=1 (i) ( n N) b n. (ii) ( M R) ( n N) n b k M. k=1 ( (i) n b n a n b n (?) (i) (ii) ) n k=2 7 n 2 1 k k 1 dx n [ ] x α = dx x 1 α n 1 x α = 1 1 α 1 1 α. n α n n 1 dx x α. n k=2 1 k α 42

(i) T n := n b k {T n } (ii) k=1 {T n } {T n } b n n 3.9 () ( ) X f {φ n } n N N N N (f, φ n ) 2 f 2 n=1 (f, φ n ) 2 b n = (f, φ n ) 2, M = f 2 n=1 3.8 (f, φ n ) 2 f 2 ( Bessel ) n=1 3.3 ( c = c ζ := z c, ζ := z c a n ζ n ζ = ζ n ζ < ζ ζ c = ) 3.1 a n (z c) n z = z z c < z c n= z lim n a n (z c) n = ( lim a n =. s := n a n, s n := n=1 s n 1 s n a n = a n : n=1 n a k s n s k=1 n 1 n a k a k = s n s n 1 s s =.) k=1 k=1 ( M R)( n N {}) a n (z c) n M. 43

n b n := M z c z c ( n N {}) a n (z c) n = a n (z c) n z c z c n M z c z c {b n } (z c)/(z c) < 1 : 3.11 b n = n= M 1 z c / z c. a n (z c) n n= n = b n. a n (z c) n z = z z c > z c z n= z 1 c < z 2 c z 2 z 1 3.12 ( ) a n (z c) n (i) z = c z (ii) z (iii) ρ z c < ρ z c > ρ ( ) { } A := z C a n (z c) n n= c A (z = c n 1 a n (z c) n = ) 3 n= (i) A = {c}. c (ii) A = C. (iii) (i), (ii) 44

(iii) (i) z c A \ {c}. r := z c c z c < r (ii) z d C \ A. R := z d c z c > R < r < R ρ := r + R 2 c + ρ A z c < ρ r 1 := ρ, R 1 := R c + ρ A z c > ρ r 1 := r, R 1 := ρ z c < r 1 z c > R 1 r r 1 < R 1 R, R 1 r 1 = R r 2 {r n }, {R n } {r n } {R n } n r n < R n, R n r n = R r 2 n. n z c < r n z c > R n {r n } {R n } ρ ρ r > ρ >. z c < ρ z c > ρ (i) ρ =, (ii) ρ = ρ ( ρ ) z c < ρ z c > ρ 3.13 (, ) ρ {z C z c < ρ} 3.14 () (c =, a n = 1 ) z n z z < 1 z 1 1 D(; 1) = {z C z < 1}. n= 36. ( ) 3.15 ( ) {z C z c = ρ} z 45

z n 1 z = 1 n= z n 1 z = 1 n2 n=1 n=1 z n n 1 z = 1 z = 1 ( ) {a n } ρ () 3.16 (, ratio test, ) a n (z c) n n a n lim n a n a n+1 a n c = lim = ρ z < ρ z > ρ n a n+1 z < ρ z < R < ρ R (ρ < R := z + ρ ρ = 2 a n R := z + 1 ) lim = ρ ( N N) ( n N: n N) n a n+1 a n > R ( a n+1 < 1 R ). a n+1 N 1 m a N+m z N+m = a a N+1 N an+2 a N+m z N z m a N a N+1 a N+m 1 a N z N ( ) m z. R n N a n a n z n a N z N 46 ( ) n N z. R n=

b n := a n z n ( n N 1) ) a N z N n N (n N) ( z R n N a n z n b n, b n = n= N 1 n= a n z n + a n z n n= a N z N 1 z /R ( ). z > ρ z > R > ρ R (ρ = R := z + ρ a n ) lim = ρ ( N N) ( n N: n N) 2 n a n+1 a n < R ( a n+1 > 1 R ). a n+1 () n N a n a n z n a N z N ( ) n N z. R a n z n a n z n n= 3.17 n=1 z n n 1 a n = 1 n lim n a n a n+1 = lim n + 1 n n ( = lim 1 + 1 ) = 1. n n {z C z < 1}. z n 1 n2 n=1 z n n! (exp z Taylor ) a n = 1 n! n= lim n a n a n+1 = lim (n + 1)! n n! C. n!z n n= = lim n (n + 1) =. 47

n= ( 1) n (2n)! z2n (cos z Taylor ) ζ = z 2 n= ( 1) n (2n)! z2n = n= ( 1) n (2n)! ζn. ζ a n = ( 1)n (2n)! lim n a n a n+1 = lim (2(n + 1))! n (2n)! = lim n (2n + 2)(2n + 1) =. ζ ζ z. C. a n a n+1 a n+1 a n a n ( ) n z ρ ( z < ρ ) ρ 3.18 (Cauchy-Hadamard ) a n (z c) n ρ 1 =, 1 = lim sup n n= n an = 1 ρ. {a n } lim sup n a n lim sup ( C.1.1(p. 168) )?? 3.19 n=1 z n2 = z + z 4 + z 9 + (n a n = 1, a n = ) 1 Cauchy-Hadamard 1 n= 48

3.2 2 n z n, n= 3 n z n 1 2, 1 3 n= (2 n + 3 n ) z n 1 3 n= lim n 2 n + 3 n 2 n+1 + 3 n+1 = 1 3 d Alembert 1 3. 2 37. a n z n, n= b n z n R 1, R 2 < R 1 < R 2 < n= (a n + b n )z n R 1 38. (1) (2) n= a n z n, n= b n z n R n= (a n + b n )z n R n= (a n + b n )z n R n= 3.4 (C ) ( ) 3.21 ( ( )) K f : K C, {f n } K ( n f n : K C ) {f n } f (K ) (converge pointwise) ( x K) lim f n (x ) = f(x ) n () 49

