1. ## Proof: if f holomorphic then f(z)=λz+c

Hello. I need a bit of help or a tip maybe..

I am to show that if f is a holomorphic function that is of the form f(x+iy) = u(x) + i*v(y) where u and v are real functions,
then f(z) = λz+c where λ is a real number and c is a complex one.

How would I begin to prove this?

2. ## Re: Proof: if f holomorphic then f(z)=λz+c

Originally Posted by seijo
I am to show that if f is a holomorphic function that is of the form f(x+iy) = u(x) + i*v(x) where u and v are real functions,
then f(z) = λz+c where λ is a real number and c is a complex one.
How would I begin to prove this?
You can't prove it, because it isn't true (unless $\lambda = 0$).
Notice that, when considered as f(z) = f(x,y), the assumption is that f(x,y) is holomorphic and $\partial{f}/\partial{y} = 0$.
But your f(x+iy) = λ(x+iy)+c varies with y (unless $\lambda = 0$).
The thing to prove is that the function must be constant (i.e. it actually is true, but only with $\lambda = 0$).
You can prove that by a straighforward application of the C-R equations.

3. ## Re: Proof: if f holomorphic then f(z)=λz+c

I see your point. But I made a typo. It's meant to be f(x+iy) = u(x) + i*v(y) not f(x+iy) = u(x) + i*v(x).

I suppose in this case it actually does make sense? Would the C-R equations still be the way to go?

4. ## Re: Proof: if f holomorphic then f(z)=λz+c

Yes, it drops straight out:
Spoiler:

$u_x(x) = v_y(y) \implies u_x(x) = v_y(y) = \lambda$ for some $\lambda \in \mathbb{R}$, because $u_{xx} = \partial v_y(y) / \partial x = 0$.

Thus $u(x) = \lambda x + b_0, v(y) = \lambda y + b_1$, so $f(z) = f(x+iy) = u(x) + iv(y)$

$= (\lambda x + b_0) + i(\lambda y + b_1) = \lambda (x+iy) + (b_0 + ib_1) = \lambda z + c$, where $c=b_0 + ib_1$.

Thus $f(z) = \lambda z + c$ for some fixed $\lambda \in \mathbb{R}, c \in \mathbb{C}$.