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Questions tagged [entire-functions]

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2
votes
1answer
58 views

Solving functional equation $\sin f = f^2-3if+\pi$

I want to find all (entire) solutions to the equation $$\sin f = f^2-3if+\pi.$$ Using the identity theorem, I was able to show that if a solution exists, it must be constant. Therefore, all that is ...
0
votes
1answer
61 views

Given $g\in L^1(-a,a)$ and $f(x+iy):=\int_{-a}^ag(t)e^{2\pi i(x+iy)t}dt$, is it true that $|f(z)|=o(e^{2\pi a|z|}), z\rightarrow \infty$?

If $a>0$ and $g\in L^1(-a,a)$, define: $$f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto \int_{-a}^ag(t)e^{2\pi izt}\operatorname{d}t.$$ Is it true that: $$|f(z)| =o(e^{2\pi a|z|}), z\rightarrow\...
2
votes
1answer
52 views

If $f$ is entire and $f(z)\to \infty$, for $z\to \infty$ , then $f(z)$ is polynomial

If $f$ is entire and $f(z)\to \infty,$ for $z\to \infty$ , then $f(z)$ is polynomial. As $f(z)\to \infty, for z\to \infty$ then $\exists M,$ such that $|z|>M$ implies $|f(z)|>1$ COnsider ...
1
vote
1answer
27 views

Is it possible to argue that non constanst entire function doesnot take only values in strip only Without using Picard's Theorem? [duplicate]

By Picard theorem ,Only Possible that non constant entire function only leaves at most 2 point in $\mathbb C$. But Is it possible to argue without using that theorem to show that it is not possible ...
4
votes
0answers
47 views

Example of entire function which can take every value except …

Example of entire function which can take every value except 1) Finite Number of Points 2) Countable Number Of points I know about example of function which does not take only single value and entire ...
0
votes
1answer
36 views

Unable use countability argument in following problem [duplicate]

Suppose f is analytic function defined everywhere in $\mathbb C$ and such $z_0\in \mathbb C$ at least one coefficent in expansion $f(z)=\sum^\infty_{n=0}c_n(z-z_0)^n$ is equal to $0$. Then prove ...
0
votes
1answer
38 views

Example of entire function on $\mathbb C$ such that which does not take only one value in $\mathbb C$

I am intersted in Example of entire function on $\mathbb C$ such that which does not take only one value in $\mathbb C$ . I know that it is not possible for entire function to leave only some open ...
4
votes
1answer
41 views

Find the entire function $f$.

Suppose that $f:\mathbb{C}\to\mathbb{C}$ is entire and that $f(x,y)=u(x,y)+iv(x,y)$. If $u^2-v^2\geq x^2-y^2$ for all $z=x+iy$, what information can we have about $f$? It seems Liouville's theorem ...
-1
votes
2answers
33 views

Let $f$ be an entire function such that image of $f$ lies in $L=\{2+iy:y\in R\}$ if $f(2+i)=2+i$ then show that $f(z)=2+i$ for all $z \in C$

Let $~f~$ be an entire function such that image of $~f~$ lies in $L=\{2+iy:y\in R\}$ if $f(2+i)=2+i$ then show that $f(z)=2+i$ for all $z \in C$
10
votes
1answer
109 views

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 $...
1
vote
1answer
64 views

Order of the entire function $f(z)=\sin(z)$

I want to find the order of the entire function $f(z)=\sin(z)$. I have this result Let $$f(z)=\sum_{n=0}^\infty a_n z^n$$ be an entire function, non-constant and with finite order. Then, the ...
1
vote
3answers
59 views

Prove if $k^\text{th}$ derivative of an entire function $f$ is polynomial, then $f$ itself is polynomial. Where's my mistake?

The exact wording of the question is as follows: Let $f$ be an entire function. Suppose there exists a positive integer $k$ such that $k^\text{th}$ derivative $f^{(k)}$ is a polynomial. Prove that $...
9
votes
1answer
98 views

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 ...
5
votes
1answer
70 views

Entire function problem: translation

Let $f$ be an entire function such that $f\circ f$ has no fixed points. Prove that $f$ is a translation $$z\mapsto f(z)=z+b \qquad (b\neq 0)$$ Firstly, we prove that there exists a constant $c\in \...
1
vote
2answers
59 views

Is $f(z)$ entire?

