Suppose f:C->C is a holomorphic function of form f(x,y)=u(x)+iv(y). Show f(z) is degree 1 polynomial in z.?

C is complex field. holomorphic means analytic or complex differentiable.

Suppose f:C-%26gt;C is a holomorphic function of form f(x,y)=u(x)+iv(y). Show f(z) is degree 1 polynomial in z.?
There's a theorem about if f = u + vi is analytic, then u(sub)x = v(sub) y and u(sub)y = -v(sub)x (where by (sub) I'm meaning the partial derivative). Or that's as best I remember it (it's been a little while since I took Complex Analysis). The Cauchy theorem I think it's called.





The only way f can satisfy the first equation is if it is of the form kz + c.