## Harmonic functions and holomorphic functions

Prove that a function is harmonic if and only if there exists an entire function such that . ( if and only if it is the real part of an entire function)

## Entire Function Revisited

Prove that if is an entire function and for infinitely many points with then is constant or there exists with and some positive integer such that .

See this solution and after defining we see that for infinitely many points on the unit disk. This infinite set has an accumulation point, therefore, being equal on a set having a limit point in , we have on .

From this point continue like in the proof in the link.

See this link for further details. Looks like the function is strongly related to the Schwarz Reflection Principle.

## Injective Entire Functions

Prove that all entire functions that are also injective, take the form with and .

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## Every expansion has a 0

Suppose is entire such that for each at least one of the coefficients of the expansion is equal to 0. Prove that is a polynomial.

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## Amazing property of entire functions

Prove that if are non-constant, non-vanishing entire functions with then there exists an entire function such that are constant multiples of .

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## Entire function

Prove that if is an entire function and then is constant or there exists with and some positive integer such that .

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