I am trying to determine if the the following is entire $$f(z)= \begin{cases} e^{-z^{-4}} & z\neq0 \\ 0 & z=0\\ \end{cases} $$ My attempt: Consider $z\ne 0$. $f(z)=e^{-z^{-4}...
1
vote
1answer
43 views

Entire extension of $f(x+y)=g(x)g(y)-h(x)h(y)$

I am currently working on the following practice question for complex analysis; Assume $f(x+y)=g(x)g(y)-h(x)h(y)$ for all $x,y\in \mathbb{R}$ and some entire functions $g,h$. Show that there exists a ...
3
votes
1answer
72 views

If $f$ is entire and $\left|f\left(\frac{1}{\ln{(n+2)}}\right)\right|<\frac{1}{n}$ for every positive integer $n$ then $f=0$

Let $f(z)$ be an entire function satisfying $$\left|f\left(\frac{1}{\ln{(n+2)}}\right)\right|<\frac{1}{n}$$ for every $n\in\mathbb{N}).$ Show that $f(z)=0.$ I need some help for this ...
1
vote
2answers
125 views

Proof of Casorati-Weierstrass [closed]

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch9.2 I have questions on the proof of Casorati-Weierstrass Theorem (Thm 9.7) - If $z_0$ is ...
0
votes
2answers
56 views

Is there a characterization of entire functions with image $\Bbb C \setminus \{0\}$? [closed]

Do we have a characterisation of entire functions with image $\Bbb C \setminus \{0\}$? If not, is there an example of such a function that is not in the form of $\exp(g)$ for some entire function $g$?...
0
votes
2answers
63 views

If $f$ is entire and $f(z)/z$ is bounded, then $z = 0$ is a removable singularity of $f(z)/z$.

Let $f$ be an entire function with $\sup_{z\in\mathbb{C}}|f(z)/z|<\infty$. Show that $z=0$ is a removable singularity of $g(z):=f(z)/z$. To prove the claim, I need to show that $0 = \lim_{z\to 0}(...
12
votes
1answer
241 views

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 ...
0
votes
0answers
38 views

Order of Entire function, Shifted

Let $f$ be an entire function, and $a \in \mathbb{C}$. Prove that order of $f(z+a)$ is equal to the order of $f(z)$. My attempt: Denote $g(z)=f(z+a)$. I tried to take bounds for $f$ and $g$ in ...
0
votes
1answer
38 views

Is it possible for a vanishing entire function to have constant modulus “for a while”?

I am wondering if it is possible for an entire function $\phi$ to admit a direction $v\in\Bbb C$ such that $\phi(tv)\to0$ for $t\to+\infty$ but $|\phi|$ is constant "for a while" in that direction, ...
3
votes
2answers
60 views

Prove that an entire function with $Imf\le (Ref)^2$is constant

This is an old question from Ph.D Qualifying Exam of Complex Analysis. Without using Picard's theorem directly, prove that if $f$ is an entire function such that $\text{Im}f(z)\le (\text{Re}f(z))^2$ ...
2
votes
1answer
79 views

Extension of real or imaginary part of entire function to complex valued functions

In this forum post on MSE we were trying to find the imaginary part of an entire function given it's real parte. One of the answers called for this method $$f(z) = 2u\left({z\over 2},-{{iz}\over 2}\...
0
votes
1answer
32 views

Entire functions with controlled growth in an angular region and bounded elsewhere

I am trying to construct a holomorphic function $U \to L^2(\mathbb R)$ for some open $U \subseteq \mathbb C$, which is not locally bounded by an $L^2$ function. (I believe this should exist.) To do ...
4
votes
0answers
40 views

Is an entire function upper bounded by a polinomial a polynomial? [duplicate]

I whant to know if the following proposition holds: If $f$ is entire, and $p$ is a polynomial such that $$|f(z)| \leq |p(z)|,\forall z \in \mathbb{C}, $$ then there is $c \in \mathbb{C}; f(z)=cp(z)$...
-1
votes
1answer
64 views

$f$ and $g$ entire and $f(1/n) = g(1/n)$ $\forall n\in \Bbb{N}$, then $f=g$ in $\Bbb{C}$?

If functions $f$ and $g$ are entire such as: $f(1/n) = g(1/n)$ $\forall n\in \Bbb{N}$, then $f=g$ in $\Bbb{C}$ Is it true or not? In order to answer this question I want use a statement which says: ...
1
vote
1answer
27 views

Problems to understand complex infinity limits.

I was thinking a problem like this: Imagine you have an entire function $f(z)$. Then u can write $f(z)=\sum_{n\ge0}a_nz^n$ for Taylor; Wich is a serie. Then i consider $\lim_{z \to \infty}f(z)$. And ...
4
votes
1answer
56 views

entire function with bounded multiplicity is a polynomial [duplicate]

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function. Let $n\in\mathbb{N}$ and suppose that $$\forall w\in\mathbb{C}:\#\{z\in\mathbb{C}:f(z)=w\}\leq n$$ In words, every complex value is attained ...
2
votes
1answer
19 views

entire function that is non trivially lower bounded outside a ball is constant

Let $f$ be an entire function. Let $m:[0,\infty)\to\mathbb{R}$ be a function defined by $m(r)=\min\limits_{|z|=r}|f(z)|$. Suppose that $\lim\limits_{r\to +\infty}m(r)$ exists and equals to a ...
1
vote
0answers
44 views

Existence of entire function s.t. for all $z\in\mathbb{C}$ with $|z|\geq 100:|f(z)|=|z|+1$ [duplicate]

