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Posts Tagged ‘entire’

Harmonic functions and holomorphic functions

May 3, 2010 1 comment

Prove that a function u:\mathbb{R}^2 \to \mathbb{R} is harmonic if and only if there exists an entire function such that u=\text{Re} f. ( if and only if it is the real part of an entire function)

Entire Function Revisited

April 29, 2010 Leave a comment

Prove that if f :\mathbb{C} \to \mathbb{C} is an entire function and |f(z)|=1 for infinitely many points z with |z|=1 then f is constant or there exists c \in \mathbb{C} with |c|=1 and n some positive integer such that f(z)=cz^n.

See this solution and after defining g we see that f(z)=g(z) for infinitely many points on the unit disk. This infinite set has an accumulation point, therefore, f,g being equal on a set N \subset \Omega having a limit point in \Omega, we have f=g on \Omega.

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

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

Injective Entire Functions

October 21, 2009 Leave a comment

Prove that all entire functions that are also injective, take the form f(z)=az+b with a,b \in \mathbb{C} and a \neq 0.
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Every expansion has a 0

October 13, 2009 Leave a comment

Suppose f is entire such that for each x_0 \in \mathbb{C} at least one of the coefficients of the expansion f=\sum\limits_{n=0}^\infty c_n (z-z_0)^n is equal to 0. Prove that f is a polynomial.
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Amazing property of entire functions

October 12, 2009 Leave a comment

Prove that if p,q,r are non-constant, non-vanishing entire functions with p+q+r=0 then there exists an entire function h such that p,q,r are constant multiples of h.
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Entire function

October 1, 2009 1 comment

Prove that if f :\mathbb{C} \to \mathbb{C} is an entire function and |z|=1 \Rightarrow |f(z)|=1 then f is constant or there exists c \in \mathbb{C} with |c|=1 and n some positive integer such that f(z)=cz^n.
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