Questions tagged [entire-functions]

This tag is for questions relating to the questions on entire functions. The polynomials which form a special and important class of entire functions, can be characterized as those entire function which have at most a pole as a singularity at infinity.

Definition: In mathematical analysis and, in particular, in the theory of functions of complex variable, an entire function, also called an integral function, is a function that is holomorphic in the whole complex plane (except, possibly, at the point at infinity).

It can be expanded in a power series $$f(z)=\sum_{k=0}^{\infty}a_k~z^k~,\qquad a_k=\frac{f^{(k)}(0)}{k!}~,\qquad k\ge 0$$which converges in the whole complex plane,$$\lim_{k\to \infty}|a_k|^{\frac{1}{k}}=0\qquad\text{or,}\qquad\lim_{n\to\infty}\frac{\ln|a_k|}{k}=-\infty~.$$

Examples:

  • Examples of entire functions are polynomial and exponential functions. All sums, and products of entire functions are entire, so that the entire functions form a $\mathbb C$-algebra. Further, compositions of entire functions are also entire.

  • All the derivatives and some of the integrals of entire functions, for example the error function $erf$, sine integral $Si$ and the Bessel function $J_0$ are also entire functions.

  • In general, neither series nor limit of a sequence of entire functions need be an entire function.

  • The inverse of an entire function need not be entire. Usually, inverse of a nonlinear function is not entire. (The inverse of a linear function is entire). In particular, inverses of trigonometric functions are not entire.

References:

https://en.wikipedia.org/wiki/Entire_function https://m.tau.ac.il/~tsirel/dump/Static/knowino.org/wiki/Entire_function.html https://www.encyclopediaofmath.org/index.php/Entire_function

575 questions
29
votes
4 answers

$f(ax)=f(x)^2-1$, what is $f$?

Suppose $f(ax)=(f(x))^2-1$ and suppose that $f$ is analytic in some neighborhood of $x=0$. Expanding in power series, we get $a=1+\sqrt{5}$ or $1-\sqrt{5}$. We take positive $a$. If $f\neq{\rm const}$ then $f'(0)\neq0$ - it can be any non-zero…
20
votes
2 answers

Proving that a doubly-periodic entire function $f$ is constant.

Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: $$f(z+\omega_1)=f(z)=f(z+\omega_2), \quad \forall\,z\in \Bbb…
19
votes
3 answers

Fermat's last theorem for entire functions

Let $f,g,h$ be entire functions, i.e., holomorphic functions on $\mathbb{C}$. Suppose $f^n+g^n=h^n$ for some integer $n\geq2$. What can we say about $f,g,h$? Clearly this is Fermat's last theorem for entire functions. I did a little search on the…
Colescu
  • 7,816
  • 2
  • 19
  • 60
19
votes
1 answer

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…
user228168
16
votes
0 answers

Is an entire function "determined by" its maximum modulus on each circle centered at the origin?

Let $f$ be an entire function, and $$M_f(r)=\max_{|z|\leq r}|f(z)|$$ denotes its maximum modulus on the circle centered at the origin with radius $r>0$. It's clear that for any entire functions $f(z)$ and $g(z)=e^{i\varphi}f(e^{i\theta}z)$,…
lzk
  • 175
  • 4
15
votes
2 answers

Entire function $f(z)$ grows like $\exp(x^\pi)$ as $x\to+\infty$

Does there exists an entire function $f(z)$ such that $\lim_{x\to+\infty}f(x)/\exp(x^\pi)=1$ (along the real axis)? I have successfully constructed $f(z)$ when $\pi$ is replaced by a rational number $\frac pq$. For…
Kemono Chen
  • 8,273
  • 1
  • 16
  • 60
13
votes
3 answers

When does a multivariate power series define an entire function?

In the single variable case, the power series $$\sum_{n=0}^\infty a_n z^n $$ defines an entire function, provided that $$R^{-1}:=\limsup_{n \to \infty} |a_n|^{1/n}=0. $$ Moreover, if $R^{-1} >0$, the series converges for $|z|
user1337
  • 23,442
  • 6
  • 51
  • 136
12
votes
1 answer

Measure of set where holomorphic function is large

Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is a non-constant entire function. By Liouville's theorem, we know that $f$ must take on arbitrarily large values. However Liouville doesn't say anything about what this large set must look like. …
12
votes
2 answers

Is there an entire function with $f(\mathbb{Q}) \subset \mathbb{Q}$ and a non-finite power series representation having only rational Coeffitients

I'm trying to answer the following question: Is there an entire function $f(z) := \sum \limits_{n=0}^\infty c_nz^n$ such that $f(\mathbb{Q}) \subset \mathbb{Q}$ $\forall n: c_n \in \mathbb{Q}$ $f$ is not a polynomial ? I'm trying to show that no…
11
votes
2 answers

How to Prove that if $f(z)$ is entire, and $f(z+i) = f(z), f(z+1) = f(z)$, then $f(z)$ is constant?

So the problem states that if $f(z)$ is entire, and satisfies the relation $f(z+i) = f(z)$ and $f(z+1) = f(z)$, show that $f(z)$ is constant. So I was thinking that since any point in $\mathbb{C}$ can be written as $\alpha * 1 + \beta * i $ we can…
I Love Cake
  • 1,163
  • 1
  • 14
  • 19
11
votes
3 answers

If $f$ is a nonconstant entire function such that $|f(z)|\geq M|z|^n$ for $|z|\geq R$, then $f$ is a polynomial of degree atleast $n$.

I have a question in my assignment : If $f$ is a nonconstant entire function such that $|f(z)|\geq M|z|^n$ for $|z|\geq R$ for some $n\in\mathbb N$ and some $M$ and $R$ in $(0,\infty)$ show that $f$ is a polynomial of degree atleast $n$. Now ,…
11
votes
2 answers

If $f$ is an entire function and $f(z) \not \in [0,1]$ for every $z$, then $f$ is constant

I want to prove that if $f$ is an entire function and $f(z) \not \in [0,1]$ for every $z$, then $f$ is constant. If it was written that $|f(z)| \not \in [0,1]$ I would have used the fact that $\frac{1}{f}$ is entire function and Cauchy's formula to…
11
votes
1 answer

Entire function satisfying an iteration formula

I hope to figure out that what is the entire function $f$ that satisfies the following iteration formula $$f(z+1)-f(z)=Ce^{-z}$$ for some constant $C$. Actually, I guess that $f$ has to be the form $f(z)=e^{-z+a}+be^{i2\pi{z}}+c$ where $a,b,c$ are…
F.G.
  • 820
  • 6
  • 18
10
votes
2 answers

Does the set of entire functions have the same cardinality as the reals?

I've recently been thinking about entire functions and the Weierstrass factorisation theorem and it got me thinking about the cardinality of the set of entire functions. Clearly $e^{cz}$ is entire, for all $c \in \mathbb{C}$, so the cardinality of…
h4tter
  • 389
  • 1
  • 7
9
votes
1 answer

Non-constant entire functions

Question: If $g$ is a non-constant entire function does it follow that $G_1(z)=g(z)-g\left(z+e^{g(z)}\right)$ is non-constant? The reason I care is it would imply Prop 3 below, which in turn implies Prop 1, giving a proof that might seem better…
David C. Ullrich
  • 84,497
  • 5
  • 61
  • 132
1
2 3
38 39