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.

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Fourier transform of compact supported function is entire

Suppose $f\in L^1(\mathbb{R})$ has compact support, say $\operatorname{supp}f\subset[-r,r]$. I want to show that its Fourier transform $$ \hat{f}(z) = \int_{-a}^{a} f(t) e^{-2\pi itz}dt$$ for $z\in\...
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1answer
47 views

An entire function $f $ such that $f(x+iy)=f(3x+i2y).$ [closed]

Let $f(z)$ entire function suppose for any nonzero z satisfied the equation $f(z)=f(x+iy)=f(3x+i2y)$ , then f must be constant
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2answers
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Existence of holomorphic function $f$ such that $f^{(n)}(0) = n^{2n}$

I need help with the following problem: Is there a holomorphic function $f$ in an open disk around $0$ such that \begin{align} f^{(n)}(0) = n^{2n} \end{align} for all $n \in \mathbb{N}$? I thought ...
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1answer
81 views

Evaluate $\int_0^\infty f(x)dx$ when ,$f(x)=e^{-e^{x}}e^{x}\sin x$ by residues or otherwise

Evaluate improper integral $\int_0^\infty f(x)dx$ . $$f(x)=e^{-e^{x}}e^{x}\sin x$$ My try- Can I have a semi circular path from $0$ to $R$ and counterclockwise semi-circle?
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30 views

Condition for an entire function to be a polynomial

I need a hint to solve the following problem. Shows that if $f:\mathbb{C}\rightarrow\mathbb{C}$ is an entire function that satisfies $\lim_{n\rightarrow\infty}|f(z_n)|=\infty$ provided that a sequence ...
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1answer
21 views

Characterization of complex exponential in terms of real identities

It is well known (or at least easy to see) that the complex exponential function $\exp:\mathbb{C}\to \mathbb{C}$ can be charaterized as the unique holomorphic function $f:\mathbb{C}\to \mathbb{C}$ ...
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0answers
36 views

Find all $f$ such that $\Re f+\Im f$ is bounded

Find all entire functions such that $\Re f + \Im f$ is bounded. So there exists $M \ge 0$ such that $\Re f + \Im f \lt M$. We now try to find $g$ such that the given bounded expression is similar to ...
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Finding specific entire functions proof check

Find all entire functions $f$ such that $f\left( e^{-\frac{i\pi}{n}} \right) = e^{\frac{2\pi i}{n}} ,\ \forall n \in \mathbb{N}$ My proof: Consider a sequence $a_n = \left\{ e^{ -\frac{i\pi}{n}} \...
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1answer
24 views

Properties of entire function

I met a problem, which says if $f$ and $g$ are all entire functions on $C$. And $f(g(z)) \in C/[- \infty,0]$, for all $z \in C$. Then both $f$ and $g$ must be a constant. I have no idea about how to ...
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1answer
47 views

A reflection question about a complex function

Let's say we have an entire complex function $f(z)$ such that: $f(z)$ is real when $z$ is real $f(z)$ is purely imaginary when $z$ is purely imaginary. SO basically this entire function maps real/im ...
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Entire non-constant function $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f(cz) = f(z)$ [duplicate]

I would like to find an entire, non-constant (complex) function $f$ with $f(cz) = f(z)$ $\forall z \in \mathbb{C}$ for some $c \in \mathbb{C}$. This is pretty straightforward if $|c| \neq 1$, but I am ...
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0answers
26 views

Prove that $f(z)$ is entire. [duplicate]

Suppose $f(z)$ is analytic in a neighborhood of the origin and $\sum_{i=1}^\infty f^{(n)}(0)$ converges. Prove that $f(z)$ extends to be an entire function. From the given information I can guess ...
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1answer
165 views

$\prod_{n=1}^{\infty} (1- z/ a_n) $ is entire iff $\sum_{n=1}^{\infty} 1/(z-a_n) $ is meromorphic

This question was asked in a masters exam previous year paper and I was unable to prove it. Show that the $$\prod_{n=1}^{\infty} (1- \frac{z}{a_{n}}) $$ is entire iff $$\sum_{n=1}^{\infty} \frac{1}{z-...
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2answers
107 views

Show that This infinite product is entire

This question is from Ponnusamy and silvermann complex analysis and I am making some mistake in this question and unable to find it. Show that $\prod_{n=1}^{\infty} (1-z/n) e^{z/n +5z^2/n^2}$ is ...
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1answer
55 views

Construct an entire function whose only zeroes are following

This question is from Ponnusamy and Silvermann Complex Variables pg 436, subsection wietestrauss product theorem Question: Construct an entire function whose only zeroes are z=ln n. Weierstrauss ...
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1answer
55 views

What properties $f$ must have so that $\overline{f(z)} =f(\,\overline {z} \,)$

What properties function $f$ must have so that $\overline{f(z)} =f(\,\overline {z} \,)$ where $z\in\mathbb{C}$ My working : I can prove it for f being a polynomial in $z$ with real coefficient using ...
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1answer
22 views

