8. y = f(x) (x, y : ) w = f(z) (w, z : ) df(x) df(z), f(x)dx dx dz f(z)dz : e iωt = cos(ωt) + i sin(ωt) [ ] : y = f(t) f(ω) = π f(t)e iωt dt. : ϕ(x, y) x + ϕ(x, y) y = ( )..
. : 3. (z p, e z, sin z, sinh z, log z) 4. : 5. 6. :.3.3. x, y,... R: z : i: (i = ) Re z, Im z R: z z z: z z = x + iy = Re z + i Im z () z = x iy = Re z i Im z Re z = x = z + z z z z z, Im z = y =. i z = z z = (x + iy)(x iy) = x (iy) = x + y R + i i = ( + i)( i) ( i)( i) = ( + i)( + i) ( i)( + i) + 3i + i = = + 3i i.3. z = x + iy z z z (x, y) (r, θ) z = r(cos θ + i sin θ) = re iθ ( r = x + y = z, tan θ = y x ). r = z > z θ arg z z
π z = r(cos θ + i sin θ) = r [cos(θ + nπ) + i sin(θ + nπ)] (n =...,,,,,,... Z) Z: z = r (cos θ + i sin θ ), z = r (cos θ + i sin θ ) z z = r r [cos(θ + θ ) + i sin(θ + θ )] z z = r r = z z, arg(z z ) = θ + θ. () z z z z arg(z z ) z z = r (cos θ + i sin θ ) r (cos θ + i sin θ ) = r r [cos θ cos θ sin θ sin θ + i (sin θ cos θ + cos θ sin θ )] = r r [cos(θ + θ ) + i sin(θ + θ )]. z = r [cos(θ θ ) + i sin(θ θ )] z z r = r = z r z, arg z ( z z ) = θ θ. (3) () z = z = z z = r [cos(θ) + i sin(θ)] ( (), (3) ) z n = r n [cos(nθ) + i sin(nθ)] (n Z) (4) n n n z: n z (( n z) n = z) n z = z /n (4) z n n z [ ( n z = r /n θ cos n + m ) ( θ n π + i sin n + m )] n π (m Z) (5) m Z n n 3
' Finite tfi! L± (a) n =3 n = z lr3@oggy.o.n ' ' (b), n =3 : n = z ' lr3@gga.n', z = r(cos θ + i sin θ) n z = R(cos Θ + i sin Θ) (4) z = ( n z) n r(cos θ + i sin θ) = R n [cos(nθ) + i sin(nθ)] r = R n, cos θ = cos(nθ), sin θ = sin(nθ). R, Θ R = r /n, nθ = θ + mπ (m Z) z n n z = R(cos Θ + i sin Θ) (5) m π n z = r(cos θ+i sin θ) /n ( n z = r /n ) /n (arg z = θ + mπ) n n n (r = =, θ = arg = ) n = cos ( m n π ) ( ) m + i sin n π (m Z). 3 4 ( ) ( ) ( ) ( ) 3 4 4 =, cos 3 π + i sin 3 π, cos 3 π + i sin 3 π ± 3i =, ( ) ( ) ( ) ( ) ( ) ( ) 4 4 4 4 4 =, cos 4 π + i sin 4 π, cos 4 π + i sin 4 π cos 4 π + i sin 4 π =, i,, i. e iθ = cos θ + i sin θ (6) 4
. n ' ' E i. i E i E n=. 43yd. n=i.gr " ',., O s > Isi i j : 3 4 + ( 3i = cos π 3 + i sin π ) = e iπ/3 3 = e πi = e 4πi /3 =, e πi 4πi 3, e 3 ( ) e z = e x+iy e x e x x n = n! = + x +! x + 3! x3 + 4! x4 + 5! x5 +. x = iθ cos θ, sin θ e iθ = + iθ +! (iθ) + 3! (iθ)3 + 4! (iθ)4 + 5! (iθ)5 + (7) = + iθ! θ 3! iθ3 + 4! θ4 + 5! iθ5 + (8) =! θ + (θ 4! + + i 3! θ3 + ) 5! θ5 + (9) = cos θ + i sin θ. (). z = ( ) 6 + 8i 4 3i z 4, π ( ) ( ) ( ) 6 + 8i (6 + 8i)(4 + 3i) 5i = = = (i) = 4 = 4 (cos π + i sin π). 4 3i (4 3i)(4 + 3i) 5 (x, y) = ( 4, ). 3 3 + i 5
( 3 + i = 3 + i ) = /6 3 cos π 4 + i sin π 4 ( π = [cos /6 + nπ ) ( π + i sin 3 + nπ )] 3 (n Z) 3 /6 θ = π, π + π = 3π, π + 4π = 7π 3 4 3 6
. y = f(x) (x, y R) x y = f(x) w = f(z) (w, z ) z w = f(z) f(z) u(z) v(z) w = f(z) = u(x, y) + iv(x, y) (z = x + iy, u(x, y), v(x, y) R) f(z) u(z), v(z). f(x) f(x) lim f(x + x) = f(x ) x f(x) x = x f(x) f(x + x) f(x ) lim x x f(x) x = x df dx (x ) = f (x ) () x f(x) f(x) = x dx (x + x) x dx = lim x x=x x x + x x + x x = lim x x = lim x x + x = x. f(x) x = x f(x + x) = f(x ) + f (x ) x + f (x ) y = f(x) x = x 7
実関数の場合を参考に 複素関数 f (z) についても連続性と微分を以下のように定義する 複素関数 f (z) について other ' Zzf. at eother a IL lim f (z + Z z) = f (z ) z I が満たされるとき f (z) は z = z で連続であるという at eil ' Zzf I 複素関数a f (z) について 次の極限 Z " f (z + z) f (z ) df lim (z ) = f (z ) z z dz. ' " () ' が存在するとき f (z) は z = z で微分可能という I 実数関数の微分と一見同じ形をしているが 今回は複素平面上で z をどの方向からゼロに I 近づけても同じ値に収束することが必要になる L L Itri 上記の点だけ注意すれば 計算自体は実関数と同様に計算できる i i T E 例) f (z) n= = z の (複素) 微分は ^ n ^,r on=i two / n dz dz E L. i i T. z + zi z +, z z (z + z) z, = lim = lim z + z = z. = lim / z z z I z z,r on=i two z=z Itri n= L.,,. = ; 微分可能な関数 f>(z) は z = z で近傍で次のように近似できる Isi = ; > f (z + z) =Isif (z ) + f (z ) z + ) ) ) ox<. l i. i. ) j j (3) he ^Y=fH ^y=fh ^Y=fH ^y=fh T.IE#it:YnK..a.fzo T.IE#it:YnK..a.fzo goofy ox< goofy. l i. :$ i.:$ i.i. > io io '! ' ', > ; '! ' ',, >,, x > x,,, ( ;, ; ; i i,. xotoxxotox ) ( ox ( >ox )> ( o o ) (a) 実関数の微分 > >x. Xotox Xotox x ^ ^ he Zotoz oz oz x x Zotoz zotoz zotoz Toto Toto > > ) (b) 複素平面上の極限 図 3: (a): x = x における実関数の微分は 式 () の極限 ( x ) を取ることで得られる x の 正の側 ( x > ) と負の側 ( x < ) から近づく 通りの極限の取り方がある (b): 複素微分の定 義 () の極限 z は 複素平面上の様々な方向から取ることができる その全てについて式 () の左辺が同じ値に収束するとき 関数 f (z) は z = z で微分可能となる 8