3.22 ( ) f n (x) = tan 1 (nx) (x R, n N) 1 (x > ) f(x) = (x = ) 1 (x < ) x R lim f n (x) = f(x). (x = f n (x) = n lim f n(x) =. x > n nx tan 1 (nx) 1. x < n ) {f n } f (R ) f n f 3.23 ( (witch s hat), ) n N f n : R R n 2 x ( x < 1 ) n f n (x) := n 2 x + 2n ( 1 x < 2 ) n n (x < x 2 ) n x R lim f n(x) =. n {f n } f(x) = (R ) ( ) ( ε > ) ( N N) ( n N: n N) f n (x) f(x) < ε (a) x : n N f n (x) = N = 1 (b) x > : N > 2 x N N 8 n N f n (x)dx = 1 2 2 n n = 1 R R lim f n (x) dx = 1. n R f(x) dx = lim f n (x) dx n R R lim f n(x) dx. n 8 : N 5

lim f n (x) dx = lim f n(x) dx n K K n {f n } K (term by term integration) 3.24 ( ) K f : K C, {f n } K {f n } f (K ) lim sup f n (x) f(x) = n x K K {a n } n s n (x) := a n (x) (K ) n= n a k (x) {s n } n N (K ) k= ( ) x K f n (x ) f(x ) sup f n (x) f(x) x K ( ) 9 3.22 sup f n (x) f(x) = 1, x R 3.23 sup f n (x) f(x) = n x R sup f n (x) f(x) = x R 3.25 () K C, f : K C, {f n } K {f n } f K f K x K ε {f n } f K ( N N) ( n N: n N) sup f n (y) f(y) < ε y K 3. f N K ( δ > ) ( x K: x x < δ) f N (x) f N (x ) < ε 3. 9 {f n } f {f n } {f n } f f 51

x K, x x < δ f(x) f(x ) = f(x) f N (x) + f N (x) f N (x ) + f N (x ) f(x ) f(x) f N (x) + f N (x) f N (x ) + f N (x ) f(x ) sup y K f(y) f N (y) + f N (x) f N (x ) + sup f N (y) f(y) y K ε 3 + ε 3 + ε 3 = ε. f x 39. 3.22 {f n } f x = 3.25 () 1 Weierstrass M-test 11 ( ) 3.26 (Weierstrass M-test) K {a n } K {M n } (i) ( n N) ( x K) a n (x) M n. (ii) M n n=1 a n (x) n=1 a n (x) K ( K a n (x) K ) n=1 n N, x K n T n := M k, S n (x) := k=1 n=1 n a k (x), s n (x) := k=1 x K, n, m N n a k (x) k=1 ( ) s n (x) s m (x) S n (x) S m (x) T n T m 1 11 ( ) ( ) ( ) ( ) Weierstrass M-test 52

( ) {T n } Cauchy x K ( ) {S n (x)}, {s n (x)} Cauchy T = lim n T n, ( ) m S(x) = lim n S n (x), s(x) = lim n s n (x) s n (x) s(x) S n (x) S(x) T n T. x K sup x K s n (x) s(x) sup S n (x) S(x) T n T. x K n {S n } S {s n } s K ( a n (x) a n (x) n n ) ( 3.12 ) 3.27 ( ) a n (z c) n ρ < R < ρ R a n (z c) n K := {z C z c R} c = R < ˆr < ρ ˆr (ρ < ˆr = R + ρ, ρ = ˆr = R + 1) a n z n 2 n= z = ˆr lim a n nˆr n =. {a nˆr n } n N M R. z R M := sup a nˆr n n N a n z n = a nˆr n zˆr n M ( ) n R. ˆr ( ) n R M n := M z K a n z n M n, R < ˆr ˆr n= Weierstrass M-test a n z n K n= ( ) 53 n= n= M n

3.1 () D(c; ρ) (C ) R < ρ {z C z c R} D(c; ρ) D(c; ρ) ( uniformly convergent on every compact set ) 3.5 3.28 (, Abel) f(z) = ρ f D(c; ρ) a n (z c) n n= f (z) = na n (z c) n 1 n=1 (z D(c; ρ)). ρ ( 1 Abel Abel ) c = g(z) := na n z n 1 g(z) ( n lim sup n n=1 na n z n ) Cauchy-Hadamard n nan = lim n n lim sup n n n an = 1 ρ = ρ. < R < ρ R f D(; R) f (z) = g(z) ε g(z) z R k a k R k 1 < ε 3 k=n+1 N N z D(; R) z + h D(; R) h f(z + h) f(z) g(z) h = a n (z + h) n a n z n na n z n 1 h n= n=1 N ( ) (z + h) k z k a k kz k 1 (z + h) k z k + a k h h k=1 k=n+1 + ka k z k 1. k=n+1 54

2 ( z k) = kz k 1 h N ( ) (z + h) k z k a k kz k 1 < ε h 3. k=1 ( z < R, z + h < R ) (z + h) k z k = (z + h z) [ (z + h) k 1 + (z + h) k 2 z + + (z + h)z k 2 + z k 1] h ( z + h k 1 + z + h k 2 z + + z + h z k 2 + z k 1) h kr k 1 2 (z + h) k z k a k h k=n+1 k=n+1 a k kr k 1 < ε 3. 3 ka k z k 1 k a k R k 1 < ε 3. k=n+1 k=n+1 f (z) = g(z) f(z + h) f(z) h g(z) < ε. 3.2 () Cauchy-Hadamard f g Cauchy-Hadamard ( ) 12 a n (z c) n, na n (z c) n ρ 1, ρ 2 ρ 1 = ρ 2 n= n=1 { A = r R r >, n= } { a n r n <, B = r R r >, n=1 n a n r n < ( ρ 1 = sup A A = (, ρ 1 ) A = (, ρ 1 ] A = ( sup A = ρ 1 )) 12 [6] ( Cauchy-Hadamard Cauchy-Hadamard ) } 55