Existence of analytic function with certain conditions

$1$. Does there exists an analytic function $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f(z)=z$ for all $|z|=1$ and $f(z)=z^2$ for all $|z|=2$. $2$.There exists an analytic function $f: \mathbb{...
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Pole of composition of entire functions

$f_{1}, f_{2}$ are entire functions and $\lim_{z \rightarrow \infty} f_{1}(f_{2}(z))= \infty $. I now have to show that both $f_{1}, f_{2}$ are polynomials. I have tried to replicate the technique ...
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1answer
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When is this norm defined ( continuous functions and entire functions)

Note : This question was asked 7 hours ago but closed so I thought about it again and asking it with my attempts . Hope , it on -site now. The following question was asked in a masters exam for which ...
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40 views

Prove that $f=0$ if $\varphi(y)=\int_\mathbb{R} |f(x+iy)|^2 d x$ is a bounded function.

Suppose that $f$ is an entire function of exponential type and \begin{equation} \varphi(y)=\int_{-\infty}^\infty |f(x+iy)|^2 d x \end{equation} is a bounded function on $\mathbb{R}$. Prove that $f=0$. ...
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How to prove that if an entire function has uncountably many zeroes then it must be constant

I am solving assignment questions of complex analysis which might not be discussed due to pendamic and I was unable to solve this particular question. Let f be an entire function that has uncountably ...
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Entire functions on two nonparallel lines belongs to $L^2(-\infty,\infty)$

Suppose that $f$ is an entire function of exponential type. If its restriction on two nonparallel lines belongs to $L^2(-\infty,\infty)$, show that $f=0$. I am sorry that I don't how to write the ...
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Entire Function with Finite Order and the Growth of its Coefficients. [Stein, Chapter 5, Problem 4 (b)]

This post is about part (b) of Stein Complex Analysis, Chapter 5, Problem 4. Part (a) has been asked and solved here: Problem on entire function of finite order [Stein, Chapter 5, Problem 4] (Conard ...
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1answer
53 views

Problem on entire function of finite order [Stein, Chapter 5, Problem 4]

I am working on the part (a) of Stein Complex Analysis, Chapter 5 Problem 4, which states as follows: Let $F(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be an entire function of finite order. Suppose $|F(z)|\...
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An entire function of exponential type on nonparallel lines.

Suppose that $f$ is an entire function of exponential type, i.e., there exist constants $A$ and $B$ such that \begin{equation*} |f(z)| \le A e^{B|z|} \end{equation*} for all complex $z$. Prove that if ...
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1answer
48 views

Prove that $f$ can have at most finitely many zeros.

Let $f : \Bbb C \longrightarrow \Bbb C$ be an entire function such that $$\lim\limits_{z \to 0} \left \lvert f \left (\dfrac {1} {z} \right ) \right \rvert = \infty.$$ Then show that $f$ can have at ...
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Taylor series of $\frac{1}{f(z)}$ exists (simply)

I was attempting to simplify a standard proof of the fundamental theorem of algebra for undergraduates who know about Taylor series and some complex numbers. Normally, the proof would look like: ...
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1answer
23 views

Inverse image of $e^z$ is bounded for finite circle or not?

I got struck on this option of a question in my complex analysis quiz and couldn't correctly solve it. So, I am asking for help here. Consider entire function $g(z)=e^z $ $ z\in \mathbb{C}$. Then is ...
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1answer
35 views

Show that for all $z \in \mathbb{C}$, $f(z)=zf(1)$, if $f$ has the additive property.

Suppose $f$ is an entire function such that $f(z_1+z_2)=f(z_1)+f(z_2)$ for all $z_1,z_2 \in \mathbb{C}$. Show that for all $z \in \mathbb{C}$, $f(z)=zf(1)$. Here, by an entire function, we mean a ...
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1answer
72 views

Entire function can be approximated uniformly by polynomials with rational coefficients on every compact set.

I am working on Stein Complex Analysis Chapter 2 Problem 5. The question has a hint that I cannot prove. Basically, the hint states as follows: Let $p_{1}, p_{2},\cdots$ denote an enumeration of the ...
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39 views

If $f(z)$ is entire and $|f''(z)| \leq 5$. Show that $f$ must have the form of $az^2 +bz +c$

This might be a completely incorrect approach, but since we have that $|f''(z)| \leq 5$, can we integrate both sides twice and obtain that $|f(z)| \leq \frac{5z^2}{2} +cz +c$ for c being some ...
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65 views

Prove that $f$ is a constant function.

Let $f : \Bbb C \longrightarrow \Bbb C$ be an entire function i.e., analytic everywhere in $\Bbb C.$ Suppose $$\lim\limits_{\left \lvert z \right \rvert \to \infty} \dfrac {f(z)} {z} = 0.$$ Prove that ...
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1answer
91 views

Prove that an entire and bounded function is constant

Let $f$ an entire function. Suppose that $|f(z)|\leq 1 + |z|^{1-\alpha}$ for $\alpha \in (0,1)$ and for all $z \in \mathbb{C}$. Prove thar $f$ is constant. I need to prove this statement without ...
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1answer
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Prove checking: Show that an entire function is constant.