.3 x x x x z ( z ) z () f(z) f(z) 4: z z = z () f(z) = u(x, y) + iv(x, y) (z = x + iy, x, y, u, v R) (). [ z = x, x ( x R)]: z () z = x + iy f(z + z) f(z ) lim z z [u(x + x, y ) + iv(x + x, y )] [u(x, y ) + iv(x, y )] = lim x [ x u(x + x, y ) u(x, y ) = lim + i v(x ] + x, y ) v(x, y ) x x x = u x (x, y ) + i v x (x, y ). (4). [ z = i y, y ( y R)]: z f(z + z) f(z ) lim z z [u(x, y + y) + iv(x, y + y)] [u(x, y ) + iv(x, y )] = lim y i y [ u(x, y + y) u(x, y ) = lim + i v(x ], y + y) v(x, y ) y i y i y = i u y (x, y ) + v y (x, y ). (5) 9
df/dz(z ) (4) (5) u x (x, y ) = v y (x, y ), u y (x, y ) = v x (x, y ) (6) (6) () z = x + i y x, y f(z) f(z + z) = u(x + x, y + y) + iv(x + x, y + y) (7) u(x, y ) + u x (x, y ) x + u [ y (x, y ) y + i v(x, y ) + v x (x, y ) x + v ] y (x, y ) y ( ) ( ) u u = u(x, y ) + iv(x, y ) + x + i v x + x y + i v y. (8) y u/ y v/ x v/ x u/ x ( ) ( u f(z + z) = u(x, y ) + iv(x, y ) + x + i v x + v ) x x + i u y x ( ) u = u(x, y ) + iv(x, y ) + x + i v ( x + i y) x = f(z ) + () ( u x + i v x f(z + z) f(z ) lim z z ) z. (9) = u x (x, y ) + i v x (x, y ) () z ( y/ x). f(z) z = z. z = z f(z) = u(x, y) + iv(x, y) (6) f(z) f(z) f(z) f = u + iy u x + u y =, v x + v =. () y
f u(x, y) (6) u x = x u x = v x y = y v x = u y y u x + u y = 4 (6) 3 v(x, y) f(z) f(z) z z z x = (z + z)/ y = (z z)/i x z =, x z = i = i f = u + iv z f z u(x, y) v(x, y) = + i z z = u x x z + u ( y v y z + i x x z + v ) y y z = u x + i u y + i v x + i i v y = ( u x v ) + i ( u y y + v ) =. x f f/ z = f z
....... '. 3 ( ) f(x) x z f(z) 3. : z n m f(z) = c n z n = c + c z + c z + + c m z m ityotoy ) ^ f (z) = i Yo zoi ) oy o Xo ^ ^ i zoi Yo, i i o f(z) = az + b (a, b ): > ( o W=iz '. > LZ W=iz iq.fi ^ w^=izti.... i. f: > >. > LZ. w=z 4 A Y TIM, > (a) f(z) = iz +,, # 4. a ' ' > 4 ityotoy ) ^ ^ i o > ( o m c n nz n oy o, d LZ iq.fi d f(z) = z : z = re iθ f(z) = r e θ ^ a. r... θ i. Xo Xotosc f: 5: ^ i. > Xotosc w^=izti > >. >. LZ w=z Y TIM, > z = r(cos θ + i sin θ) (4) ( θ f(z) = z /n = r [cos /n n + πmi ) ( θ + i sin n n + πmi )] n πmi n n A. 4 #,, 4 (b) f(z) = z (m Z). ' ' > 4
: p(z), q(z) f(z) = p(z) q(z) = c + c z + c z + + c n z n d + d z + d z + + d n z m (q(z) ) f(z) f(z) = az + b cz + d (a, b, c, d ) 3. f(x) = e x (x R). f(z) = u(x, y) + iv(x, y). f(z) f(x) = e x y = u(x, ) = e x, v(x, ) = () 3. u x = v y, u y = v x () u(x, y), v(x, y) f(z) () e x e iθ = cos θ + i sin θ ( ) e z = e x+iy = e x (cos y + i sin y) (3) (3) e z = u(x, y) + iv(x, y) u(x, y) = e x cos y, v(x, y) = e x sin y. u x = ex cos y, u y = ex sin y, v x = ex sin y, v y = ex cos y. (6) 3
e iy = cos y + i sin y. (4) ( e iy = cos y + i sin y = cos y + i sin y =.) y = π/ exp(iπ/) = i, y = π exp(iπ) = e z πi e z+πni = e z (n Z). ( e z+πni = e z e πni = e z (cos(nπ) + i sin(nπ)) = e z = e z.) : e z = 3. z = x + iy e z = 3 e x cos y = 3, e x sin y =. z = log 3 + nπi (n Z) 3.3 sin x, cos x cosh x, sinh x (4) e iy = cos y + i sin y, e iy = cos( y) + i sin( y) = cos y i sin y. (y R) ( ) sin y, cos y cos y = eiy + e iy, sin y = eiy e iy i, tan y = sin y cos y = eiy e iy e iy + e iy. y R z ( ) cos z = eiz + e iz, sin z = eiz e iz i, tan z = eiz e iz. (5) e iz + e iz (4) e iz = cos z + i sin z 4
( ) e cos z + sin iz + e iz ( ) e iz e iz z = + = (eiz + e iz ) (e iz e iz ) i 4 = 4 4 = d dz cos z = d e iz + e iz dz d dz sin z = d e iz e iz dz i = ieiz + ( i)e iz = ieiz ( i)e iz i = eiz e iz = sin z i = eiz + e iz = cos z cos z, sin z cos x, sin x z = iy (y R) ( ) ei iy + e i iy cos(iy) = ei iy e i iy sin(iy) = i = e y + e +y = e y e +y i = cosh y, = i ey e y = i sinh y. (6) z = x + iy ( ) cos(x + iy) = ei(x+iy) + e i(x+iy) = e y + e y sin(x + iy) = ei(x+iy) e i(x+iy) i = e y e y i = eix e y + e ix e y cos x + i e y e y cos x + i e y + e y i = e y (cos x + i sin x) + e y (cos x i sin x) sin x = cos x cosh y i sin x sinh y = eix e y e ix e y i = e y (cos x + i sin x) e y (cos x i sin x) i sin x = sin x cosh y + i cos x sinh y (6) y z ( ) cosh z = ez + e z, sinh z = ez e z. (7) 5
