z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z

Tips KENZOU 28 6 29 sin 2 x + cos 2 x = cos 2 z + sin 2 z = OK... z < z z < R w = f(z) z z w w f(z) w lim z z f(z) = w x x 2 2 f(x) x = a lim f(x) = lim f(x) x a+ x a

z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z z { } f(z) (3) lim = A z z g(z) B, B..2 D w = f(z) D z lim z z f(z) = f(z ) f(z) z () f(z) g(z) z f(z) ± g(z) f(z) g(z) f(z)/g(z) g(z = ) z (2) F (w) w = w w = f(z) z w = f(z ) F (f(z)) z.2 D w = f(z) D z df dz = lim f(z) f(z ) f(z + z) f(z ) = lim z z z z z z (.) f(z) z f (z ) D D f(z) z z z f(z) f(z) z = z z () {f(z) ± g(z)} = f (z) ± g (z) (2) f(z) g(z) = f (z)g(z) + f(z)g (z) (3) { } f(z) = f (z)g(z) f(z)g (z) g(z) g(z) 2 2

. z θ (.) df dz = lim r z = re iθ (.2) f(z + re iθ ) f(z) e iθ (.3) r f(z) z (.3) θ f(z) [Example] { f(x) = x 2 (x ) x 2 (x < ) (.4) f(x) f (x) = { 2x (x ) 2x (x < ) (.5) y y x x f (x) f () = ) f (x) x = f (x) x = f (x) x =.2. f(z) (.3) df/dz θ z z = x + i y, ( z = x + iy ) (.6) f(z) = u(x, y) + iv(x, y) f(z + z) f(z) = u(x + x, y + y) u(x, y) +i{v(x + x, y + y) v(x, y)} (.7) 3

u, v.7 x y θ r u(x + x, y + y) = u(x, y) + u u x + y (.8) x y v(x + x, y + y) = v(x, y) + v v x + y (.9) x y f(z + z) f(z) = x u u v v + y + i x + i y x y x y ( u = x + i v ) ( ) u x + x y + i v y (.) y df dz = lim f(z + z) f(z) = z z cos θ = e iθ i sin θ df dz = u x + i v [ u x + y + v x i x = r cos θ, y = r sin θ, z = re iθ (.) ( u x + i v ) ( ) u cos θe iθ + x y + i v sin θe iθ (.2) y ( u x v y )] sin θe iθ (.3).3 θ [ ] = = u x = v y, u y = v x (.4).4 f(z) w = z 2 2 z = x + iy z 2 = (x + iy) 2 = x 2 y 2 + 2ixy = u + iv u v = 2x = x y, u v = 2y = y x w = z 2 2 f(z) = z 2 () (2) z = x + iy f(z) = z 2 = (x + iy)(x iy) = x 2 + y 2 = u + iv... u = x 2 + y 2, v = u x = 2x, u y = 2y, v y = v x = z = x = y = f(z) () z = (2) 2 f(z) z = z f(z) z 4

.3 f(z) z f(z) 3 () 2 (2) () (2) (3) (4).3. f(z) z f(z) z f(z ) = lim z z f(z) (.5) f(z) z = z z = z f(z) = z e z z = f(z) z = z lim f(z) = lim z z e z = z = f(z) f() = z =.3.2 f(z) z = z z z lim f(z) = (.6) z z z = z f(z) z = z (z z ) n z f(z) N N z z (z z ) 2 (z z ) 3 2 3 4 f(z) = z 2 +, f(z) = (z i)(z + i) f(z) z = i i 3 5

.3.3 f(z) z = z z = z z = z z = z 4 f(z) = e z z = x(x > ) lim f(z) = lim e x = x + x z = x(x < ) lim f(z) = lim e x = x x + z = x 5 f(z) = e z a, a f(z) = e z a = N = 6 f(z) = (z + i)e z+i, f(z) e z = n! (z a) n z n n! (z + i) n f(z) = (z + i)e (z+i) = (z + i) n! =, ( z + i > ) n!(z + i) n z = i.3.4 (z z ) b b log (z z ) z = z 2π z = z w = z b z z = e iθ θ 2π θ θ 2π z e iθ e i2π = w e ibθ e i2πb θ 2π z w b = n e i2πn b θ = 2π w z w Q&A 4 z z e z sin z z = 6

f(z) = cosec(/z) z = /nπ (n = ±, ±2, ) n z = z = z = y = /x x x + x 2 + f(z) = /z r e iθ r θ k 2 z z 2 f(z) z z 2 2 5 I = f(z)dz (2.) I = f(z)dz (2.2) 5 contour( 7

