橡複素関数.PDF

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1 = x+iy =r(cosθ+isinθ) arg =θ+2nπ (n=0,,q) r y r= x 2 +y 2 = θ 0 x a,b =x+iy =a+ t(b-a) (t: a r -a =r, =a+r(cosθ+isinθ) =r (cosθ +isinθ ), 2 =r 2 (cosθ 2 +isinθ 2 ) 2 =r r 2 cos(θ +θ 2 )+isin(θ +θ 2 ), arg 2 =arg + arg 2 r = cos(θ -θ 2 )+isin(θ -θ 2 ), arg =arg 2 r2 - arg 2 2 cosθ+isinθ n =cos nθ + isin nθ (n: n a=r(cosθ+isinθ) n = a = n θ+2kπ θ+2kπ r cos +isin (k=0,,q,n-) n n D w w w= f() D

2 x,y =x+iy,w=u+iv u,v u=u(x,y), v=v(x,y) w D w w w w w=f() w f() 0 f() 0 f()-f( 0 ) f '( 0 )= $ 0-0 f() D f() D D f '() f() f() 0 f() 0 P()=a 0 +a +a 2 2 +Q+a n n P'()=a +2a 2 +Q+na n n- P(),Q() R()= Q() Q() 0 P() D f '()=0 f() D aushy-rieman =x+iy, f()=u(x,y)+iv(x,y) f() v(x,y) D u v =, x y u v =- y x D u(x,y), aushy-rieman r,θ aushy -Rieman () ()

3 u = r r v, θ v =- r r u θ u v v u f '()= +i = -i x x y y (2) u(x,y),v(x,y) D f()=u(x,y)+iv(x,y) D () aushy-rieman h(x,y) D 2 h 2 h Wh= x 2 + y 2 =0 (3) h(x,y) D u(x,y),v(x,y) aushy-rieman () v(x,y) u(x,y) D u(x,y),v(x,y) () u(x,y),v(x,y) D (2) f()=u(x,y)+iv(x,y) D v(x,y) u(x,y) E x e r c i s e f()=e -y ( cos x+i sin x) =x+iy u(x,y)=x 2 -y 2-2x v(x,y) u=ax 3 +bx 2 y+3xy 2 +y 3 a,b v(x,y) u(x,y) -u(x,y) v(x,y) u(x,y),v(x,y) f()=(u y -v x )+i(u x +v y )

4 f=u+iv u=e -y cos x, v=e -y sin x u x =-e -y sin x, u y =-e -y cos x, v x =e -y cos x, v y =-e -y sin x, aushy-rieman () u x =v y, u y =v x f() (2) f '()=u x +iv x =-e -y sin x+ie -y cos x=i(e -y cos x+ie -y sin x)=if() u x =2x-2, u xx =2, u y =-2y, u yy =-2 Wu=u xx +u yy =0 u(x,y) v(x,y) aushy-rieman () v x =-u y =2y Q, v y =u x =2x-2 Q v=2xy+φ(y), v y =2x+φ'(y)=2x-2 X φ'(y)=-2, X φ(y)=-2y+ v(x,y)=2xy-2y+ φ(y) y y (: 3) u x =3ax 2 +2bxy+3y 2, u xx =6ax+2by, u y =bx 2 +6xy+3y 2, u yy =6x+6y Wu=u xx +u yy =6ax+2by+6x+6y=6x(a+)+2y(b+3)=0 a=-,b=-3 4) v u u x =v y, u y =-v x v x =(-u) y, v y =-(-u) x -u v 5) p=u y -v x, q=u x +v y p x =u yx -v xx, p y =u yy -v xy, q x =u xx +v yx, q y =u xy +v yy Wu=u xx +u yy =0, Wv=v xx +v yy =0, u yx =u xy, v xy =v yx p x =q y, p y =-q x aushy-rieman f=p+iq

5 aushy 3. b a b =(t)=x(t)+i y(t), (α t β) (α)=a,(β)=b a b L β β L= '(t) dt= α α (x'(t)) 2 +(y'(t)) 2 dt a f(t)=u(t)+iv(t) t(α t β) β β β f(t)dt= α u(t)dt+i α v(t)dt α F'(t)=f(t) α β f(t)dt=[ F(t) ] β α =F(β)-F(α) a b =(t), (α t β, (α)=a,(β)=b)