a n r n n=1 n a n r n B A. n=1 < r < ρ 1 r A, r > ρ 1 r A { { (A = ) ρ 1 = sup A (A ). ρ (B = ) 2 = sup B (B ). ρ 1 = A =. B A B =. ρ 2 = ρ 1 = ρ 2. ρ 1 > A (A = B ) ( ) ( r A)( r : < r < r) r B. r ( ) > 1, lim n n = 1 ( N N) ( n N: n N) n n r r n r. n (r/r ) n n a N r n ( r r ) n an r n = a n r n. n a n r n n=n a n r n <. n=n r B (( ) ). B. B A ρ 2 = sup B sup A = ρ 1. ρ 2 < ρ 1 ρ 2 < r < r < ρ 1 r, r r < ρ 1 r A. ( ) r B. ρ 2 = sup B r. ρ 2 < r ρ 2 = ρ 1. ( ) Taylor Taylor ( ) 3.29 ( Taylor ) a n (z c) n n= f D(c; ρ) a n = f (n) (c). n! f (n) (c) f(z) = (z c) n (z D(c; ρ)). n! n= D(c; R) k N f (k) (z) = n=k n(n 1) (n k + 1)a n (z c) n k. f (k) (c) = n!a k. 56

4. c C, r >, {a n } n {b n } n a n (z c) n = n= b n (z c) n ( z c < r) n= a n = b n (n =, 1, 2, ) ( 3.29 z = c (z c) ) ( ) 3.3 a n (z c) n ρ ρ > f(z) n= D(c; ρ) F a n F (z) := n + 1 zn+1 ρ F (z) = f(z) n= 3.31 ( ) f(z) = f() = 1, f () = ( ) f (z) = f (z) = na n z n 1, n=1 n(n 1)a n z n 2 = n=2 f(z) a n z n f (z) = f(z), n= (n + 2)(n + 1)a n+2 z n. n= ( n Z : n ) (n + 2)(n + 1)a n+2 = a n. a n a n+2 = (n + 2)(n + 1). f() = 1 a = 1, f () = a 1 = a 2k 1 = (k N), a 2k = ( 1)k (2k)! f(z) = k= (k =, 1, 2, ). 57 ( 1) k (2k)! z2k.

( ) f(z) = cos z. ( ) ( ) Gauss x(1 x)y + (γ (α + β + 1) x) y αβy = Bessel Bessel Bessel x 2 y + xy + ( x 2 α 2) y = e z = exp z := cos z = sin z = n=1 cosh z = sinh z = n=1 n= n= z n n!. ( 1) n (2n)! z2n. ( 1) n 1 (2n 1)! z2n 1. n= 1 (2n)! z2n. 1 (2n 1)! z2n 1. C z R 1 z R C ( ) e z = exp z ( e x+iy = e x (cos y + i sin y) ) ( π ) ( ) ( ) 41. (e z ) = e z, (cos z) = sin z, (sin z) = cos z 58

42. (e z ) x, y R e x+iy = e x (cos y + i sin y) 43. z C cos z = eiz + e iz 2, sin z = eiz e iz 2i 44. (1) sin z = (2) sin z = 2 () ( ) α (1 + z) α = z n ( z < 1). n ( n= ( ) α 2 : n ( ) α := n α(α 1)(α 2) (α n + 1). n! 1 + z = (1 + z) 1/2 ) tan 1 z = log(1 + z) = n= ( 1) n 2n + 1 z2n+1 ( z < 1). n= ( 1) n n + 1 zn+1 ( z < 1). ( ) tan 1 z, log(1 + z) tan 1 x = x dt, log(1 + x) = t 2 + 1 x dt t + 1 tan z ( tan z = sin z cos z ) 3.6 ( ) 2 59

3.32 K R n Jordan {f n } K {f n } n K f f K f n (x) dx = f(x) dx. lim n K K () f n (x) dx f(x) dx = (f n (x) f(x)) dx f n (x) f(x) dx K K K K sup f n (x) f(x) dx x K = sup f n (x) f(x) (K ). x K K 3.33 R I = [a, b] C 1 {f n } n N 2 (1) {f n } n N n f I (2) {f n} n N n g I f I C 1 f = g x [a, b] f n (x) = f n (a) + n f(x) = f(a) + x a x a f n(t) dt g(t) dt. g f (x) = g(x). 45. d dx x a g(t) dt = g(x) 6

3.7 Abel, Abel z n z = 1, z 1 z n n=1 π 4 = 1 1 3 + 1 5 log 2 = 1 1 1 2 + 1 3 3.34 (Abel ) {α n } n, {β n } n s n := n n 1 α k β k = s n β n + s k (β k β k+1 ) k= k= a = s, a k = s k s k=1 (k 1) n α k β k = s β + k= = s β + = s β + = s β + n (s k s k 1 ) β k k=1 n s k β k k=1 n s k 1 β k k=1 n 1 n s k β k s k β k+1 k=1 ( n 1 k= ) ( ) n 1 s k β k + s n β n s β 1 + s k β k+1 k=1 n 1 = s (β β 1 ) + s k (β k β k+1 ) + s n β n k=1 n 1 = s k (β k β k+1 ) + s n β n. k= k=1 n α k 3.35 {α n } n {β n } n α n β n n= k= 61

s n := n α k ( M R) ( n N) s n M. k= n n 1 αβ k = s k (β k β k+1 ) + s n β n. k= k= n s n β n s n β n Mβ n. n s k (β k β k+1 ) M (β k β k+1 ), M (β k β k+1 ) = Mβ Mβ n+1 Mβ k= n 1 s k (β k β k+1 ) 3.36 k= n α k β k k= n=1 z n n z = 1, z 1 z α n := z n, β n := 1 n n z n = k=1 z(1 z n ) 1 z z (1 + zn ) 1 z = 2 1 z {α n } {β n } n=1 z n n Abel 3.34 3.37 {α n } n, {β n } n n N s m := (m n) m α k β k = s m β m + k=n 3.34 m 1 k=n s k (β k β k+1 ) α n β n = n=1 m k=n α k 62

