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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

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

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

Categories: Complex analysis

Every expansion has a 0

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.

Amazing property of entire functions

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 $h$.
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 $f(z)=cz^n$.