4 ( ) ln z = ln z + i arg z z p = e p ln z w = az + b cz + d 4. z z = x + iy = re iθ = r (cos θ + i sin θ) (r = z = x + y, x, y R) z z θ arg z z : e z = e x+iy = e x e iy = e x (cos y + i sin y) (8) z e iθ = cos θ + i sin θ z = re iθ = r (cos θ + i sin θ) z n (n Z) z z /n z n = r n z n = (re iθ ) n = r n e inθ = r n [cos(nθ) + i sin(nθ)]. (9) [ cos ( θ n + mπ ) ( θ + i sin n n + mπ ) ] (m Z). (3) n z /n n z ((z /n ) n = z) ( ) { [ ( z /n n = r θ n cos n + mπ ) ( θ + i sin n n + mπ ) ]} n n = ( { ) [ ( r n θ n cos n n + mπ )] [ ( θ + i sin n n n + mπ )] } n [ ( ) ( )] = r cos θ + mπ + i sin θ + mπ = r (cos θ + i sin θ) = z. (3) z /n mπ/n sin(θ + mπ) = sin θ, cos(θ + mπ) = cos θ 6
4. y = ln x (x, y R) y = e x y = e x x = ln y ( e ln x = x, ln(e x ) = x) (3) ln x e w = ln w (z, w ) ln z z = e w w = ln z (33) w = ln z = u + iv z z = re iθ z = e w z = e w z ( = re iθ) = e w = e u+iv = e u e iv. (34) ln z = u + iv u v z = re iθ r = e u, e iθ = e iv. (35) (35) (35) r = e u u = ln r, (36) e iθ = e iv v = θ + nπ (n Z). (37) e nπ = (n Z) e iθ = e iv v nπ (36), (37) z = re iθ ( ) ln z ln z = ln ( re iθ) = ln r + i (θ + nπ) (n Z) (38) = ln z + i arg z + nπi (39) ln z z ln z arg z (39) nπi Ln z Ln z = ln z + i arg z ( π < arg z π) (4) ln z (39) arg z π π ln( ) = (n + )πi =..., 3πi, πi, πi, 3πi, 5πi,..., Ln( ) = πi. 7
ir it. ^ LZ ir, ^ L w=lnz r ' minty, it r I. ; io ;,, ft, ; 6: w = Ln z z r z = re iθ w w = ln r + iθ ( π < θ π) (39) ( ) ln(z z ) = ln z + ln z, z = z = : ( z ) ln = ln z ln z. (4) z ln(z ) = ln(z ) = ln( ) = πi + nπi ln(z ) + ln(z ) = πi + nπi, ln(z z ) = ln [( ) ( )] = ln = + nπi. (n Z) nπi d ln z dz = d Ln z dz = z. (4) (39) ln z z Ln z z = x + iy Ln z = ln z + i arg z = ln x + y + i arctan y x u + iv. (43) Ln z u v ( ) y v u x = x x + y, u y = x + y, u x = v y, y x + y, v y = x x + y. (44) x = u y = v x. (45) d Ln z/dz d Ln z/dz d Ln z = Ln z dz x = u x + i v x = x x + y + i y x + y = x iy x + y = z z z = z. (46) 8
4.3 z z n z /n p z = e Ln z z p = (e ln z ) p = e p ln z. (47) z p p ln z z p z p i i z p = e p Ln z (48) i i = e i ln i = e i[( π +nπ)i] = e ( π +nπ). (n Z) (49) i =, arg i = π/ ln i = ( π + nπ) i (n Z) ln i nπ i i Ln z = πi i i = e i Ln i = e i πi = e π (5) 4.4 ( ) w = az + b (a, b, c, d, ad bc ) (5) cz + d ad bc = w = ( ) ad bc : (5) w = z + b : b = x + iy w = az : a = re iθ r θ w = /z : z = w = /z z = re iθ w = z = re iθ = r e iθ (5) w = r = z z arg w = θ = arg z arg z z w 9
' ' z = x + iy z A(x + y ) + Bx + y + D = (A, B,, D R) (53) x = z+ z, y = z z w = /z i Az z + B z + z + z z i + D = A w w + B w + w + w w + D = i w = ˆx + iŷ (ˆx = $ A + B w + w w+ w, ŷ = + w w i + Dw w = (54) w w ) i A + Bˆx ŷ + D (ˆx w + ŷ ) =. (55) ' ' w yr ) z = x + i (x R) Z=teio )O w = z w ˆx + ( ŷ + ) = ) o z = x + i (x R) y = (56) trap (53) A = B =, =, D = w = /z (55) ŷ (ˆx ' ' io + ŷ ) ( e = ˆx + ŷ + ( ) W=zI=r. = (57) ) 4 w i $ ' ' teio O ) ) o Z= yr try # ^ i i x, *i I > ^ o E w=' > W= =r ' e io i (a) z = re iθ (b) z = x + i (x R) ^ # 7:,i w = /z (a) z = re iθ w = r e iθ, ati ^w=' : O. > w + i/ = / i/ / > x r θ (b) z = x + i (x R) Ẹ i
: (5) w = az + b cz + d = a ( ) z + d c ad + b c cz + d + b cz + d + a c = ad c K cz + d + a c.. f (z) = cz + d (c d ). f (z) = z ( ) 3. f 3 (z) = Kz + a c (K a/c ) (58) ( ) ad bc K c (58) w = f 3 (f (f (z))) f 3 f f (z) (59) (58) f,,3 (z) z : (5) 4 a, b, c, d a, b, c, d αa, αb, αc, αd (α ) w w = az + b cz + d = αaz + αb αcz + αd 3 3 z 3 w z =,, w =, 5, 8 (5) z =,, w =, 5, 8 w = az + b = b cz + d d, 5 = a + b c + d, 8 = a + b (6) c + d a = 3d, b = d, c =. (6) w : w = 3dz + d z + d (6) = 3z +. (63) (5) z = d/c w z = d/c w w = w = /z z z = w = z z = w z =
5 ( ) f(z)dz = (f(z): ) 5. a x b f(x) n b a a = x < x < < x n < x n = b f(x) lim n n f(ˆx i ) x i ( x i x i x i ) (64) i= x i n ˆx n x n ˆx n x n f(x) n f(x) [a, b] z(t) t z(t) = x(t) + iy(t) (a t b) t [a, b] n a = t < t < < t n < t n = b. f(z) n f(z)dz = lim f(ẑ i ) z i (z i z(t i ), z i z i z i ) (65) n i= z i n f(ẑ i ) i ẑ i = z(ˆt i ), t i ˆt i t i 5. : [k f (z) + k f (z)] dz = k f (z)dz + k f (z)dz : z f(z)dz = z z z f(z)dz :, f(z)dz = f(z)dz + f(z)dz
