MHB Proof: if f holomorphic then f(z)=λz+c

[Seijo]

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

I am to show that if f: C -> C 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.

 

[Sudharaka]

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

I am to show that if f: C -> C 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.

Hi Seijo, :)

Try using the Cauchy-Riemann equations since \(f\) is holomorphic.

Kind Regards,
Sudharaka.