f(z)是解析函数,已知u(x,y)=1/2ln( - 知识问答

f(z)是解析函数,已知u(x,y)=1/2ln(x^2+y^2),f(1+i)=1/2ln2,求v(x,y)

1个回答

由柯西-黎曼条件：对u(x,y)=1/2ln(x^2+y^2)求x的偏导x/(x^2+y^2),
对u(x,y)=1/2ln(x^2+y^2)求x的偏导y/(x^2+y^2),
f'(z)=x/(x^2+y^2)-yi/(x^2+y^2)=1/z
f(z)=lnz+C
f(1+i)=1/2*ln2,C=-π/4*i
f(z)=lnz-π/4*i

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