= fix ) Meta AY > x Erie? ' 8: z i f(ẑ i ) 5.3 t z = z(t) dz = dz(t) dt dt t z z = z = + i t z(t) = t + it ( t ) dz dt dz = dz d(t + it) dt = dt = ( + i)dt. (66) dt dt [ ] z dz = (t + it) ( + i)dt == ( + i) 3 t dt = ( + i) 3 t3 = 3 + i. (67) 3 z z = z z z = + i : z(t) = t ( t ), : z(t) = + it ( t ) : dz = dz(t) dt = dt, dz z dz = z dz + z dz = [ ] [ = 3 t + i t + i t 3 t3 t dt + ] : dz = dz(t) dt = i dt dz ( + it) idt = ( 3 + i + i ) 3 (67) = 3 + i. (68) 3 3
/z θ z(θ) = e iθ = cos θ + i sin θ ( θ π) (69) dz dz = dz(θ) deiθ dθ = dθ dθ dθ = ieiθ dθ (7) π z dz = e iθ ieiθ dθ = (z z ) n (n Z) z z = ρ θ (z z ) m dz = π idθ = i [θ] π = πi. (7) z(θ) = z + ρe iθ ( θ π) (7) dz = dz(θ) dθ dθ = d ( z + ρe iθ) dθ = iρe iθ dθ (73) dθ π ( ) π ρe iθ n iρe iθ dθ = iρ n+ e i(n+)θ dθ. (74) n π π [ ( ) ( )] iρ n+ e i(n+)θ dθ = iρ n+ cos (n + )θ + i sin (n + )θ dθ = iρn+ n + n = n + = i π (z z ) m dz = [ sin ( (n + )θ ) i cos ( (n + )θ )] π =. (75) dθ = i [θ] π = πi (76) { πi (n = ) (n ). (77) 5.4 D f(z) D f(z)dz = (78) 4
" " life.. o, >! idea go.ee 9: (67), (68), (4), (77) D D D f(z) = u(x, y) + iv(x, y), dz = dx + idy f(z)dz = (u + iv)(dx + idy) = ( ) ( Vy x V ) x dxdy = y (udx vdy) = j (udx vdy) + i (udy + vdx). (79) (V x dx + V y dy) (8) ( v x u ) dxdy. (8) y D f(z) u x = v y, u y = v x (8) (8) 5.4., = f(z)dz = f(z)dz f(z)dz f(z)dz = f(z)dz. (83) 5
, f(z) f(z) D D f(z) = /z (/z z = ) " " ; Hiatt : 5.4. ( ) f(z)dz + f(z)dz + = (84),... f(z) = /z " 4*9%4 : D, 6
5.4.3 f(z) = z z = z = + i z f(z) = z z (/3)z 3 +i z dz = [ 3 z3 ] +i = ( + i)3 3 = 3 + i. (85) 3 f(z) = cos z z = z = i cos z i cos zdz = [sin z] i = sin(i) sin = i sinh (86) sin = sin(i) = ei i e i i i = e e i = e e i = i sinh (87) 7
6 ( ) f(z)dz = (f(z): ) z z dz = πi f(z ) = πi f (n) (z ) = n! πi f(z) dz z z f(z) (z z ) n+ 6. a b (b a) b a f(z)dz = a b f(z)dz. a b c : b a f(z)dz = c a f(z)dz + b c f(z)dz. f(z) f(z) f(z)dz = [f(z) : ] a b A, B A B f(z) A f(z) B f(z) 8
( ) a A b B a f(z) b a b b = f(z)dz = f(z)dz + f(z)dz = f(z)dz f(z)dz. a A b a A b B f(z)dz = b a B a A f(z)dz a B, f(z), f(z) ( ) (),.. 3. 4. f(z) () = f(z)dz = f(z)dz + f(z)dz + f(z)dz + f(z)dz. (88) 3 4 3 f(z)dz = f(z)dz, f(z)dz = f(z)dz = f(z)dz. (89), 3 4 f(z)dz = f(z)dz. (9) 4,, 9
(88) = f(z)dz + f(z)dz + f(z)dz + f(z)dz 3 4 = f(z)dz + f(z)dz f(z)dz f(z)dz,, = f(z)dz f(z)dz,,, f(z)dz =, f(z)dz. f(z) " " im#i*,,, : f(z) 6. z = z f(z ) D f(z) f(z) πif(z ) = dz. (9) z z z D z D z = z /(z z ) πi : dz = πi. z z f(z) f(z) dz = πi f(z ) z z 3
z = z f(z ) ( ) (9) f(z) = f(z ) + (f(z) f(z )) z f(z ) f(z) f(z ) f(z) f(z ) f(z) f(z ) dz = dz + dz (9) z z z z z z z = z z = z ρ z = z + ρe iθ ( θ π) (9) πif(z ) z = z + ρe iθ ( θ π) f(z ) π dz = f(z ) dz = f(z ) z z z z ρe iθ iρeiθ dθ = f(z ) i θ z(θ) = z + ρe iθ dz = dz dθ dθ = iρeiθ dθ π dθ = πif(z ). f(z) f(z ) z z ρ f(z) f(z ) z z ϵ/ρ f(z) f(z ) z z dz < f(z) f(z ) z z dz < ϵ ρ πρ = πϵ ϵ. ab < a b (a, b ) f(z) f(z ) dz = z z (93) z z z = z f(z) z z z = z f(z) f(z) z z z = z + ρe iθ ( θ π) f(z) f(z ) f(z) ϵ > δ > z z < δ f(z) f(z ) < ϵ ρ < δ f(z ) f(z) z z < ϵ ρ z = z + ρe iθ ( θ π) 3
6.. (9) (78) f(z) dz (f(z): ) z z z = z f(z) dz = πif(z ). z z z = z f(z) z z ( f(z) f(z)dz = ) f(z) z z dz =. [ ] z + z (a) z = (b) z = (c) z = (d) z = i ^ (d) LE ^ i (a) i > ' or 7 ^ ( c ) ( b ) 3: (a) z =, (b) z =, (c) z =, (d) z = i [ ] z + z = z + (z + )(z ) z =, z = 3
(a), (b) z = z = z z z + + z dz = z+ z dz = πi z + z + = πi + z= + = πi. (a), (b) (a), (b) (c) z = z + z + z dz = z + z + z dz = πi z + z = πi ( ) + z= = πi. (c) (c) (d) z = ± z + z z + dz =. z (d) 6.3 z = z f(z ) dn f dz n (z ) f (n) (z ) D f(z) D f(z) D z = z f (n) (z ) f (n) (z ) = n! πi f(z) dz (n =,,,...) (94) (z z ) n+ z D z D f(z) ( f (z ) ) f(z) f(x) 33