I 6 z z 2 z 2 z f(z)dz = I (2.3) 2. z z 2 z = z(t) = x(t) + iy(t), (α t β) z(α) = z z(β) = z 2 t α t β f(t) = u(t) + iv(t) z2 z f(t)dt = z2 z z2 u(t)dt + i v(t)dt (2.4) z f(z) D D z z 2 [ ] f z2 (z)dz = f(z) = f(b) f(a) (2.5) z y z 2 y Ex-7 y 2 z 2 = x 2 + iy 2 3 2 y Ex-8 z 2 = + i z x x 2 x 2 x 7 I n = zdz n (n =, 2, 3) x x 2 z 2 I 2 I = zdz + z = x 2 + iy H H P zdz = zdz H P H xdx = 2 x2 ( y y 2 ) dz = idy y2 P zdz = i (x 2 + iy)dy = ix 2 y 2 2 y2 2 I = 2 x2 + ix 2 y 2 2 y2 2 = 2 (x 2 + iy 2 ) 2 = 2 z2 2 (2.6) 2 2 y = (y 2 /x 2 )x z = x + iy = x + i y ( 2 x = + i y ) 2 x x 2 x 2 dz = ( + i y 2 x 2 I 2 = zdz = 2 ) dx x2 6 ( + i y ) 2 2 dx = x 2 2 (x 2 + iy 2 ) 2 = 2 z2 2 8

3 y2 zdz = ydy = 2 y2 2 zdz = P x2 H (x + iy 2 )dx = 2 x2 + ix 2 y 2 I 3 = 2 x2 2 2 y2 2 + ix 2 y 2 = 2 z2 2 I = I 2 = I 3 f(z) = z 8 + i 2 x = y 2 () f(z) = z, (2) f(z) = z 2 () x(t) = t 2 y(t) = t t ) z(t) = x(t) + iy(t) = t 2 + it dz = (2t + i)dt z(t) = x(t) iy(t) = t 2 it zdz = (t 2 it)(2t + i)dt = i 3 2 2 x = t, y = t ) x = y = t t ) zdz = 2 H zdz + zdz = P tdt + i( it)dt = + i f(z) = z z = (2)f(z) = z 2 2.5 +i [ z 2 dz = z 2 dz = z 2 dz = 2 3 z3 ] +i = 2 ( i) 3.9 a n { (z a) n 2πi n = dz = n z a = re iθ dz = ire iθ dθ 2π 2π (z a) n dz = i (re iθ ) n re iθ dθ = ir n+ e i(n+)θ dθ = I n... n = I n = i n = 2π dθ = 2πi I n = ir n+ [ e i(n+)θ n + ] 2π = (2.7) 2. f(z) f(z) f(z)dz = (2.8) 9

7 F ig. F ig.2 2 F ig.3 F ig.4 D n D 2 z D 2 z 2 [ ] z = x + iy dz = dx + idy f(z) f(z) = u + iv f(z)dz = (u + iv)(dx + idy) = {(u + iv)dx + (iu v)dy} = (P dx + Qdy) P = u + iv Q = iu v 8 P Q x, y (P dx + Qdy) = D ( Q x P ) y (2.9) 2.9 Q x P ( u y = y + v ) ( u + i x x v ) y.4 2.8 9 f(z) D F ig. f(z)dz = (2.) f(z) 2 n D F ig.2 f(z)dz = f(z)dz + f(z)dz + + 2 f(z)dz n (2.) f(z) D z z 2 D 2 D F ig.3 f(z)dz = f(z)dz 2 (2.2) f(z) D D 2 Fig4 D f(z)dz = f(z)dz (2.3) 2 7 8.8 7 f(z) 8 http://www2.plala.or.jp/ksp/vectoranalysis/greenstheorem/ 3 9