3.38 (Abel (Abel s continuity theorem)) f(z) = a n z n z = R (R > ) K { } Ω K := z C 1 z/r z < R, 1 z /R K n= f Ω K {R} f Ω K {R} lim f(x) = f(r). x [,R) x R n n α n := a n R n α j = a j R j : j= n ( M R) ( n N) α j M. j= ( z ) n, z Ω K β n := fn (z) := R a n z n = α n β n. β n β n+1 = n= z n z 1 = R R n= j= n a k z k k= 1 z/r 1 z /R K. lim β n = ( = z n R < 1 ). m, n N, m > n m m m f m (z) f n (z) = a k z k = α k β k = s k := k j=n+1 k=n+1 α j k s k = α j j= n j= f m (z) f n (z) sup k>n k=n+1 k=n+1 s k (β k β k+1 ) + s m β m. α j = f k (R) f n (R) sup f k (R) f n (R). k>n f k (R) f n (R) m k=n+1 K sup f k (R) f n (R) + s m β m. k>n 63 β k β k+1 + s m β m

(6) f m (z) f n (z) K sup f k (R) f n (R) + s m β m. k>n z n {f n (z)} Cauchy f(z) (6) m f(z) f n (z) K sup f k (R) f n (R). k>n {f n } Ω K a n z n 3.3 (Abel ) Niels Henrik Abel (182 1829, ) 1. α (1 + x) α ( 2 ) 2. 5 3. () n= 4 ( ) 4.1 log z log : ( ) log log: (, ) R exp y R, x (, ) y = log x x = e y. log Taylor ( 1) n ( 1) n 1 ( ) log (1 + x) = n + 1 xn+1 = x n ( x < 1) n n= 64 n=

1 1 + x = ( x) n = n= ( 1) n x n ( x < 1) n= ( ) log (1 + z) = ( 1) n 1 z n ( z < 1) n n= 1 log lim log x = + ( lim log(1 + x) = + ) x + x 1+ ( ) x = 1 ( x dt log x = t ) 1 z C z = e w w w = u + iv (u, v R) e w = e u e iv z z = re iθ (r, θ [, 2π)) z = e w re iθ = e u e iv r = e u e iθ = e iv. (: re iθ = e u e iv r = e u re iθ = e u e iv e iθ = e iv ) r = e u u = log r r = z ( r = z > ) z = e w u = log r v θ (mod 2π) u = log r ( n Z) v = θ + 2nπ ( n Z) w = log r + i (θ + 2nπ). 65

e w = z z C \ {} w e w = z z z = re iθ (r >, θ [, 2π)) w = log r + i (θ + 2nπ) (n Z) w = log z + i arg z 4.1 ( ) z e w = z e w = ( e w e w = e w w = e = 1 ) e w = 1 w = log 1 + i arg 1 = 2nπi (n Z). e w = 2 w = log 2 + i arg 2 = log 2 + 2nπi (n Z). x > e w = x w = log x + 2nπi (n Z). 2nπi e w = 1 w = log 1 + i arg( 1) = + (2n 1)πi = (2n 1) πi (n Z). e w = 2 w = log 2 + i arg( 2) = log 2 + (2n 1)πi (n Z). e w = i w = log i + i arg i = + (2n + 1/2)πi = (2n + 1/2) πi (n Z). e w = 2i w = log 2i + i arg(2i) = log 2 + (2n + 1/2)πi (n Z). log z e w = z z arg z (i) e w = z w log z := log z + i arg z = log r + i (θ + 2nπ) (n Z). z log z log ( z 1 ) ( ) z log (ii) z Arg z ( ( π, π] ) Log z : Log z := log z + i Arg z = log r + iθ (z = re iθ, r >, θ ( π, π] r, θ). Log: C \ {} C ( ) N := {z C z < } (x z x 66

Im Log z π, z x Im Log z π. z x Log z ) Mathematica Mathematica Log[] Im Log z Plot3D[Boole[x^2 + y^2 < 4] Im[Log[x + I y]], {x, -2, 2}, {y, -2, 2}] (Boole[] ) Re Log z log z Im log z C \ {z C z } ( ) Ω ( π,π] := {w C π < Im w π} f : Ω ( π,π] C, f(w) = e w (w Ω π,π ) ( ) z = f(w) dz dw = ew = z dw dz = 1 z. d dz Log z = 1 z 13 (iii) z [, 2π) ( ) z = re iθ, r >, θ [, 2π) log z := log r + iθ log: C \ {} C ( ) P := {z C z > } C \ {z C z } ( ) Ω [,2π) := {w C Im w < 2π} g : Ω [,2π) C, g(w) = e w (w Ω π,π ) z = g(w) dz dw = ew = z dw dz = 1 z. d dz log z = 1 z 13 Log (Log z) = 1 z 67

(ii) Log (iii) log log x (, ) log x = Log x = log x. ( log ) (?): (ii) Log N (iii) log P 4.2 z α a, b a b a b ( ) a > a b = e b log a z α z α p(z, α) (p (power) ) z C \ {}, α C p(z, α) := e α log z log z p(z, α) z z = re iθ (r >, θ R) p(z, α) = e α log z = e α(log r+i(θ+2nπ)) (n Z). α R e α log r = ( e log r) α = r α p(z, α) = r α e i(αθ+nα 2π). (a) α Z nα Z e i(nα 2π) = 1. p(z, α) = r α e iαθ = ( re iθ) α = z α. ( z α α > α z α < α z ) (b) α Q \ Z (7) α = q p (p N, q Z, p q ) p(z, α) = r α e iαθ nq 2πi e p. p q nq p, 1,..., p 1 p(z, α) = r α e iαθ e ik 2π p (k =, 1,..., p 1). 68