z = z f(z) f(z) = n! f (n) (z )(z z ) n = f(z )+f (z )(z z )+ f (z )(z z ) + 3! f (z )(z z ) 3 + { (z z ) dz = πi (n = ) n (n ) (94) f(z) dz (z z ) (97) n+ f(z ) + f (z )(z z ) + = f (z )(z z ) + f (z 3! )(z z ) 3 + dz (z z ) n+ [ f(z ) = dz (z z ) + f(z )(z z ) + f (z )(z z ) + + ] f (n) (z )(z z ) n + n+ (z z ) n+ (z z ) n+ n! (z z ) n+ [ f(z ) = dz (z z ) + f(z ) n+ (z z ) + f (z ) n (z z ) + + ] f (n) (z ) + n n! z z = πi n! f (n) (z ). (98) (96) πi (n ) z z (z z ) n (94) (95) [ (94) ] n = f (z ) = πi (95) (96) f(z) z z dz (99) f (z ) [ f (z ) = lim z z [f(z f(z) + z) f(z )] = lim z πi z z (z + z) f(z) ] z z [ f(z) z z (z + z) f(z) ] = z z [ ] f(z) (z z ) + z f(z) dz. (z z z)(z z ) z (94) z d z z z z (z + z) d 34
z f(z) (z z z)(z z ) dz z < f(z) L (z z z)(z z ) dz < z d max f(z). 3 () L = dz max f(z) f(z) f(z) z () (94) 3.6 [ ] cos(πz) z = (z ) 3 [ ] z = cos(πz) z = (z ) 3 (94) n =, z = cos(πz) πi dz = [cos(πz)] = πi (z ) 3! z= [ π cos(πz) ] z= = π 3 i cos(π) = π 3 i. z = 35
7 [ 3. 3. ] z = z f(z) = a n (z z ) n = a + a (z z ) + a (z z ) + () ( ) : f(z) = n! f (n) (z )(z z ) n = f(z ) + f (z )(z z ) + f (z )(z z ) + 7. () f(z) a n z z z z f(z) z f(z) ( z = ) f N (z) = N z n = + z + z + + z N = zn+ z () f N (z) N z N+ N z = r = e iθ z N z N = (re iθ ) N = r N e inθ = r N N ( z = r < ) ( z = r = ). ( z = r > ) e inθ = () 3 N f N (z) = z n = ( z < ) zn+ z N ( ) ( z = ) z ( z = r > ) (3) z < z 3 z = r = () f N (z) z=e iθ = ei(n+)θ [cos(nθ) + i sin(nθ)] e iθ = e iθ N f N (z) 36
e z e z = n! zn = + z +! z + 3! z3 + (4) z 7. 3. z, z,... = {z n } n c ϵ > N > z n c < ϵ ( n > N) (5) lim z n = c n lim n z n = n z n z n R z n n= z n = z + z + {z n } f N (z): f N (z) = N z n = z + z + + z N n= N n= z n = z + z + z n = z + z + n= n= z n 37
( ) 3 n= z n n= b n z n b n (n =,,...) (6) n= z n n= b n 4 z n n= z n lim n z n+ z n = L (a) L < (b) L = (c) L > z lim n+ n z n = L n z n + z n+ + z n+ + = z n ( + L + L + ) L < zn L > L L = 5 4 [ ] n= z n n= z n n= z n < n= b n < N n= z n N N 5 (b) L = () z n = n, () z n = n (7) n= z z n () L = lim n+ n+ n z n = lim n n =, () L = lim n (n+) n = () () z n = n= n= n= z n = n < + n= n = + + 3 + > x dx = + x dx = [ln x] =, [ x ] = + = () () π /6) 38
n= z n lim n n z n = L (a) L < (b) L = (c) L > lim n n z n = L z n L n ( ) L < L > 6,7 7.3 7. ( ) () a n (z z ) n (a n ) z = z n= z z < z z () z = z z z > z z z = z z = z z z z = z z z z = z [ ] z n= a n(z z ) n 7. () z () 3. () () 6 (7) (), () L = lim n n z n = 7 z lim n+, n n z limn n z n zn+ z n 3., n z n q < 39
^ ^ Zl zol Zz Zo Zo/ Zz \ P " > > 4: n= a n(z z ) n z = z z = z z z z ( ) z z ( ) ( ) R z z > R a n n ( n= a n(z z ) n lim a n+ n a n = L R R = L = lim a n (8) L n a n+ = R = lim n a n+ a n (L = ) R = z = z [ ] 7. n= a n(z z ) n a n+ (z z ) n+ = a n+ a n (z z ) n a n z z n L z z L L L < z z < /L L > z z > /L 4
: lim n n a n = L R = /L a n+ /a n 8 R = lim n n= (n)! (n!) (z 3i)n R /4 (n)! (n!) [(n + )]! [(n + )!] n= R = lim n (n)! [(n + )!] = lim n [(n + )]! (n!) n n! (z 3)n n n! (n+) (n + )! n (n + )! = lim n (n+) n! = lim n (n + )(n + ) (n + ) = 4. = lim n n =. 8 a n /a n+ ( ) 4
8 [ 3.3, 3.4 ] a nc n f(z) z = z f(z) = n! f (n) (z )(z z ) n = f(z ) + f (z )(z z ) + f (z )(z z ) + n f(z) z = z f(z ) f (n) (z ) 8. (z = ) z z z : a nz n, b nz n z < R a n z n = b n z n a n b n a n = b n (n =,,,...) z = f(z) = a nz n, g(z) = b nz n z < R f(z) ± g(z) = (a n ± b z )z n ( ) ( ) f(z)g(z) = a n z n b m z m = a b + (a b + a b ) z + (a b + a b + a b ) z + = m= ( n ) a m b n m z n. (9) m= 4
f(z) = a nz n 9 ( f (z) = d ) a n z n d = dz dz (a nz n ) = n a n z n () n a nz n (8) R R = lim a n n a n+ () R R = lim n a n n (n + )a n+ = lim n n n + lim a n n a n+ = lim n R a n a n+ = R ( ) ( ) f(z)dz = a n z n dz = a n z n dz = n + a nz n+ f(z) 8. f(z) f(z) z z < R f(z) f(z) = n! f (n) (z ) (z z ) n () f (n) (z ) (94) n! f (n) (z ) = f(z ) πi (z z ) n+ dz () f(z) z D 9 a nz n () [ f(z + z) f(z) (z + z) lim f n (z) = lim z z a z n ] n n a n z n = z z 3.3 43
[ () ] (9): f(z) = πi f(z ) z z dz (3) z z z = z q = + q + q + (4) z z z z = z z (z z ) = z z [ = z z + z z z z + z z z z ( ) ] z z + = z z (3) f(z) = dz f(z ) πi (z z ) (z z ) n = n+ = ( πi (z z ) n+ (z z ) n. ) f(z ) (z dz (z z z ) n+ ) n n! f (n) (z ) (z z ) n. (94) (), () f(z) f(z) z = z z = z z z f(z) f(z) (z = ) f(z) z = z ( 3.4 ) 44