. z = Fig.5 z 2 = r 2 e iθ2 I = z dz I 2 = 2 z dz 2 z 2 F ig.5 z 2 θ 2 r 2 θ 2 2π r 2 2 f(z) = z z = I y = f(z) = /z = /x r2 I = x dx = log r 2 z = r 2 e iθ dz = ir 2 θe iθ I = I = I + I z2 r 2 z dz = i θ2 I = log r 2 + iθ 2 r 2 e iθ r 2 e iθ dθ = iθ 2 I 2 I 2 = r2 z2 I 2 = r 2 z dz = i I 2 = I 2 + I 2 x dx = log r 2 θ2 2π I 2 = log r 2 + i(θ 2 2π) I 2 = I 2πi 2 dθ = i(θ 2 2π) F ig.ex F ig.ex 2 2i Γ a i Γ Γ 2 i 2

Ex- a Fig.Ex- () dz = 2πi (2) dz = (n > ) z a (z a) n Ans a Γ Γ (z a) 2 2.3 2.7 Γ dz = 2πi, z a (z a) n dz = Γ Γ (z a) n dz dz = (n > ) (z a) n z Ex-2 z 2 dz z = 2 Fig.Ex-2 + Ans z z 2 + = ( 2 z i + ) z + i z = ±i z = ± Γ Γ 2 z z 2 + Γ Γ 2 2. z z 2 + dz = z z 2 + dz + z z 2 + dz Γ Γ z z 2 + dz = ( ) 2 Γ z i dz + Γ z + i dz /(z + i) Γ (2.7 dz = 2πi Γ z i Γ 2 Γ 2 Γ Γ 2 z z 2 dz = iπ + z z 2 dz = πi + z z 2 dz = 2πi + 2.. f(z) D D a f(a) = f(z) dz (2.4) 2πi z a a Γ (Fig Ex- Γ f(z) z a 2.3 f(z) z a dz = f(z) dz (2.5) Γ z a 2

Γ z z = a + re iθ f(z) 2π Γ z a dz = i f(a + re θ ) re iθ re iθ dθ = i r + 2.4 [ ] 2π lim i f(a + re iθ )dθ = if(a) r 2π 2π f(a + re iθ )dθ (2.6) dθ = 2πif(a) (2.7) 2 2 D f(z) a D f(a) = 2πi f(z) z a dz f(z) dz (2.8) 2πi 2 z a 2 a D 2.8 h a + h 2.4 f(a + h) f(a) = ( f(z) h 2πhi z (a + h) f(z) ) dz = f(z) z a 2πi (z a h)(z a) dz h f(a + h) f(a) lim = f (a) = lim h h h 2πi f(z) (z a h)(z a) dz = f(z) dz (2.9) 2πi (z a) 2 f(z) f (z) f (a) = 2! f(z) dz (2.2) 2πi (z a) 2 f (n) (a) = n! f(z) dz (n =, 2, ) (2.2) 2πi (z a) n+ f(z) n Q&A 3

( f(x) x f(z) z f(z) f(z) u v f(z) http://collie.low-temp.sci.yamaguchi-u.ac.jp/ ashida/work/ 2 4 3 3. {a n } a n = a + a + + a n + z z n f(z) = c n z n = c + c z + c 2 z 2 + + c n z n + (3.) c n n c n z n n c n z n > c n+ z n+ (3.2) 4

