, . , . . , , .
f(z) DÌC
. f(z) z0 z0
.
. f¢(z0) , () z0
Dw = A Dz + a Dz, (A = f¢(z0)), a - .
. , f(z) = u(x,y) + i v(x,y) () z0 , u, v , -:
: Dz = Dx, f¢(z0) = ux +ivx. Dz = iDy, f¢(z0) = uy +i vy = vy -i uy. , .
: Dw = Du + i Dv = uxDx + uyDy +a|Dz|+i(vxDx + vyDy) +ib|Dz|= uxDx + uyDy +i(-uy Dx + ux Dy)+g|Dz|= (ux -iuy) Dx + (ux -iuy)iDy+g|Dz|=(ux -iuy) Dz+g|Dz||=(ux -iuy) Dz+g Dz=ADz+eDz.
1. u v
Dw = uxDx + uyDy +i(-uy Dx + ux Dy)+eDz
2. ,
uxDx + uyDy +i(vxDx + vyDy) = .
: , , Df=uxDx+uyDy+ i(vxDx+vyDy)+g|Dz|,
f=u(, )+iv(, ),
,
,
3. - .
4. , f ,
Dz=|Dz| eiq.
q, . .
f(z),
.
. z0 . γ, z0. γ . φ , γ z0 z, , w0=f(z0) w ( , ). α = φ γ z0 .
|
|
z z0 z Î γ. .
γ z0
, , a k γ ( ).
w = f(z) , z0 z0.
D, D.
, .