6 f() β f()d= f( (t) ) '(t)dt α [ ] () ( f()+g() ) d= f()d+ g()d (2) k ( k f() ) d=k f()d a b b a *b - - f()d=- f()d - a b b c 2 a c + *c 2 b f()d= f()d+ f()d a *a f() M L f()d α β f( (t) ) '(t) dt ML f() D D a b f '()d=[ f() ] b a =f(b)-f(a) a n -a n d = 2π i (n=-) 0 (n -)

7 Y -a=re iθ d=rie iθ dθ (-a) n d= 0 2π X n=- I n =i dθ =2π i 0 2π re iθ n rie iθ dθ = ir n+ o 2π e i(n+)θ dθ = I n n - I n =ir n+ e i(n+)θ n+ 2π 0 =0 + i 2 () f()= + i (2) f()= 2 x= y 2 2 * 0, 2, 3 i () (+i)d + (2) d 2 2 () 2 i d -i (2) 3 2 d

8 ) = (t)= t 2 + it (0 t ), d=(2t+i)dt t=0 =0( t= =+it( d= (t 2 i -it)(2t+i)dt= = t (0 t ), d= dt =+ i t (0 t ), d= idt X d= 0 tdt+ 0 i(-it)dt=+i 2 f()= 2 = 3 3 ' 2 d= 2 d= i 0 = -2+2i 3 2) = (t)= t(+i) (0 t ), d=(+i)dt (0)=0, ()=+ i = (t)= (-t)+ i (0 t ), d = -dt (0)=+i, ()=i (+i)d= d + i = i = 2 2 =e iθ d i+i (t-it)(+i)dt + (-t-i)(-dt) 0 ( 4 3 π θ 4 5 π), d=ie iθ dθ 2 5 π + 4 d= + ie iθ π 3 π e iθ dθ = - 2+ i 2 4 3) d=2π i, n - n d=0

9 X 2 i d=2π = = X 3 2 d= d=0 4. aushy f() D f()d=0 D,, Q, f() n, D f()d = f()d+q+ f()d D n n f() D a,b D a b D, D *b 2 D 2 f()d= f()d a* 2 f() D D, 2 D 2 D

10 f()d= f()d 2, 2 a 2 n () (-a) n d=0 (2) 2 (-a) n d=0 (-a) n n aushy (-a) n d=0 (-a) n =a a 2 2 aushy 2 (-a) n d=0 < a () d=2π i (2) -a (-a) n d=0 (n>) a Γ Γ /(-a) n (n (-a) d= n Γ(-a) n d a G 2- Γ d=2π i, -a d=0 (n>) Γ(-a) n = = + (-i)(+i) 2 < d =2 + -i +i =i =-i Γ Γ 2 Γ Γ =i =-i 2 +

11 Γ Γ d= Γ 2 + Γ 2 d= + 2 d + Γ2 2 d + d+ Γ -i Γ y d 2i +i Γ +i aushy i Γ d=0 Γ +i d=2π i -i Γ2 Γ -i Γ 2 d= 2π i=π i, + 2 Γ 2 2 d= 2π i=π i + 2 X 2 d =2π i + 2 < a Γ Γ Γ d=2π i a -a a =a Γ -a a Γ d= -a d -a d=2π i -a Γ d=2π i -a f() D D a f()

12 F()= a f( ) ζ dζ F() f() F() F'()=f() D f() F'()=f() F() f() D f() F() D a,b a b f()d=[ F() ] b a =F(b)-F(a) D f(),g() D a,b a b f()g'()d=[ f()g() ] b b a - f '()g()d a < π i e - d ( π i,e - '=, e - =(-e - )' π i e - d= -e - π i π i + e - 2 d= + +π i e r an n 0 ζ an n n=0 n=0 ζ < r ζ d= an n a n=0 d= n ζ n+ n=0 n+ r 0

13 : - =/2 () (2) =a a -a a 0<ε<r ε e i d- r e i r d=2i ε sin x dx x aushy -a a () 2 d +2 a (2) sin(-2) d i ) = + 3 3(3-2) X i, 2/3 d=2π i -2/3 2 4 d= d+ 3 9 d= π i -2/ d= d=-2π i e i 2) Γ =0 aushy f()= i

14 Γ ε r e i d=0 e ix x dx+ r e i d+ -r -ε e ix x dx- ε e i d=0 Γ -r ε -ε e ix r x dx=- ε e i d- r e -ix dx ε x e i r d= ε =2i ε r e ix -e -ix dx -r -ε ε r x sin x dx x 3) 2 +2=0 =" 2i / d=0 +2 aushy sin(-2)=0 =2+nπ(n: /sin(-2) aushy 0

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =

橡複素関数.PDF

橡複素関数.PDF