3.2 n c n z n > c n+ z n z z < c n c n+ (3.3) 3. c n R = lim n c n+ (3.4) z < R (3.5) R z. R + 2z + 4z 2 + 8z 3 + f(z) = c n z n = 2 n z n 2 n R = lim n 2 n+ = 2 /2 z < /2.2 e z e z = c n = /n! R z n n! = + z! + z2 2! + z3 3! + c n (n + )! R = lim = lim = lim n c n+ n n! n = n R = z < z.3 f(z) = + z + 2z 2 + 3z 2 + 4z 2 + f(z) = + nz n n 2 R = lim n n + = z <.4 z + z 2 + z 3 + z > z = + h, (h ) z n = ( + h) n > + nh > nh lim n z n > lim > nh = n 5

z < a = / z > lim n z n > lim n > /an = / = z = lim n zn =.5 + z + z 2 + z 3 + z < /( z) S n = + z + z 2 + + z n = zn z z < zn /(z ) 3.2 D f(z) D a D 2 f(z) = A n (z a) n = f (n) (a) (z a) n (3.6) n! A = f(a), A = f (a), A 2 = 2 f (a),, A n = n! f (n) (a) (3.7) F ig.5 z a D z a R r ζ [ ] Fig 5 a R 3 a r < R r f(z) r z a r < R (3.8) z 2.4 f(z) = f(ζ) dζ (3.9) 2πi ζ z z ζ, a z ζ ζ a = r (3.) z 2 z a < r (3.) z a ζ z < (3.2) 2 3 R 6

3.9 /(ζ z) 3.2.6 ζ z = ζ a (z a) = ζ a z a ζ a = ζ a ( ) n z a (3.3) ζ a (3.9 f(z) = (z a) n 2πi f(ζ) dζ (3.4) (ζ a) n+ f(z) = f (n) (a) (z a) n (3.5) n! 3.8 z a < R R 4.6 z = 3.7 f(z) = z + f (z) = ( + z) 2, f 2 (z) = ( + z) 3,, f (n) (z) = ( )n n! ( + z) n z = f (n) () = ( ) n n! + z = z + z2 z 3 + (3.6) z = R z = R =.7 f(z) z = a f(z) = z a + b z = a z a + z a b f(z) = = z a b f(z) = b z a b 2 + z a b z a + b = b + z a b ( ) 2 ( ) 3 z a z a + + b b (z a)2 (z a)3 b 3 b 4 + < z a < b b.8 f(z) z = 2 f(z) = (a + z) 2 4 a 7

z = z f(z) = a 2 ( + z a) 2 z/a < f(z) f(z) = a 2 2z a 3 + 3z2 a 4 4z3 a 5 + = ( ) n n + c n = ( ) n (n + )/a n R a n c n R = lim n c n+ = lim a n + n n + 2 = a f(z) z = a R = a zn 4 a a D f(z) z = a a f(z) f(z) = A + A (z a) + A 2 (z a) 2 + + A z a + A 2 (z a) 2 + = A n (z a) n A n + (z a) n = A n (z a) n (4.) n= n= a A ( f(z) z = a f (a) A n A n = f(ζ) 2πi (ζ a) n+ dζ, A n = f(ζ) dζ 2πi (ζ a) n+ (z a) (z a) k f(z) k f(z) z = a k z = a f(z).9 f(z) = z = z(z + ) f(z) 6 f(z) = z z + = z + z z2 + z f(z) z = 8

5 5. a A z = a f(z) f(z) a z a A Res(f, a) Res(a) z = a f(z) = F (z) z a f(z) Res(a) = lim z a (z a)f(z) (5.) f(z) = A z a + A n (z a) n (z a) z a { } lim (z a)f(z) = lim A + A n (z a) n+ = A z a z a z = a k f(z) = f(z) Res(a) = Res(a) = lim z a (z a)f(z) F (z) (z a) k [ d k { (k )! lim z a dz k (z a) f(z)} ] k (5.2) f(z) = A k (z a) k + A k+ (z a) k + + A (z a) + A n (z a) n (z a) k (z a) k f(z) = A k + (z a)a k+ + (z a) 2 A k+2 + + (z a) k A + A n (z a) n+k A z k z a [ d k { lim z a dz k (z a) f(z)} ] [ ] k = lim (k )! A + A n (n + k)(n + k ) (n + 2)(z a) n+ z a = (k )! A [ d k { } ] Res(a) = (k )! lim z a dz k (z a) k f(z) f(z) = Q(z) P (z) P (a) = P (a) Res(a) = Q(a) P (a) (5.3) 9