Question Let $f:\Bbb{C}\rightarrow \Bbb{C}$ be an entire function such that $$\lim_{|z|\rightarrow\infty} \dfrac{f(z)}z=0.$$ Show that $f$ is constant function. Attempt: Since $\lim_{|z|\rightarrow\...
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Reference Request: Identity Principle of Real Analytic functions (Multivariate)

I'm using a (basic result in complex analysis), the identity principle a variant of which is discussed in this post, in a paper I'm writing. However, I can't seem to find a reference to a ...
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2answers
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Modulus of Infinite Product

We know that for complex (entire) functions $f,g$ we have $|f(z)g(z)|=|f(z)||g(z)|$, where $|.|$ means complex modulus. What about if we have an infinite product? Is it true that $$\bigg| \prod_{k=1}^...
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1answer
34 views

Uniqueness Theorem to show non-existence of entire function

I am having trouble with the details of the following question: Does there exist an entire function such that $f(\frac1n)=e^{-n}$ for all positive integers $n$? I think this is supposed to be a ...
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78 views

Prove the existence of an entire function satisfying these conditions

This question was asked in an assignment that I am trying to solve. Let $f:\mathbb{R} \to \mathbb{C}$ be continuous. Then prove the existence of an entire function $g$ such that $|f(x)-g(x)|< 1$ ...
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1answer
45 views

Growth rate of entire function

Let $f(z)$ be an entire function such that $|f(z)|\geq |z|^a$ in $\overline{D}$ for some $a\in\mathbb{R}$ with $0<a<1$. Prove that $|f(0)|\geq 1$ I am thinking of the function $g(z) = \frac{z^a}{...
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Write an entire function in terms of 2 entire functions [duplicate]

This question was asked in complex analysis end term of previous year and I am trying to solve it to gain some experience. I need help with following question: If f and g are entire functions with no ...
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1answer
50 views

Proving this continuous function is entire if a related function is given entire

This question was asked in a masters exam for which I am preparing and I was unable to think about it. So, I am asking it here. Question: Let f be a continuous function from $\mathbb{C} \to \mathbb{C}...
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0answers
34 views

Prove that entire function satisfying this is constant [duplicate]

This question was part of an assignment which I am trying to solve but couldn't. Let f be an entire function such that $\frac{f(z) } {z} \in 0$ as $ |z|\to \infty$ . Show that f is constant. Entire ...
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2answers
48 views

Prove that every entire function satisfies this particular property

This question was asked in an assignment given to me and I could not solve it . Question: If $f$ is an entire function such that $|f(z)|\to \infty$ as $|z| \to \infty$ prove that $|f(z)|\geq c|z|$ ...
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1answer
42 views

Entire functions and fractional growth [closed]

I ran into the following problem: is it possible that a function $f$ is the ratio of two entire functions, $$ f=\frac{G}{E}$$ but at the same time it is asymptotic to $1/\sqrt{x}$ for $x\to \infty$, $$...
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1answer
45 views

Why is this composite function entire?

Let $f: \mathbb{C}^n \to \mathbb{C}$ be an entire function and let $z,w \in \mathbb{C}^n$ be fixed. Then, why is $g: \mathbb{C} \to \mathbb{C}$ with $g(\lambda) = f(z + \lambda w)$ is entire? (By $f$ ...
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1answer
67 views

True or false? If $f$ and $g$ are entire functions such that $f(z) g(z) =1$ for all $z$, then $f$ and $g$ are constants

This question was asked in complex analysis quiz and I was unable to solve it at that time , so I am asking it here for help. State true/false with proper explanations: If $f$ and $g$ are entire ...
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1answer
62 views

Davenport's proof of the Hadamard theorem

In Multiplicative Number Theory, H. Davenport proves the following theorem in the paragraph §11 (page 74) : Theorem : (Hadamard) Let $f:\mathbb{C}\to\mathbb{C}$ be entire of order $\leqslant1$ (that ...
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2answers
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If $g(z) = f(1/z)$ and $g$ has a pole at $0$ and $f$ entire.Then there is a $z_0$ such that $f(z_0) = 0$.

I figured I can somehow use Liouville's theorem. I don't really know what the method is for solving or where to begin. Suppose that $f: \mathbb{C} \to \mathbb{C}$ is an entire function. Let $g(z) = f(\...
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1answer
38 views

An entire function satisfying $Im(f(z)) > 0$ for all $z \in \mathbb{C}$ is constant [duplicate]

This particular question was part of a multiple choice question asked in my quiz. I contradicted other $3$ options but this one is true and I have no idea how to prove it. Assume that $f$ is entire ...
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1answer
35 views

A question based on existence of entire functions [closed]

This is an assignment question of my complex analysis class . Due to COVID-19, the topic was not discussed. So, I am asking for help here. Consider $f:\{z \in \mathbb{C}\, : |z|>1 \}\rightarrow \...

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