Thread: Simple question about a holomorphic function

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?

Thanks to everyone in advance.

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 ).
Notice that, when considered as f(z) = f(x,y), the assumption is that f(x,y) is holomorphic and .
But your f(x+iy) = λ(x+iy)+c varies with y (unless ).
The thing to prove is that the function must be constant (i.e. it actually is true, but only with ).
You can prove that by a straighforward application of the C-R equations.

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?

Yes, it drops straight out:

Spoiler:

for some , because .

Thus , so

, where .

Thus for some fixed .