P (z) z = a P (a) = P (z) = P (a)(z a) + P (a) (z a) 2 + 2! f(z) = Q(z) z a P (a) + 2 (z a)p (a) + (z a) z a { } Q(z) Res(a) = lim (z a)f(z) = lim z a z a P (a) + 2 (z a)p (a) +.2 f(z) = e /z = Q(a) P (a) f(z) z = f(z) f(z) = e /z = n! z n = + z + 2! z 2 + A Res() =.2 z3 + 5 z(z ) 3 z = Res() = lim z zf(z) = 5 z = 3 Res() = lim z 2! dz 2 {(z )3 f(z)} = 6 d 2 5.2 f(z) z = a f(z) = A + A (z a) + A 2 (z a) 2 + + A z a + A 2 (z a) 2 + (5.4) z = a.9 { (z a) n 2πi n = dz = n (5.5) 5.4 z = a { f(z)dz = A + A (z a) + A 2 (z a) 2 + + A z a + A 2 (z a) 2 + }dz = 2πiA = 2πi Res(a) (5.6) f(z) a k (k =, 2,, n) D f(z)dz = n k= k f(z)dz = 2πi n Res(a k ) (5.7) 5.7 k= 2

F ig.6 n a n 2 a a 2 D 5.2. Ex- I = x 2 dx (a > ) + a2 Ans x z = x + iy r I = lim r x 2 dx = lim + a2 r z 2 dz = lim f(z)dz (5.8) + a2 r r F ig.7 y F ig.8 y F ig.9 y ia r ia r ia x r r x r x r ia ia ia f(z) = /(z 2 + a 2 ) f(z) = z 2 + a 2 = (z ia)(z + ia) (5.9) z = ±ia f(z) Fig.8 5 y Fig.9 5.8 5 I = I + I 2 = lim f(z)dz + r f(z)dz (5.) 2

5. 2 I 2 = f(z)dz = 2πiRes(ia) Res(ia) = lim (z ia)f(z) = lim z ia z ia z + ia = 2ia I 2 = π a I I = lim f(z)dz lim r r z = re iθ ( θ π) π z 2 dz = lim + a2 r 6 π { ie iθ } I = dθ lim = r 2r rie iθ r 2 e 2iθ + a 2 dθ Q&A I = x 2 + a 2 dx = I + I 2 = π a (5.) 4 843 A f(z) 6 http://www.osakac.ac.jp/labs/mandai/writings/bi-m3.pdf 22

6 6. f(z) z f(z) f(z) D D f(a) = f (a) = f (a) = = f (n ) (a) =, f (n) (a) (6.) a n n a f(z) f(z) = f (n) (a) (z a) n (6.2) n! a n f (n ) (a) f(z) = f (n) (a) (z a) n + f (n+) (a) n! (n + )! (z a)n+ + f (n+2) (a) (n + 2)! (z a)n+2 + = (z a) n { f (n) (a) n! + f (n+) (a) (n + )! (z a) + f (n+) } (a) (n + 2)! (z a)2 + = (z a) n g(z) (6.3) 6.2 D 2 f (z) f 2 (z) D D D f (z) = f 2 (z) D f (z) = f 2 (z) D f (z) = f 2 (z) D f (z) = f 2 (z) D D D f(z) f (z) = f 2 (z) D f(z) D a n f(z) = (z a) n g(z) g(z) = g(a) + g(z) g(a) g(z) g(z) g(a) 23

z a g(z) g(a) ɛ g(z) > g(a) ɛ > g(z) a z z a g(z) 7 a a f(a) = a f(z) = D f(z) D f(z) = D f(z) = f(z) = f (z) f 2 (z) f(z) f (z) f 2 (z) D f (z) f 2 (z) D f (z) = f 2 (z) D f (z) = f 2 (z) cos x = ( ) ( ) n x 2n (6.4) 2n! z ( ) cos z = ( ) n z 2n (6.5) 2n! cos z 8 6.4 6.5 z ( cos x = sin x + π ) ( = cos z = sin z + π ) 2 2 cos 2 x + sin 2 (x) = = cos 2 z + sin 2 (z) = 6.3 D f(z ) D or ) D D or ) f(z) = f (z) f(z) D or ) f(z) D f (z) (A) F ig.7 (B) D r a a b r a r b b D 7 8 cos z z = D 24

