已知v=2xy+3x,求以v为虚部的解析函数f(z)=u(x,y)+iv(x,y).

问题描述:

已知v=2xy+3x,求以v为虚部的解析函数f(z)=u(x,y)+iv(x,y).

(1)
∂v/∂x=2y+3
∂u/∂x=∂v/∂y=2x (柯西黎曼方程)
f‘(z)=∂u/∂x+i(∂v/∂x)=2x+2iy+3i=2z+3i
→f(z)=z²+3iz+C,由f(0)=-i得C=-i
∴f(z)=z²+3iz-i