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

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In summary, the conversation is about proving that if a holomorphic function is of the form f(x+iy) = u(x) + i*v(y) where u and v are real functions, then it can be expressed as f(z) = λz+c where λ is a real number and c is a complex one. The suggestion is to use the Cauchy-Riemann equations since f is holomorphic.
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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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  • #2
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.
 

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

What does it mean for a function to be holomorphic?

Holomorphic refers to a function that is complex differentiable at every point within its domain. This means that the function has a well-defined derivative at every point and can be represented by a power series.

Can a function be holomorphic at some points but not others?

Yes, a function can be holomorphic at some points within its domain but not others. This is because the function may only be complex differentiable at certain points, while at others it may have a singularity or not be defined at all.

What is the significance of λ and c in the proof?

λ and c are constants that represent the slope and y-intercept, respectively, of the line that the function f(z) is being compared to in the proof. λ is the complex number that defines the slope, while c is the complex number that defines the y-intercept.

Is this proof specific to a certain type of function?

Yes, this proof is specific to holomorphic functions. It cannot be applied to all types of functions, as the conditions for holomorphicity must be met in order for the proof to be valid.

Are there any exceptions to this proof?

Yes, there are some exceptions to this proof. For example, if the function f(z) is constant, then the proof does not hold. Additionally, if the function is not holomorphic, then the proof cannot be applied.

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