f(z) x z x 6.3. P (z : a) a P (z : a) = A n (z a) n (6.6) r a z a < r a b P (z : a) b Q(z : a) = B n (z b) n (B n = f (n) (b)/n!) (6.7) b r b z b < r b 6.7 Fig.7 A) b z b < r a b a (B) r b r a b a 2 P (z : a) = Q(z : b) Q(z : b) P (z : a) z b < r b z a < R (B) P (z).22 9 f(z) = + iz log 2i iz (6.8) g(z) = tan z (6.9) z = ±i f(z) = ±i z z = ±i f (z) = + z 2 g(z) = tan z,g(z) g (z) = + z 2 f() = g() = 2 f(z) g(z).23 D = {z : z < } f (z) = + z + z 2 + z 3 + (6.) ( z < ) D 9 z 25

D f(z) z f (z) = z (6.).23.24 D D H G i D F z = D z = D 6. D 6. [ ] f(z) z 2? z < i D 2 D a b D f(z) f (z) 2 f (z) = a z a = a ( ) n z a z a / a < (6.2) a f (z) a a 6. 6.2 /(z ) /(z ) z = f (z) D f (z) D f (z) a D D D D D f (z) = f (z) f(z) 2 X 2 /( x) = x 2 ( x < ) f(z) = { f (z) z < f (z) z a < a 26

f(z) D D f(z) = /( z) f (z) f(z) f(z) f (x) D 2 b f 2 (z) = b ( ) n z b z b / b < b D D D 2 f(z) f(z).24, f (z) = 2z 2 + 4z 4 8z 6 + 6z 8 + f (z) = ( 2) n z 2n c n = ( 2) n z 2n 3.3 n c n z 2n > c n+ z 2(n+) z 2 < 2 z < 2 R = / 2 D z < / 2 6.3 2z 2 f (z) = + 2z 2 = ( i 2z)( + i 2z) (6.3) z = ±i/ 2 6.3.25 f(z) = ( z) n, F {z; z < } g(z) = i ( + iz) n, G {z; z i < } () g(z) f(z) G (2) H = {z; z + < } H h(z) g(z) H h(z) ()z F z < f(z) = z G ( z) n = + ( z) + ( z) 2 + + ( z) n + = ( z) = z g(z) g(z) = i + iz = i( i + z) = z i < ( + iz) n + ( + iz) + ( + iz) 2 + + ( + iz) n i + = ( + iz) = z 27

f(z) g(z) D = F G f(z) = g(z) g(z) f(z) (2)h(z) g(z) G H h(z) = g(z) = z H h(z) = z.26 D Re(z) f (z) = sin z D f (z) f (z) = sin z (Re(z) ) f (z) z Re(z) f (z) = sin z = sin(z + 2π) f (z) = sin(z + 2π) (6.4) 2π Re(z) < + D 6.4 f (z) Re(z) f (z) 6.4 f (z) f (z) f 2 (z) f 2 (z) = sin(z + 4π), D 4π Re(z) < + Re(z + 2π) f (z) f 2 (z) f (z).27 x f(x) = x f(x) = { x x x x < f(x) D D x z x < z 2 6.3.2 Γ(z) Γ(z) = t z e t dt (6.5) Re(z) > (6.6) 28

refeq-23 D 6.6 6.5 z [ ] zγ(z) = z t z e t dt = (t z ) e t dt = t z e t + t z e t dt = t z e t dt = Γ(x + ) (6.7) t t z e t ) Re(z) > Γ(z) = Γ(z + ) z (6.8) < Re(z) Γ(z).26 Γ() = n 6.8 Γ(n + ) = nγ(n) = n(n )Γ(n ) = n(n )(n 2) Γ() = n! 22 Q&A f (z) = f 2 (z) K 22 29