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

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Seijo
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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.
 
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Seijo said:
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.
 
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...
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