ω := e i 2π p ω 1 p p(z, α) = r α e iαθ ω k (k =, 1,..., p 1). z = r α p α = 1 p z p (c) α C \ Q p(z, α) ( ) p(z, α) z α (a) α Z (b) α Q \ Z α (7) p p p (c) α C \ Q ( z ) 5 ( ) 5.1 ( : 1 ) ( ) ( Ahlfors ) C R 2 1. 2. 3. 69

Cauchy 2 5.1 ( ) Ω C f : Ω C C : z = φ(t) (t [α, β]) Ω C 1 C f β (8) f(z) dz := f (φ(t)) φ (t) dt. C C α 5.2 Ω C f : Ω C C : z = φ(t) (t [α, β]) Ω C 1 β (9) f(z) dz = f ds := f (φ(t)) φ (t) dt. ( f 1 C C C ds C ) C 1 F (t) := f (φ(t)) φ (t) U(t) := Re F (t), V (t) := Im F (t) β α F (t) dt = β α α (U(t) + iv (t)) dt := β α β U(t) dt + i V (t) dt α ( Fourier ) 5.3 f(z) = z 2, C φ(θ) = e iθ (θ [, π]) f(z) dz = π ( ) π [ e iθ 2 e ie iθ dθ = i e 3iθ 3iθ dθ = i 3i C ] π = e3πi e 3 = 1 1 3 = 2 3. 5.4 ( ) t φ (t) (8) φ: [α, β] Ω C 1 φ: [α, β] Ω {t j } n j= s.t. α = t < t 1 < < t n = β, φ I j := [t j 1, t j ] φ j := φ Ij I j C 1 tj t j 1 f (φ j (t)) φ j(t) dt 7

( ) (8) C f(z) dz := n j=1 tj t j 1 f (φ j (t)) φ (t) dt. ( ) 5.5 ( ()) Ω C F : Ω C C Ω C 1 a, b F (z) dz = [F (z)] z=b z=a = F (b) F (a). C C z = φ(t) (t [α, β]) φ C 1 C F (z) dz = β α F (φ(t))φ (t) dt = β = F (φ(β)) F (φ(α)) = F (b) F (a). α d F (φ(t))φ(t) dt = [F (φ(t))]t=β t=α dt φ C 1 C 1 C F (z) dz = n j=1 tj t j 1 F (φ(t))φ (t) dt = n (F (φ (t j )) F (φ (t j 1 ))) j=1 = F (φ(t n )) F (φ(t )) = F (b) F (a). 5.6 φ() = 1, φ(π) = 1, C ( z 3 3 ) = z 2 [ z 3 f(z) dz = 3 ] z= 1 z=1 = ( 1)3 1 3 3 () = 2 3. Cf. f : [a, b] R f F s.t. F = f. 71

F (x) := x a d dx x f(t) dt b a a f(t) dt = f(x) f(x) dx = [F (x)] b a = F (b) F (a) 5.7 f(z) = 1 z, z Ω := C \ {z C z }, C : z = φ(t) = eiθ (θ [, 2π]) C f(z) dz = 2πi. f F f(z) dz = F (φ(2π)) F (φ()) = F (1) F (1) = C f ( ) f Ω := C\{z C z } F (z) = log z = log(re iθ ) = log r + iθ ( θ (, 2π) ) f < ε < 1 C ε : z = φ(t) = e iθ (θ [ε, 2π ε]) C ε f(z) dz = F (φ(2π ε)) F (φ(ε)) = (log 1 + i(2π ε)) (log 1 + iε) = (2π 2ε)i. ε + 2πi n Z, f(z) = z n n 1 F (z) := zn+1 n + 1 F (z) = f(z) f(z) dz = F (b) F (a). C (n ) 72

1 log ( z a z z 2 + 1 1 2 1 ) z a z α = e z log α ( 1 z + i + 1 ) z i x = Re z, y = Im z, z, Arg z Riemann : b a f(x) dx = lim n f(ξ j ) x j, x j := x j x j 1, ξ j [x j 1, x j ], := max 1 j n (x j x j 1 ). C f(z) dz = lim j=1 n f(ζ j ) z j. j=1 z j := z j z j 1, ( ζ j z j 1 z j, :=? z j, ζ j α = t < t 1 < < t n = β ξ j [t j 1, t j ] (j = 1,..., n) z j = φ(t j ), ζ j := φ(ξ j ) f(ζ j ) = f (φ(ξ j )), z j = φ(t j ) φ(t j 1 ) φ (ξ j ) t j, t j := t j t j 1 n f(ζ j ) z j j=1 n f(φ(ξ j ))φ (ξ j ) t j. j=1 73

t f(φ(t))φ (t) Riemann := max (t j 1 j n t j 1 ) β C f(z) dz = C α f (φ(t)) φ (t)dt f ds = lim n f(ζ j ) z j. (f 1 Riemann ) j=1 74

5.2 5.8 Ω C C : z = φ(t) (t [α, β]) Ω ( φ: [α, β] Ω ) φ {φ(t) t [α, β]} C ( ) C C 1 φ C 1 1 φ C C 1 φ C 1 t [α, β] φ (t) C C 1 [α, β] {t j } n j= [t j 1, t j ] φ φ [tj 1,t j ] C 1 (t j ) C C 1 {t j } n j= [t j 1, t j ] φ φ [tj 1,t j ] C 1 C (closed curve) φ(α) = φ(β) C (Jordan ) ( ) φ ( t 1 [α, β])( t 2 [α, β]) t 1 t 2 φ(t 1 ) φ(t 2 ) ( ) φ [α, β) ( t 1 [α, β))( t 2 [α, β)) t 1 t 2 φ(t 1 ) φ(t 2 ) ( ) C C ( ) C 1 2 (Jordan ) 75