^ * 6 T R Z *, Zo * > 4..5.5..5 4 x + x + x + x (a) (b) 5: (a) z = z f(z) z = z f(z) R (b) z = z = x+i z 8.3 f(z) (), () 8. ( ) z = z = z n = + z + z + z 3 + (5) z z = z = c z = c z = [ + z ( z ) ] c c + + c c z = + z + z 4 + z 6 + z z z = e z = n! zn = + z + z + 3 z3 + 45
e z ( ) z = (8.3) z iz (4) e iz = cos z + i sin z (6) sin z = (n + )! zn+ = z 3! z3 + 5! z5 (7) cos z = (n)! zn =! z + 4! z4 (8) cosh z = ez + e z, sinh z = ez e z (8.3) Ln ( + z) z = ( ) n Ln ( + z) = z n = z z n + z3 3. (9) n= Ln( + z) = Ln( + z) + i arg z Ln( + z) z = 8... ) ( z) z = : ( z) z ( z) = d dz z ( z) z (5) ( z) = d dz + z = d ( + z + z + z 3 + ) dz = + z + 3z + 4z 3 +. 46
( z) = ( z ) (5) (9) z = ( z) z z = Ln( + z) z = Ln( + z) +z Ln( + z) = z + z dz (5) z z Ln(+z) = z + z dz = z dz ( z + z z 3 + ) = z z + 3 z3 4 z4 + Ln( + z) (9) z = z = +z Ln( + z) z = ( z) n (n Z) z = ( z) n z ( z) n = (n )! d n dz n z (5) ( z) n = (n )! = = (n )! (n )! d n dz n z = d n (n )! dz n z m = m= m(m ) [m (n ) + ]z m (n ) m=n m=n m! [m (n )]! zm n+ = m= (n )! m= ( m + n )! z m = (n )! m! d n zm dzn n m z m. m= 47
9 [ 3.4 4. ] f(z) z = z f(z) = a n (z z ) n b n + (z z ) = + b n (z z ) + b +a +a (z z )+a (z z ) + z z n= z = z (z z ) n z = z z = z f(z) 9. f(z) z = z (): f(z) = n! f (n) (z ) (z z ) n = f(z ) + f (z )(z z ) +! f (z )(z z ) + 3! f (z )(z z ) 3 + () ( ) z = z z = Ln z z = i Ln z = Ln z + (Ln z) (z z ) + z=z! (Ln z=z z) (z z ) + 3! (Ln z=z z) (z z ) 3 + = Ln i (z i) + z=i (z i) z=i (z i) 3 + z z=i! z 3! z 3 = Ln i i (z i) +! i (z i) 3 i (z 3 i)3 + = πi + i(z i) (z i) i 3 (z i)3 + f(z) f(z) z = z Ln z z = z=z z = i z = R = i = 48
z = z = z = z n = + z + z + z 3 + ( R = ) e z = n! zn = + z + z + 3! z3 + ( R = ) ( ) n cos z = (n)! zn = z + 4! z4 ( R = ) ( ) n sin z = (n + )! zn+ = z 3! z3 + 5! z5 ( R = ) ( ) n Ln( + z) = z n = z n z + 3 z3 4 z4 ( R = ) n= z = z z = R = Ln( + z) z = R = = e z, cos z, sin z R = cosh z, sinh z cos z, sin z + cosh z = + z + 4! z4 +, sinh z = z + 3! z3 + 5! z5 + (R = ) f(z) z w f(z) z w ) + iz = = + z + z + = iz 4 z 8 + 4 z z iz 4 z iz 4 ( π ) sin z = z 3! z3 + 5! z5 = π z π z z ( π ) 3 6 z ( π ) 5 + z + e iz = cos z + i sin z cos z = eiz + e iz, sin z = eiz e iz i e z z ±iz a n R = lim n a n a n+ 49
f(z) + g(z) f(z)g(z) (9) 3 z + z = + z + z + + ( + z + (z) + ) = + 3z + 5z + z sin(z) = z (z 3! (z)3 + ) 5! (z)5 = z 3 43! z5 + 5 5! z7 f (z) f(z)dz z (+z) = : ( + z) = d ( ) = d ( z + z z 3 + z 4 ) = z + 3z 4z 3 + (R = ) dz + z dz z+3 z = : [ z + 3 = 3 ( ) = + z ( ) ] z z 3 3 + + = 3 3 3 ( 3 z + 4 ) 9 z + = 3 9 z + 4 7 z3 + 3 e z cosh z = ez + e z = [ + z + z + 3! z3 + ( 4! z4 + + + ( z) + ( z) + 3! ( z)3 + )] 4! ( z)4 + = [ + z + z + 3! z3 + ( 4! z4 + + z + z 3! z3 + )] 4! z4 + = + z + 4! z4 + (R = ) sinh z = ez e z = [ + z + z + 3! z3 + ( 4! z4 + z + z 3! z3 + )] 4! z4 + = z + 3! z3 + 5! z5 + (R = ) 5
3+z z = 3 z = z = 3 R = 3 9. 3z + z 3z + = = z + z = + z + z + + [ + z + (z) + ] = + 3z + 5z + z = z 9.. f(z) z = z, f(z) f(z) = a n (z z ) n + = + n= b n (z z ) n b (z z ) + b z z + a + a (z z ) + a (z z ) + () f(z) a n = f(z ) πi (z z ) n+ dz (n =,,...), b n = (z z ) n f(z )dz (n =,,...) πi () cos z z z = ( cos z z ) z = z z cos z = z ( + z 4! z4 ) = z z + 3. (3) z cos z z 5
f(z) f(z) f(z) = f(z ) πi z z dz f(z ) πi z z dz, z z z = z (), () () a n, b n { :z (z z ) dz = πi (n = ) (4) n (n ) () f(z) f(z ) πi (z z dz (n =,,...) ) n+ f(z ) πi (z z ) n+ dz = [ ] dz a πi (z z ) n+ m (z z ) m b m + (z z m= m ) m = = [ ] dz a m πi (z z ) + b m. (5) m+n+ (z z ) m +n+ m= (4) ( z z ) m + n + = m = n m + n + n + > (5) πi f(z ) (z z ) n+ dz = πi dz [ m=, m = a m (z z ) + m+n+ m = b m (z z ) m +n+ a n () f(z) πi (z z ) n f(z )dz (n =,,...) (z z ) n f(z )dz = [ ] dz (z z ) n a m (z z ) m b m + πi πi (z z m= m ) m = = [ ] dz a m πi (z z ) + b m. m n+ (z z ) m n+ m= m = ] = a n (6) m n + m n + m = n (6) (z z ) n f(z )dz = [ ] dz a m πi πi (z z ) + b m = b m n+ (z z ) m n+ n m= b n m = (7) 5
