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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Derivation of an complex integral

In the proof of a lemma of the book I study, the derivative of the following integral is calculated : $\frac{d}{dz}\frac{1}{2\pi} \int_{0}^{2\pi}\operatorname{Re}\left(\frac{re^{i\theta}+z}{re^{i\...
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Problem using Liouville's Theorem

True/ False There does not exists function $f$ that is analytic in $D$ such that $|f(z)| \leq 1, f\left(\frac{1}{3}\right)=0$ and $f\left(\frac{-2}{3}\right)=\frac{5}{6}$ If $D=\mathbb C$ then by ...
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Showing an entire function is identically zero

Suppose $f$ is an entire function with the property that $|f(z)|\leq C\exp(|z|^\rho)$ with $\rho\geq0$. And $f(\log(n))=0 \forall n\geq3 $ Now I have to prove that $f$ is identically $0$. I was ...
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Question about Hadamard Factorization Theorem and canonical product

Let $P(z)=\prod_{n=1}^{\infty}E_{p_{n}}\left(\frac{z}{a_{n}}\right)$ where $E_{p}(z)=(1-z)\exp \left(z + \frac{z^{2}}{2}+\ldots +\frac{z^{p}}{p}\right)$ for $p\geq 1$. In the proof of Hadamard’s ...
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Definition of order of entire function

The definition, used in the book I study, of the order of an entire function is as follows : An entire function $f$ is of finite order if there is a positive constant $a$ and an $r_{0}$ such that $$\...
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Question about the order of entire function $f+g$

My question is based on this old question : Order of entire function $f + g$. I try to better understand the inequality we get for the order of $f+g$. It is obvious that there are two cases. The one ...
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2 votes
1 answer
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If $f$,$g$ are entire and $g′=f(g)$, then $f$ is linear or $g$ is constant.

If $f$,$g$ are entire functions such that and $g′(z)=f(g(z))$ everywhere, then show that $f$ is linear or $g$ is constant. This claim was mentioned without proof in this related question (Applications ...
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Entire function is zero.

Let $a\in\mathbb{R}$, $f$ be an entire function on $\mathbb{C}$ and the inequality $$\int\limits_0^{2\pi} |f(re^{i\theta})|\text{d}\theta\le r^a$$holds for all $r>0$. Can we prove that $f\equiv0$? ...
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3 answers
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to prove a Holomorphic function to be constant

Suppose that $f$ is holomorphic on $\mathbb{C}$, and suppose that the function $g(z) = f(z)/z$, defined for $z \neq 0$, satisfies $g(z) → 0$ as $|z| \to \infty$. Prove that $f$ is constant. I have ...
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Entire function with finite number of zeros in the closed upper half complex plane

Let $g_+:\mathbb R \rightarrow \mathbb C $ be a function such that $\int_{-\infty}^{\infty}|g(t)|dt$ is finite, $g_+$ is not the zero function nor the the function zero except in a zero measure set, ...
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Question about proof of Liouville's theorem [duplicate]

I am trying to work through a proof of Liouville's theorem here and ran into an issue. In his answer (as I understand it) Eremenko first establishes that $|f(0)|\leq \max_{|z|=r}|f(z)|$ for entire ...
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On invertibility of entire functions

If we have an entire function f that's locally invertible at every point z, then should the function be invertible on the entire complex plane C? I am trying to solve this problem with the help of ...
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5 votes
1 answer
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What is the minimal order of a “sigmoid-like” entire function?

