Articles of holomorphic functions

Describing all holomorphic functions such that $f(n)=n$ for $n \in \mathbb{N}$

This question is inspired by a somewhat simpler one. The question is: how can we classify all holomorphic functions $f:\mathbb{C}\rightarrow\mathbb{C}$ satisfying the property $\forall n \in \mathbb{N} \quad f(n)=n $? If we have $g:\mathbb{C}\rightarrow\mathbb{C}$ such that $g\big|_\mathbb{N}\equiv 0$, then $f(z)=z+g(z)$ satisfies the criterion. Conversly, given such $f$ and defining $g(z)=f(z)-z$, we get $g\big|_\mathbb{N}\equiv 0$. So, […]

Show that for any $w \in \mathbb{C}$ there exists a sequence $z_n$ s.t. $f(z_n) \rightarrow w$

I want to prove the following: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be holomorphic and non-constant. Then for $w \in \mathbb{C}$ there exists a sequence $(z_n)_{n \in \mathbb{N}} \subset \mathbb{C}$ with $lim_{n\rightarrow w}f(z_n) = w$. Which theorem can I use here? I know that by Liouville $f$ must be unbounded but does that help me? Can […]

If $g(x)=\sum_{n} f_n x^n=\frac{2x^2+x^3}{1-x-x^2}$,then find the general expression for the coefficients,$f_n$.

As evident $f_n=\frac{1}{n!}\frac{d^n}{dx^n}g(x)(at x=0)$.If I use Cauchy’s integral formula to find the $nth$ derivative,then I’m stuck,because there also the derivative crops up while finding the residue.

Show that $f$ is a polynomial of degree $\le n$

Let $f$ be an entire function and suppose that $\exists$ constants $M,R>0$ and $n\in \Bbb N$ such that $|f(z)|<M|z|^n$ for $|z|>R$. Show that $f$ is a polynomial of degree $\le n$. To show that $f$ is a polynomial of degree $\le n$,we show that $f^{(n+1)}(z)=0$ has uncountably many zeros in $\Bbb C$. Since $f$ is […]

Is every entire function is a sum of an entire function bounded on every horizontal strip and an entire function bounded on every vertical strip?

Is it true that every entire function is a sum of an entire function bounded on every horizontal strip (horizontal strip is a set of the form $H_y:=\{x+iy : x \in \mathbb R \}$ ) and an entire function bounded on every vertical strip (vertical strip is a set of the form $V_x:=\{x+iy:y\in \mathbb R […]

Unnecessary step, normally convergent product (p10, Remmert)

I am really confused by the above proof as the following argument seems a lot more simple: (i) If $\prod f_n$ is normally convergent in $G$, then $\prod f_n$ is compactly convergent. (ii) By definition, $\prod f_n$ is compactly convergent if for all compact subset $K \subseteq X$ exists index $m=m(K)$ such that $$ p_{m,n} […]

Prove that a holomorphic function injective in an annulus is injective in the whole ball

Let $f: B(0,1) \rightarrow \mathbb{C}$ be holomorphic and suppose $\exists\ r \in (0,1)$ such that $f$ is injective in $A = \{z \in \mathbb{C} : r < |z| < 1\}$. Prove that $f$ is injective. I tried using RouchĂ© theorem, or the identity theorem, but I don’t know what to do. Any hints? 🙂