z = z (6) ^ * * ^ i, [ z * Zo Zo * X X ) ) 6: z = z z = z z = z z = z z = z 9.. (3) (z z ) n ( ) z = z (z z ) n z( z) z = z z z z = z( z) = z z = z ( + z + z + z 3 + ) = z + + z + z + z < z = z( z) z = ( ) z z z z = 4 z( z) = z = z = z = z ( z [ ) ( ) + ) + z z= z z=(z [ z= (z ) + z= (z ) z z= z z 3 3! [ (z ) + (z ) (z ) 3 + ] = z + (z ) + (z ) + ( ) ) z z=(z 3! ( ) z z= 3 z 4 z= (z ) + z < z = ] (z ) + ] 4 z z = z = + z + z + z = [ (z )] = + [ (z )] + [ (z )] + [ (z )] 3 + = (z ) + (z ) (z ) 3 +. 53
[ 4. 4.3 ] f(z) z = z z z Res f(z) z=z : f(z)dz = πi Res f(z) z=z. f(z) z = z f(z) = a n (z z ) n b n + (z z ) = + b n (z z ) + b +a +a (z z )+a (z z ) + z z n= z = z f(z) z = z f(z) 3 5. f(z) z = z z = z f(z) f(z) = a n (z z ) n b n + (z z ) = + b n (z z ) + b +a +a (z z )+a (z z ) + z z. (m ) n= f(z) z = z z = z f(z) f(z) = a n (z z ) n + m n= b n (z z ) n = b m (z z ) m + + b z z + a + a (z z ) + (b m ) (z z ) m z = z m 5 f(z) f(z) 54
3. z sin z z = z = sin z z = z z sin z = ( z z 3! z3 + ) 5! z5 + = 3! z + 5! z4 + z = sin z z = z z z = z f(z ) f(z) = a n (z z ) n = a + a (z z ) + a (z z ) + f(z) z = z z(z ) + 3 z = z = 5 z = 5 (z ) z z(z ) + 3 5 (z ) = 3z + 64 +, z = (z ) z(z ) + 3 5 (z ) = (z ) 5 4(z ) + 4 e z z = ( e z = n! z ) n = + z + ( ) + z 3! ( ) 3 + z z (z = x x ) e /z = e /x, (z = x x ) e /z = e /x e /z z = 6 6 55
.. f(z) z = z z = z f(z ) = z = z f(z) f(z) = a n (z z ) n = a m (z z ) m + a m+ (z z ) m+ + n=m m z = z f(z) m.. z z w w = z z w z w = w = z = {z } z = ( ) f(z) = z z w f(z) = = w w = z f(z) = z = z f(z) = z f(z) = /w z = e z z w f(z) = e/w w = e z z = 7 z =. f(z) z = z m f(z) z = z f(z) z = z f(z) = :z b m (z z ) m + + b z z + a + a (z z ) + (z z ) n { (z z ) dz = πi (n = ) n (n ) 56
. l = toit. Z* of 7: {z } z z z z = f(z) [ b m f(z)dz = dz (z z ) + + b ] + a m + a (z z ) + = πi b (8) z z z z b z z f(z) z = z Res f(z) = b, z=z (8) 7.. f(z) z z Res f(z) = b z=z b f(z) z = z f(z) = b z z + a + a (z z ) + b (z z ) z z [ ] b lim (z z )f(z) = lim (z z ) + a + a (z z ) + z z z z z z [ = lim b + a (z z ) + a (z z ) + ] z z = b. 7 (8) 57
m (m > ) Res f(z) = b = lim (z z )f(z). (9) z=z z z f(z) z = z m m > f(z) = b m (z z ) m + + b z z + a + a (z z ) + (b m ) (9) (z z ) n> b (z z ) m m z z lim z z = lim z z = lim d m dz (z z ) m f(z) m [ d m dz (z z ) m m b m (z z ) m + + b z z + a + a (z z ) + d m [ bm + b z z dz m m (z z ) + + b (z z ) m + a (z z ) m + a (z z ) m+ + ] = lim z z = (m )! b. [ (m )! b + m! a (z z ) + (m + )! a (z z ) + m ] ] (3) m Res z=z f(z) = b = (m )! lim z z d m dz m (z z ) m f(z) (3) f(z) z = z (8) ( ) f(z) = πi Res z=z f(z) (3).. f(z) = 5z z = 4 z = (z + 4)(z ) 58
z = 4 (9) 5z 5 ( 4) Res f(z) = lim (z + 4)f(z) = lim = = 8. (33) z= 4 z 4 z 4 (z ) ( 4 ) z = 4 f(z) (3) f(z)dz = πi Res f(z) = 6πi. (34) z= 4 f(z) = Res f(z) = lim z= z 5z z = (3) m = (z + 4)(z ) d dz (z z ) f(z) = lim z d dz 5z z + 4 = lim z 5 z + 4 5z (z + 4) = 5 5 5 5 = 8. z = f(z) f(z)dz = πi Res f(z) = 6πi. (35) z= f(z) = cos z z = z z = cos z z cos z ( ) z (9) Res z= f(z) = lim z f(z) = lim z z z z cos z = lim cos z =. (36) z f(z) (3) f(z)dz = πi Res f(z) = πi. (37) z= 59
[ 4.3 4.4 ]... f(z) z = z f(z) z = z f(z) = + b (z z ) + b z z + a + a (z z ) + a (z z ) + z z b f(z) z = z Res z=z f(z) = b z = z f(z) f(z)dz = dz [ + b (z z ) + b z z + a + a (z z ) + a (z z ) + { (z z ) dz = πi (n = ) n (n ) : f(z) f(z)dz = ] = πib = πi Res z=z f(z). 8 = f(z)dz + f(z)dz = f(z)dz f(z)dz,,,, f(z)dz = f(z)dz,, z = z ϵ z = ϵe iθ + z θ z = z, z,... 8 = f(z)dz + f(z)dz + f(z)dz + = f(z)dz f(z)dz f(z)dz,,,,,, f(z)dz = f(z)dz + f(z)dz +,, ϵ,,... n πi Res z=zn f(z) ( ) f(z)dz = f(z)dz+ f(z)dz+ = πi Res f(z) + Res f(z) + Res f(z) z=z z=z z=z n,,,, = πi n 6
f(z) = πi n Res f(z) z=z n z = z n f(z) " Dixey ; Yippy 8: z z,,3,,3.. 3 f(z) = b z z + a + a (z z ) + z = z Res f(z) = lim (z z )f(z). z=z z z f(z) = p(z) ( p(z), q(z) :, q(z ) =, p(z ) ) q(z) z = z f(z) q(z) z = z q(z) = q(z ) + q (z )(z z ) + q (z )(z z ) + = q (z )(z z ) + q (z )(z z ) + 6