The following question is motivated by Is there a complex analytic function that acts like sigmoid on the reals?: What is the minimal order of an entire function $f$ which is real-valued, increasing, ...
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Find all Entire Functions $f$ such that $u(x, y) = av(x, y) + b$ for $a, b \in \mathbb{R}$

I want to find all entire functions $f$ such that for all $z \in \mathbb{C}$, $u(x, y) = av(x, y) + b$ for $a, b \in \mathbb{R}$. My attempt is as follows.$\DeclareMathOperator{\Re}{Re}\...
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Show that non entire function is constant [duplicate]

Let $f : \Bbb C^* \to \Bbb C$ be a holomorphic function such that $\lvert f(z) \rvert \leq \sqrt{\lvert z \rvert} + \frac{1}{\sqrt{\lvert z \rvert}}$. Show that $f$ is constant. I cannot use Liouville ...
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Entire function with arbitrary zeroes $(a_n)$ but $|a_n| \to a \neq \infty$

I've been studying the Weierstrass factorisation theorem and in the proofs I see they assume that the zeroes of an an entire function always tend to infinity. Is it possible to define an entire ...
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2 votes
1 answer
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$f$ is entire, prove that $\{f_n = f(nz) | n \in \mathbb{N}\}$ is normal on the annulus iff $f$ is constant

I am studying for my exam and came across this question: Suppose $f$ is entire and $r<R$. Prove that the family $\mathcal{F} = \{f_n = f(nz) | n \in \mathbb{N}\}$ for $z \in \mathbb{C}$ is normal ...
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Is it true that if $|f(z)| \leq p(|z|)$ for some polynomial $p$, then $f(z)$ is a polynomial?

Suppose that we have an entire function $f(z)$, which is bounded by: $$|f(z)|\leq p(|z|)$$ for some polynomial $p(z)$. Is it possible to prove that $f(z)$ is a polynomial? I know that it is easy for $...
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1 vote
1 answer
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Find all entire functions $f$ of finite order such that $f(log(n)) = n$

Find all entire functions $f$ of finite order such that $f(log(n)) = n$ I am reviewing for my complex analysis final and this problem came up towards the end of Conway and I was unsure how to solve ...
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2 votes
1 answer
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If $f$ is an entire function of order $\lambda$ then $f'$ also has order $\lambda$

If $f$ is an entire function of order $\lambda$ then $f'$ also has order $\lambda$ Can someone show me how to prove this or point me in the right direction? My definition is an entire function $f$ ...
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Proof explanation for problem with Schwarz integral formula

We want to show that a non-zero entire function such that $\lvert f(z) \rvert < e^{B \lvert z \rvert + C}$ for some $B,C>0$ then $f(z)=e^{az+b}$ for some $a,b \in \Bbb C$. The proof I was given ...
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$f$ is entire, show that the order of $f'$ is less than or equal to that of $f$

Let $f$ be an entire function of finite order. For entire functions $g$ of finite order, let $o_g$ denote the order of $g$ (recall that this order is defined as the infinum of all nonnegative ...
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Relationship between the type and zeros of an entire function

I am just being curious whether there exists a specific relationship between the type of an entire function and its zeros. My intuition says that there is no relation. But I need to confirm. I can't ...
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Number of zeros of an entire function

I want to find the zeros of the function \begin{equation} e^{\gamma z}\prod_{n=1}^{\infty}\left(1+\frac{z}{n}\right)e^{-\frac{z}{n}}-1 \end{equation} for some constant $\gamma$. The function is ...
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1 vote
0 answers
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Zeros of the Bessel's function $J_0(z)$

I am particularly interested in finding a form for the zeros of the Bessel's function $J_0(z)$. I have read somewhere that it has infinite number of zeros (need reference for this). But, in order to ...
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Order of Bessel's function $J_0(z)$

It is a well known fact that a Bessel's function of the first kind $J_{\nu}(z)$ is entire for integral values of $\nu$. Further, it is given in Wikipedia that the order of $J_0(z)$ is $1$. I am trying ...
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Rank of Entire function written as product of Weirstrass's elementary factors

Let $f$ be a an entire function. Elementary Factors are defined as follows $E_0 (z) = 1 - z$ $E_n (z) = (1 - z)\exp\left(\sum\limits_{k=1}^{n}\frac{z}{k}\right)$ If we can write $$ f(z) = \prod\...
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upper bound on $L_2$-norm of a power series, in terms of coefficients?

