複素積分.pdfVIP

i i i Σeix eix + eix + · · · z z? x, y n ? 2 xy x y z x +iy 2 x, y z x + iy y 0 upper half plane y 0 lower half plane z x + iy x real part y imaginary part Rez x Imz y z Rez + iImz z x + iy |z | √z?z √ x + iy x ? iy √x2 + y2 z r, θ x r cos θ, y r sin θ z z x + iy r cos θ + ir sin θ r cos θ + i sin θ reiθ z reiθ polar form r θ r |z | √x2 + y2 , tan θ y x 1 θ z argument θ argz z ? 0 2π 1 ? n n zn zn Reiθ z i? z re zn n n in? z r e Reiθ rn ein? R rn ? r R1/n , ein? eiθ ? ? θ/n eiθ eiθ+2iπ eiθ+4iπ · · · e2iπ cos 2π + i sin 2π 1 2π m m 0, 1, 2, . . . , n ? 1 θ + 2mπ ? n zn n z z R1/nei θ+2mπ /n de Moivre cos θ + i sin θ n cos nθ + i sin nθ 2 z rei? , zn rn ein? rn cos ? + i sin ? n rn cos n? + i sin n? θ + 2mπ r R1/n , ? n eiθ n einθ cos θ + i sin θ n cos nθ + i sin nθ z z r eiθ r eiθ r cos θ + i sin θ r cos θ + i sin θ 1 2 1 2 1 1 1 2 2 2 r r cos θ cos θ ? sin θ sin θ + i cos θ sin θ + i sin θ cos θ 1 2 1 2 1 2 1 2 1 2 r r cos θ + θ + i sin θ + θ 1 2 1 2 1 2 zn reiθ n rn cos nθ + i sin nθ i zn Reiθ i eiπ/2 R 1 n 2 z rei? ei π/2+2mπ /2 ei π/4+mπ cos π + mπ + i sin π + mπ r R1/2 1 4 4 m 0 m 1 π π 1 √ m 0 : cos + i sin 1 + i 4 4 2 π π 1 √ m 1 : cos + π + i sin + π ? 1 + i 4 4 2 m 2 m 0 m 3 m 2 ?1 + i 3 zn Reiθ ?1 + i √2ei3π/4 R 21/2 n 3 z rei? 21/6 ei 3π/4+2mπ /3 3 3 m 0, 1, 2 ? z D D z f z D f z D f z f z D regular D f z df f z + ?z ? f z lim dz ?z→0 ?z 1 D 1 ?z → 0 1 z0 z0 0 |z ? z0 | r r f z z0 0 |z ? z0 | r z0 f z z0 z0 z0 r ?z → 0 z 0 z x + iy x 0, y 0 ?z → 0 i ?x → 0 , ?y 0 ii ?x 0 , ?y → 0 2 ?y 0 ?x 0 ?x 0 ?y 0 f z 2 f z D f z x, y f z + ?z ? f z f x + ?x, y + ?y ? f x, y lim lim ?z→0 ?z ?z→0 ?x + i?y i ?x → 0 , ?y 0 f x + ?x, y + ?y