f(z) Res f(z) = lim (z z ) p(z) z=z z z q(z) = lim p(z) (z z ) z z q (z )(z z ) + q (z )(z z ) + p(z) = lim z z q (z ) + q (z )(z z ) + = p(z ) q (z ). Res f(z) = p(z ) z=z q (z ). (38) m b m f(z) = (z z ) + + b m (z z ) + b + a + a (z z ) + (b m ) z z Res z=z f(z) = b = (m )! lim z z d m dz m [(z z ) m f(z)]. (z z b ),...,m z z b f(z) = + (z z ) + b + a + a (z z ) + a (z z ) + z z f(z) z z b f(z) = e /z z = z ( ) n e /z = = + n! z z + z + Res z= e/z =...3. 4 3z dz z = z = z z 4 3z Res z= z z = lim z 4 3z z z z = 4 3z z = 4, z= 4 3z Res z= z z = lim 4 3z (z ) z z z = 4 3z z =. z= 6
. (a) z =, 4 3z dz = πi z z ( 4 3z Res z= z z + Res z= ) 4 3z = πi ( 4 + ) = 6πi. z z (b) z = z = 4 3z 4 3z dz = πi Res = πi ( 4) = 8πi. z z z= z z. ( z =, ) ze πz z 4 6 ze πz z 4 6 = ze πz (z )(z + )(z + i)(z i) z = ±, ±i z = ±i (38) Res z=i Res z= i ze πz z 4 6 = lim z i ze πz z 4 6 = lim z i ze πz ze πz = lim (z 4 6) z i 4z 3 ze πz (z 4 6) = lim z i ze πz 4z 3 = ieπi 4(i) 3 = 6, (39) = ie πi 4( i) 3 = 6 (4) ze πz z 4 6 = πi ( ze πz Res z 4 6 = πi 6 ) = πi 6 4. z=±i ^ K Llc %f.me a rz. o i Le rc Ft ; x ) = ; 9:..3..3... 63
.. cos θ, sin θ F (cos θ, sin θ) θ π I = π F (cos θ, sin θ)dθ (4) z = e iθ θ π dθ cos θ = eiθ + e iθ dz = dz(θ) dθ = z + z, sin θ = eiθ e iθ i (4) I = = z z, i deiθ dθ = dθ dθ = ieiθ dθ = izdθ dθ = dz iz ( z + z F, z ) z dz i iz (4) π dθ cos θ π dθ = cos θ z = = i z+z z = dz iz = i z = z z + dz { } { }. (43) z ( + ) z ( ) ( )= z z + = z = ± ( ) 4 ± z { z + = z ( } { + ) z ( } ) (43) z = ± z = z = (43) i z = dz { } { } = πi i Res z ( + ) z ( ) z= { } { } z ( + ) z ( ) = 4π lim z z ( + ) = 4π = π. ( + ) dz 64
. 8 f(x)dx (44) = {x R x R} = {z z = Re iθ ( < θ < π)} = + f(z)dz = πi Res f(z). z=z n n R f(z)dz = f(z)dz + f(z)dz = f(x)dx + f(z)dz R R πi Res f(z) = f(z)dz = z=z n n f(x)dx + lim f(z)dz. R (44) lim R f(x)dx = f(z)dz = πi Res f(z) (45) z=z n n R ( ) R dx + x 4 8 dx + x 4 = dx + x 4. (46) 65
=d, + z ^ = # _ir a z t di R t i > r > R : dz + z = R dx 4 + x + 4 R dz + z 4 (47) R lim R R R dx + x 4 = dx + x 4 (i) (47) (ii) (47) lim R (46) (i) (47) +z 4 + z = 4 (z e iπ 4 )(z e 3iπ 4 )(z e 5iπ 4 )(z e 7iπ 4 ) z = e iπ 4, e 3iπ 4, e 5iπ 4, e 7iπ 4 9 z = e iπ 4, e 3iπ 4 ( ) (38) Res f(z) = z=e iπ 4 (z 4 ) = iπ z=e 4 4z 3 = 3πi e 4 = iπ z=e 4 4 4 e πi 4, Res f(z) = z=e 3πi 4 4z 3 = 9πi e 4 = πi e 4 iπ. z=e 4 4 4 dz + z = 4 πi (R ) ( Res z=e iπ 4 = πi i sin ( π 4 f(z) + Res ) = π z=e 3iπ 4 f(z) = ) π. ( = πi 4 e πi 4 + 4 e 9 + z 4 z = e iπ 4, e 3iπ 4, e 5iπ 4, e 7iπ 4 ( + z 4 ) = 4z 3 z = e iπ 4, e 3iπ 4, e 5iπ 4, e 7iπ 4 +z 4 z = e iπ 4, e 3iπ 4, e 5iπ 4, e 7iπ 4 ) πi 4 66
(ii) (47) lim R z = Re iθ ( < θ < π) dz = ire iθ dθ dz π + z = ire iθ π 4 + (Re iθ ) dθ = ire iθ dθ (48) 4 + R 4 e4iθ R dz + z 4 dz π + z 4 = ire iθ π + (Re iθ ) 4 dθ < R R 4 dθ = πr R. R 4 R 4 e 4iθ R 4 R (48) (48) (46) : dx + x = dx 4 + x = dz 4 + z = π 4. (49) (R ). f(x) cos(sx)dx, f(x) sin(sx)dx, f(x)e isx dx = f(x) cos(sx)dx = Re f(x) (cos(sx) + i sin(sx)) dx = f(x)e isx dx, cos(sx) k + x dx = π k e ks, lim R e isz f(x) cos(sx)dx + i f(x) sin(sx)dx = Im f(x) sin(sx)dx f(x)e isx dx. sin(sx) dx = (s >, k > ) (5) k + x k +z dz e isz dz = lim k + z R R R (i) (5) e isx e dx + lim k + x R isz dx (5) k + z 67
eisz e isz k + z = z = ±ik k > (z ik)(z + ik) z = ik e isz Res z=ik k + x = e isx (k + z ) = eisx eis ik z=ik z = z=ik iz = e ks ik. (5) e isz e isz lim dz = πi Res R k + z z=ik k + x = πe ks. k (ii) (5) lim R z = Re iθ ( < θ < π) dz = ire iθ dθ (5) lim R e isz dx = lim k + z R π e isreiθ k + (Re iθ ) ireiθ dθ. (5) e isreiθ k + (Re iθ ) ireiθ = eis[cos θ+i sin θ] k + (Re iθ ) ireiθ = eisr cos θ e sr sin θ k + (Re iθ ) ire iθ. e isr cos θ sr sin θ e ire iθ k + (Re iθ ) = eisr cos θ e sr sin θ ire iθ sin θ e sr = k + (Re iθ ) k + Re iθ R Re sr sin θ R k. (53) e isr cos θ = e iθ =, k + Re iθ R k ( e iθ =, R > k ) (5) lim e R isz k + z dx π = lim e isreiθ R k + (Re iθ ) ireiθ dθ π e isreiθ π sr sin θ sr sin θ Re πre lim R k + (Re iθ ) ireiθ dθ lim dθ = lim =. R R k R R k s >, < θ < π s sin θ > Re sr sin θ R k R (5) e isx dx = lim k + x R e isz πe ks dz = k + z k (5) cos, sin 68