Is there any upper bound on $L_2$-norm of a convergent power series (in R), in terms of coefficients? I have $f(x) = a_0+a_1\frac{x}{1!}+a_2\frac{x^2}{2!}+a_3\frac{x^3}{3!}+...$. I need something like:...
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5 votes
1 answer
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If $f$ is an entire function such that $|f(z)| \le 2 \ln (|z|+1)$ for all $z \in \Bbb{C}$, then $f(z) = 0$ for all $z \in \Bbb{C}$.

Suppose $f$ is an entire function such that $|f(z)| \le 2 \ln (|z|+1)$ for all $z \in \Bbb{C}$. Prove that $f(z) = 0$ for all $z \in \Bbb{C}$. Here is my solution which I am hoping someone could ...
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Representation exponential type functions

I am looking for a representation for the entire functions of $\pi$-exponential type. Let us suppose that $f$ is of $\pi$-exponential type and that it is equal to zero on the integers. It would mean ...
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More thoughts on if a entire function has at least one coefficient of the Taylor series being real

I have asked a question few days ago: Entire function where at each $a \in \Bbb C$, at least one coefficient of the Taylor series at $a$ is real The question is: Let $f:\mathbb{C}\rightarrow\mathbb{C}...
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Suppose $f_1,f_2$ entire functions. Produce entire $h,g_1,g_2$ such that $f_1=hg_1$ and $f_2=hg_2$ with $g_1,g_2$ no common zeros.

Suppose $f_1,f_2$ entire functions. Produce entire $h,g_1,g_2$ such that $f_1=hg_1$ and $f_2=hg_2$ with $g_1,g_2$ no common zeros. I know I have to use Weierstrass factorization theorem somehow but I’...
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Maximum value of an entire function in a closed set.

Consider the entire function $f(z)=z(z-i)$. Put $$S=\bigg\{\frac{1}{|f(z)|}\ |\ |z|\geq 2\bigg\}.$$ At what value(s) of $z$ is the maximum of the set $S$ attained? My Idea: As $f(z)$ is an entire ...
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3 votes
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Proving that the entire function which satisfies a given property is unique

Let $\phi : \mathbb{C} \to \mathbb{C}$ be an entire function satisfying the following three properties: $|\phi'(z)| \leq |\phi(z)|$ for all $z \in \mathbb{C}$ $\phi(0) = 2$ $\phi(1) = 1$ The problem ...
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1 vote
2 answers
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What can we say about $f$ if $\int_{|z|=1}\frac{f(z)}{((k+1)z-1)^n}dz=0$?

Consider $f$ an entire function that $\int_{|z|=1}\frac{f(z)}{((k+1)z-1)^n}dz=0$ for any $k\in\Bbb N$. What can we say about $f$ for $n\in\{ 1,2,3,\cdots\}$? We know that $\frac{n!}{2\pi i}\int_{|z|=...
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1 vote
1 answer
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How to deduce that there doesn't exist any sequence of polynomials $P_n$ such that $P_n(z)\to \frac 1z$

The exercise I have reads as follow: Let $f$ be an entire function. Compute \begin{equation} \frac1{2\pi i}\int_{|z|=1}\frac{1-zf(z)}{z}dz \end{equation} Prove that all $f$ entire functions verify ...
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If $f$ is an entire function, $\lim_{z\to\infty} z^{-1}\operatorname{ Re} f(z)=0$, show $f$ is constant.

If $f$ is an entire function, $\lim_{z\to\infty} z^{-1} \operatorname{ Re} f(z)=0$, show $f$ is constant. I know only the Liouville theorem: bounded entire funcion is constant. It seems that the ...
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2 answers
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What can we say about these two entire functions?

Let $g$ and $h$ be two entire functions such that $\lim_{|z|\to\infty}\frac{|g(z)|}{|z|^5} = 0$ and $|h(z)|\le |z|^4$. What can we say about $g$ and $h$? I got to both of them being polynomials of at ...
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Is a function with an essential singularity always some composition of an entire transcendental (not a polynomial) function?