2,3 5.9 c C, r > C : z = φ(t) = c + re iθ (θ [, 2π]) C C 1 C c r 46. 5.1 C : z = φ(t) (t [, 4]) z = φ(t) := t (t [, 1]) 1 + i(t 1) (t [1, 2]) 1 + i (t 2) (t [2, 3]) i i(t 3) (t [3, 4]) C, 1, 1 + i, i C C 1 C 1 5.11 (C C) C : z = φ(t) (t [α, β]) z = φ( t) (t [ β, α]) C C 5.12 ( C 1, C 2 C 1 + C 2 ) C 1 : z = φ 1 (t) (t [α 1, β 1 ]), C 2 : z = φ 2 (t) (t [α 2, β 2 ]) C 1 C 2 (φ 1 (β 1 ) = φ 2 (α 1 )) { φ 1 (t) (t [α 1, β 1 ]) φ(t) := φ 2 (t β 1 + α 2 ) (t [α 2, β 1 + β 2 α 1 ]) C 1 C 2 C 1 + C 2 ( C 2 C 1 C 1 + C 2 ) 76

5.13 (1) (f(z) + g(z)) dz = f(z) dz + g(z) dz. C C C (2) λf(z) dz = λ f(z) dz. C C (3) f(z) dz f(z) dz. C C (4) f(z) dz = f(z) dz. C C (5) f(z) dz = C 1 +C 2 f(z) dz + C 1 f(z) dz. C 2 (1), (2) (3), (4), (5) 47. (3) ((3) Riemann n n a j a j β β (u(t) + iv(t)) dt u(t)2 + v(t) 2 dt α j=1 j=1 α ) 5.14 C 5.15 C 1 : z = e iθ (θ [, π]), C 2 : z = e iπt (t [, 1]), C 3 : z = e iπt2 (t [, 1]), C 4 : z = t + i 1 t 2 (t [ 1, 1]) {z C z = 1, Im z }. ( 1, 1) f f(z) dz C j 77

48. C 5 : z = t + i 1 t 2 (t [ 1, 1]) f(z) dz C 5 C 4 f(z) dz 5.14 a b c r φ 5.3 R 2 ( ) ( ) P f =, C t Q C f dr = C Ω f(x, y)dxdy C f ds f t ds = C P dx + Q dy. (Green ) C D C f dr = rot f dx dy, C C D P dx + Q dy = ( ( ) x rot f = det f = det D (Q x P y ) dx dy. y ) P = Q x P y. Q f u, v fdz = (u + iv)(dx + idy) = (u dx v dy) + i C 6 Cauchy (1) C C g dz = C g ds. C (v dx + u dy). Cauchy ( ) Cauchy 78

6.1 Cauchy 14 f(z) dz = (a) f C Ω (b) C Ω C 1 (c) C Ω (f ) C (a) (b) (c) dz z = 2πi ( Ω = C \ {}, f(z) = 1 z, C : z = eiθ (θ [, 2π])) z =1 (c) C C (Jordan ) Jordan C C ( ) C D C D 2 (i) Ω Ω C Ω Ω C 1 C f(z) dz = C (ii) C 1 14 2 Cauchy 79

6.2 6.1 (Goursat, Pringsheim) Ω C f : Ω C Ω f(z) dz =. ( Gray [7] Goursat ( ) [8] Pringsheim ) (a) 1 (b) 1 (1 ) 2 2 M := f(z) dz M = := 4 = 1 2 3 4. j ( ) j = 1, 2, 3, 4 2 ( ) f(z) dz = f(z) dz + 1 f(z) dz + 2 f(z) dz + 3 f(z) dz. 4 4 M = f(z) dz f(z) dz j=1 j. 4 f(z)dz j j 1 M 4 f(z)dz. 1 : f(z)dz M 1 4. = 1 2 8

n N f(z)dz M n 4. n L n L n L n = L 2. n ( c C) n = {c}. c Ω f Ω f f(z) f(c) (c) = lim z c z c g(z) := f(z) f(c) f (c)(z c) n N lim z c g(z) z c =. ( ε > )( δ > )( z D(c; δ)) g(z) ε z c. n n D(c; δ) z c L n ( ) g(z) dz g(z) dz ε z c dz εl n dz = εl n n n n 2n = εl2 4. n f(c) f (c)(z c) z 1 ( ) g(z) dz = n (f(z) f(c) f (c)(z c)) dz = n f(z) dz. n 2 g(z) dz = f(z) dz n M n 4. n M 4 n g(z) dz εl2 n 4. n M εl 2. ε M =. Cauchy Ω 1 Ω ( ) 81

6.2 ( ) 6.1 f Ω f Ω 1 a Ω f(z) dz = a a (i) a (ii) a (iii) a 3: a (i) ε 3 a ε, ε f(z) dz = f(z) dz. ε ( ε 4ε ) f(z) dz f(z) dz max f(z) dz 4ε max f(z). z ε z ε ε + f(z) dz =. (ii), (iii) (i) 6.1 Cauchy [9] ( [1] [9] ) 82

6.2 2 1 ( 45 ) 6 1 3 ( ) ( ) 6.3 Q Ω C f : Ω C 3 (i) f Ω ( F : Ω C s.t. F = f) (ii) Ω C 1 C f(z) dz = (iii) f Ω (i) (ii) (ii) (i) (i) (iii) Cauchy ( ) F F 2 f = F (i) (ii) (ii) (iii) Morera ( ) (iii) (ii) Ω = C \ {}, f(z) = 1 z ( : f(z) dz = 2πi ) C 83 C