I thought about the theorem of Casorati-Weierstraß(in my lecture notes there are actually two, one deals with entire functions and the other with essential singularities) and asked myself if the ...
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If $ |f(z^{2})|\leq|f(z)| $ and $ f $ is entire, then is $ f $ constant? [duplicate]

Assume $f:\mathbb{C}\to \mathbb{C} $ entire function, and assume $ f $ satisfies $ |f(z^{2})|\leq|f(z)| $ How can I prove that $ f $ is a constant? I have tried something but not sure about my way. I'...
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0 votes
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Prove that $f(z) = f(−z)$ $∀ z ∈ \Bbb C$

Let f be an entire function such that $f (ix) = f (x)$ for all $x ∈ (1, 2) ⊆ \Bbb R$. Prove that $f(z) = f(−z)$ $∀ z ∈ \Bbb C$ I know zero sets of non-constant holomorphic functions are discrete, i.e. ...
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show that an entire function is constant [duplicate]

Let f be an entire function with the property that $ |f(\frac{1}{n})|< n^{-n}$ for $n ∈ \Bbb N$. Show that f is constant I know it follows from Liouville Theorem that if an entire function is ...
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Does there exist $z_0 \notin \{a_n\}_{n \geq 1}$ such that $f(z_0) = 0\ $?

I am reading Weierstrass Factorization Theorem from the lecture notes given by our instructor. Here I came across a theorem which is the following $:$ $\mathbf {Theorem} :$ Let $\{a_n\}_{n \geq 1}$ ...
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2 votes
1 answer
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$f,g$ entire such that $f(0)=g(0)\neq 0$ and $|f(z)|\leq |g(z)|$ for all $z\in\mathbb{C}$, then $f=g$.

Question: Suppose we have functions $f,g$ entire such that $f(0)=g(0)\neq 0$ and $|f(z)|\leq |g(z)|$ for all $z\in\mathbb{C}$, then $f=g$. My attempt: Consider function $h(z)=\frac{f(z)}{g(z)}$, ...
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Do Meromorphic Functions automatically give us Entire Functions?

Suppose I have a meromorphic function, $f$. Then, I can write $f(z)=\frac{h(z)}{g(z)}$ where $h,g$ are entire. I would really like to be able to claim that $f$ extends to an entire function by ...
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  • 2,390
1 vote
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If $f$ is an non-null entire function that satisfies $|f(z)|=2$ for all $z\in\partial \Bbb D$, then $f$ is constant? [duplicate]

I want to prove if either this statement is true or false: Let $f:\Bbb C\to \Bbb C$ be a non-null entire function which verifies that $|f(z)|=2$ for every $z$ that belongs in the circle of center the ...
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3 votes
1 answer
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$\lim_{\vert z\vert\to\infty}\frac{zf'(z)}{f(z)}=n\in\mathbb{N}$ implies $f$ is a polynomial

Let $f$ be an entire function, meaning $$f:\mathbb{C}\to\mathbb{C}$$ is holomorphic. If $f\not\equiv0$ and $$\lim_{\vert z\vert\to\infty}\frac{zf'(z)}{f(z)}=n\in\mathbb{N}_0$$ then $f$ have to be a ...
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1 answer
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Order of growth of $\prod_{n=1}^{\infty}(1-a^nz)$ for $0<|a|<1$

This question is from Conway Complex Analysis, page 287, exercise 9(a). My attempt: Write the product as $\underset{n}\prod(1-\frac{z}{b^n})$, where $b=1/a$. First note that this entire function has ...
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Which of the following functions is/are constant?

Which of the following functions is/are constant ? Let $f(z)$ be an analytic function in extended complex plane. If $g(z)$ is an entire function such that $g(z) = u + i v$ and $u^2 \leq v^2 + 2012$ ...
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2 votes
1 answer
126 views

Show that $\int_1^{\infty} t^{x-1} e^{-t} dt$ is entire.

Two Notes : The following is a half-line claim in a several pages of proof for some theorem in a book that I am studying. It is not an exercise. I can't include any "my attempt" because I ...
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