(i) (ii) (iii) (iii) (ii) Cauchy Ω (iii) (ii) Cauchy 1 ( ) (i), (ii), (iii) 6.3 ( ) 15 (i) f, (ii) f dr =, (iii) rot f =, (i) (ii) (i) ( (ii)) (iii) 6.4 (i) (ii) ( ) ( ) 6.5 f (iii) (ii) Ω (i), (ii), (iii) (ii) (i) 6.3 ( ) Ω C f : Ω C Ω C 1 C f(z) dz = C F : Ω C F = f. Ω a Ω z Ω a z Ω C 1 C z F (z) := f(ζ) dζ C z F (z) C z a z Ω 2 C z, C z C := C z + ( C z) C (ii) = f(ζ) dζ = f(ζ) dζ f(ζ) dζ C C z 15 (i), (iii) (ii), (iii) Cauchy C C z 84

F f z Ω ε > D(z; ε) Ω. h < ε h z z + h [z, z + h] Ω C z+h C z + [z, z + h] F (z + h) F (z) = f(ζ)dζ f(ζ)dζ = f(ζ)dζ. C z+[z,z+h] C z [z,z+h] [z,z+h] dζ = [ζ] ζ=z+h ζ=z = h f(z) dζ = f(z)h h [z,z+h] F (z + h) F (z) f(z) = 1 (f(ζ) f(z)) dζ. h h F (z + h) F (z) f(z) h 1 h 1 h [z,z+h] f(ζ) f(z) dζ [z,z+h] max f(ζ) f(z) ζ [z,z+h] [z,z+h] dζ = max f(ζ) f(z). ζ [z,z+h] f z h F (z) = f(z). 6.4 Cauchy Cauchy 1 ( ) Cauhcy Cauchy ( ) Cauchy 6.4 ( ) Ω C Ω (star-shaped) ( a Ω)( z Ω) [a, z] Ω [a, z] = {(1 t)a + tz t [, 1]} (a z ). a ( z Ω) [a, z] Ω Ω a Ω 1 a 85

6.5 C, C C \ {z C z }, ( ) C C \ {} ( ) 6.6 ( OK) Ω Ω Ω Ω 6.6 ( ) Ω C f : Ω C f F : Ω C F = f. (f Ω Ω 1 ) ( d x f(t)dt = f(x) dx a ) Ω a Ω z Ω [a, z ] Ω F (z ) := f(z) dz ([a, z ] a z ) [a,z ] F : Ω C F = f Ω z Ω ε D(z ; ε) Ω. < h < ε h z + h Ω. 3 a, z, z + h ( ) Ω ( ). () [a, z ] + [z, z + h] [a, z + h] () 6.1 f(z) dz + f(z) dz f(z) dz =. [a,z ] [z,z +h] [a,z +h] F (z + h) F (z ) = f(z) dz. [z,z +h] 86

( 6.3 ) F (z + h) F (z ) f(z ) = 1 f(z) dz 1 dz f(z ) h h [z,z +h] h [z,z +h] = 1 (f(z) f(z )) dz. h F (z + h) F (z ) h f(z ) 1 h [z,z +h] max f(z) f(z ) dz z [z,z +h] [z,z +h] = max z [z,z +h] f(z) f(z ). f z h F (z + h) F (z ) lim h h F z F (z ) = f(z ). = f(z ). 49. a, z, z + h ( ) Ω (z Ω, z + h D(z ; ε) Ω, Ω a ) 5. ( ) f : Ω C z Ω h 1 f(ζ) dζ f(z) h [z,z+h] ( ) 6.7 ( Cauchy ) Ω C f : Ω C Ω C 1 C f(z) dz = C (f Ω Ω 1 ) C a, b C a = b. f F f(z) dz = [F (z)] b a = F (b) F (a) =. C 87

6.8 ( Cauchy ) D C f : D C D C 1 C f(z) dz = C (f D D 1 ) 6.9 (1/z ) ( Log ) f : C\{} C, f(z) = 1 z dz ( : = 2πi ) Ω := C \ {z C z } z =1 z 1 f Ω f Ω dζ F (z) = f(ζ) dζ = (z Ω) ζ [1,z] ( Log z z = 1 ) C Ω C 1 f(z)dz =. ( 1 z a dz z c = r z a = δ z c =r z a = dz z a =δ z a Cauchy f(z) = 1 f(ζ) dζ f 1 ) 2πi C ζ z 1 6.1 ( ) a, c C, c a < r z a dz z a = 2πi. z c =r ( ) 11/24 1: z c = r, z a = δ 11/24 2: ε ( ) [1,z] C 88

c = a c a ρ = c a ρ >. a = c + ρe iϕ ϕ R ρ = c a < r δ := (r ρ)/2 δ > D(a; δ) D(c; r). ε < π ε 2 C 1,ε : z = c + re iθ C 2,ε : z = a + δe iθ (θ [ϕ + ε, ϕ + 2π ε]), (θ [ϕ + ε, ϕ + 2π ε]), Γ ε : z = [(1 t)(ρ + δ) + tr] e i(ϕ+ε) (t [, 1]), Γ ε : z = [(1 t)(ρ + δ) + tr] e i(ϕ+2π ε) (t [, 1]), C ε := Γ ε + C 1,ε Γ ε C 2,ε C ε ε > Ω := C \ {a + re iϕ r } 1 z a ε 16 (Γ = Γ ) dz z a z c =r ( 214/11/25) a c > r R := 1 D(c; R) z a C 1, z c =r dz z a = z a =δ r + a c 2 z c = r D(c; R) C ε dz z a =. C 2, dz z a =. dz z a dz z a =δ z a =. dz 2π z a = 1 δe iθ iδeiθ dθ = 2πi. r < R < a c Cauchy dz z a =. z c =r a c < r 89

4: (a) 4 C 11, C 12, C 21, C 22, Γ 1, Γ 2 C 11 + C 12, C 21 + C 22 z c = r, z a = δ (1) C 11 +C 12 (b) z c = r z dz z a = C 21 +C 22 dz z a 1 z a = 1 (z c) (a c) = 1 z c 1 1 a c z c = a c a c z c = r = n= (a c) n (z c) n+1. < 1 (z ) Weierstrass M-test z c = r dz z a = (a c) n (z c) dz = (a c) n 2πiδ n+1 n = 2πi. z c =r n= z c =r (c) Green Cauchy (Green 17 ) D C f D = D D f(z) dz =. ( D ) 16 17 Green ( ) 9 n= D

1 D := {z C z a > δ, z c < r} D z a C 1 : z c = r C 2 : z a = δ C 1 C 2 D dz = D z c = dz z c =r z a dz z a =δ z a. 51. (1) 6.7 () () Cauchy ( ) ( ) 7 7.1 Cauchy 7.1 ( Cauchy ) c C, R > D := D(c; R), C = D Ω C D Ω f : Ω C z D ( ) f(z) = 1 2πi C f(ζ) ζ z dζ. f(ζ) f(z) (ζ Ω \ {z}) g(ζ) := ζ z f (z) (ζ = z) g : Ω C Ω \ {z} Cauchy g(ζ) dζ =. C f(ζ) ζ z dζ = C C f(z) dζ = f(z) ζ z C 1 2πi C f(ζ) dζ = f(z). ζ z dζ ζ z = 2πif(z). 91

7.2 1 7.2 C C C 1 {f n } C C f f n (z) dz = f(z) dz. lim n n f n (z)dz f(z)dz C C C C C f n (z) f(z) dz sup f n (z) f(z) dz. z C C (analytic function) 7.3 ( ) Ω C f : Ω C c Ω, R >, D(c; R) Ω (11) a n := 1 f(ζ) n+1 dζ 2πi (ζ c) ζ c =R f(z) = a n (z c) n n= (z D(c; R)). D := D(c; R), C := D (C ζ = c + Re iθ (θ [, 2π]), C = {ζ C ζ c = R} ) z D Cauchy f(z) = 1 f(ζ) 2πi ζ c =R ζ z dζ. r := z c R r < 1 ζ C z c ζ c = r 1 ζ z = 1 (ζ c) + (c z) = 1 ζ c = 1 ( ) n z c. ζ c ζ c f(z) = 1 2πi C n= 1 1 z c ζ c f(ζ) ζ c ( ) n z c dζ. ζ c f C M := max f(ζ) ζ C ( ) n f(ζ) z c M ζ c ζ c R rn. 92 n=

M n := M R rn M n Weierstrass M-test n= C n= C f(z) = 1 ( ) n f(ζ) z c ( 1 dζ = 2πi ζ c ζ c 2πi f(z) = n= a n (z c) n. n= ζ c =R f(ζ) (ζ c) n+1 dζ ) (z c) n. (11) R c R >, D(c; R) Ω R a n R a n = f (n) (c) n! ( ) < R 1 < R 2, D(c; R 2 ) Ω D = {ζ C R 1 < ζ c < R 2 } ( ) 52. n! 2πi ζ c =R 2 ( ) f(ζ) n! dζ (ζ c) n+1 2πi 7.4 ζ c =R 1 f(ζ) dζ =. (ζ c) n+1 7.5 f F = f F F 2 f = F 1 7.6 Ω C f : Ω C z Ω, R >, D(z; R) Ω n N (12) f (n) (z) = n!1 f(ζ) n+1 dζ. 2πi (ζ z) ζ z =R (Cauchy n ) f (n) (c) = n!a n = n! f(ζ) n+1 dζ. c z 2πi ζ c =R (ζ c) 93

8 8.1 Cauchy ( ) Cauchy Laurent (the Laurent expansion) (residue) c R > s.t. < z c < R f c Laurent f(z) = a n (z c) n + n= n=1 a n (z c) n ( < z c < R) a n a n = n! 2πi ζ c =R f(ζ) (ζ c) n+1 dζ (Taylor ) f c Res(f; c) ( ) 1 Res(f; c) := a 1 f(z) dz. 2πi z c =R Q Laurent A ζ := 1 ζ z c 1 z c 94

8.2 Cauchy Cauchy 8.1 ( 3.2) D C D C 1 D ( D )D f f(z)dz =. D ( (a contour integral) TEX \oint ) 5: D D C 1 + C 2 + C 3 ( ) ( 8.4 ) ( ) 8.1 ( f ) 8.2 () 95

c, a C, r >, c a < r z c =r dz z a = 2πi < δ < r a c δ D := {z C z c < r, z a > δ}, C 1 : z c = r, C 2 : z a = δ D C 1 C 2 C 1 D = C 1 C 2 dz = D z a = dz C 1 z a dz C 2 z a. z c =r dz z a = C 1 dz z a = C 2 dz z a = z a =δ dz z a = 2πi. Cauchy ( ) 8.3 ( 8.1 Cauchy ) < δ < r a c δ 1 f(z) 2πi z c =r z a dz = 1 2πi = 1 2π z a =δ 2π f(z) z a dz = 1 2π f(a + δe iθ ) iδe iθ dθ 2πi δe iθ f(a + δe iθ ) dθ. 1 2πi z c =r f(z) dz f(a) z a = 1 2π f(a + δe iθ ) dθ f(a) 1 2π 2π 1 2π max f(a + δe iθ ) f(a) θ [,2π] max f(a + δe iθ ) f(a). θ [,2π] 2π 2π dθ dθ δ 1 f(z) 2πi z a z c =r dz = f(a). ( ) Cauchy 96