Is there a function $f$ such that $f$ is entire and for all $z\in\mathbb{C}$ with $|z|\geq 100:|f(z)|=|z|+1$. An observation is that $\lim\limits_{z\to\infty}f(z)=\infty$ and so $f$ must be a ...
0
votes
1answer
23 views

existence of non constant entire function that is bounded outside an annulus

Is there a function $f$ which is not constant, analytic in $\mathbb{C}\setminus\{0\}$ and $$\forall z\in\{w\in\mathbb{C}: 0<|w|<\frac{1}{100} \lor |w|>100\}:|f(z)|<800$$ Probably not. ...
1
vote
1answer
75 views

Analytic continuation on disconnected sets

Let $f(z,s)$ be a complex-valued function in two complex variables, entire in $s$, and let $$F(s)=\int_\Omega f(x+iy,s)\,dx\,dy$$ where $\Omega$ is an open subset of $\mathbb{C}$. Assume that the ...
4
votes
1answer
58 views

entire function that maps real line to itself is linear

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function be such that $\mathbb{R}=f^{-1}(\mathbb{R})$. Show that $f$ is linear. i.e. $$\exists\ a,b\in\mathbb{R}:f(z)=az+b$$ Hint I think that $f$ must ...
2
votes
2answers
79 views

If $f(z)$ is entire function such that $|f(z)f'(z)| \leq 1$ then $f(z)$ is constant

If $f(z)$ is entire function such that $|f(z)f'(z)| \leq 1$ then $f(z)$ is constant. Choose $g(z) = \frac {(f(z))^2}{2}$, $g'(z) = f(z)f'(z) \implies |g'(z)| \leq 1$(given condition) So $g'(z)$ is ...
0
votes
1answer
37 views

Find all entire functions with real part greater than 1 and imaginary part less than -1

I'm stuck on the following question: Find all entire functions $f$ such that $Re(f) >1$ and $Im(f)<-1$. Unfortunately this doesn't seem like a problem that can be quickly solved by ...
2
votes
1answer
34 views

Liouville's theorem non-entire function

I have the following problem which I don't understand. Find all entire functions $f$ such that $$|f(z)|\ge \frac{1}{1+|z|^{2017}}=g(z), \; \forall z \in \Bbb{C}$$ The answer says that $f$ has to ...
1
vote
2answers
76 views

What conclusion can we make if $f$ is a holomorphic function?

Suppose $f$ is holomorphic in an open neighbourhood of $z_0 \in \Bbb C$. Given that the series $$\sum\limits_{n=1}^{\infty} f^{(n)} (z_0)$$ converges absolutely, we can conclude that $(1)$$\ \ \ \...
2
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0answers
65 views

Finding all entire functions $f(z)$ for which $|f(z)|\leq M.|\sin z|$

I've seen the other way for this question (Entire functions $f$ for which there exists a positive constant $M$ such that $|f(z)|\le M|\cos z|$), I just want to make sure that the same argument can be ...
1
vote
1answer
56 views

a question about specific entire function

Find all the entire functions $f(z)$ for which there exists positive constants $M, \:R$ and a positive integer $n$ such that $$|f(z)|\geq M.|z|^n\:\:\text{whenever}\:\:|z|>R.$$ What if the ...
1
vote
0answers
23 views

Possible existence of entire function

Is there an entire function $f$ such that $f(z)=1/z$ for all z in the plane punctured at $0$? I am viewing it as a problem to analytically extend the function $1/z$ in the entire plane which is not ...
1
vote
1answer
72 views

Showing $f(z)$ is a rational function

I could prove that if $g(z)$ is bounded and analytic except at a finite number of points in the complex plane $\mathbb{C}$, $g(z)$ must be constant. First I thought I can use the above to prove the ...
3
votes
1answer
393 views

Entire function mapping a parallelogram onto another one is a degree 1 polynomial

I am reading Bak and Newman's Complex Analysis and I can't figure out how to do the following exercise: Prove that an entire function which maps a parallelogram onto another parallelogram, and maps ...
4
votes
1answer
58 views

Entire function having the property [duplicate]

Let $f$ be an entire function. Consider $A=\{z \in \Bbb{C} : f^{(n)}(z)=0\; \text{for some}\; n \in \Bbb{N}\}$. Then how to prove if $A=\Bbb{C}$, then $f$ is a polynomial ? This is same as proving ...
1
vote
1answer
88 views

Examples of different type of entire functions

I try to answer following question. I would like to find a general approach for part b, since I think any entire function with $n $ roots is either polynomial of degree $n$ or is a product of ...
0
votes
0answers
61 views

Uniformly continuous entire functions

Find all entire functions that are uniformly continuous on the complex plane. I think the answer must be the the linear polynomials, but nor sure, since neither of polynomials of degree greater than ...
0
votes
1answer
73 views

Entire function $f$ with infinitely many zeros [closed]

Let $f$ be an entire function, non constant with the property: The set $\{w : f(w)=0\}$ has ininite elements. Show that for all c there exist a sequence $\{z_n\}$ such that $z_n\to\infty$ and $f(z_n)\...
0
votes
1answer
106 views

Essential singularity at infinity of exponential function.

Show that if $f(z)$ is a non-constant entire function, then $e^{f(z)}$ has an essential singularity at $z=\infty$. This is my approach: By Liouville's theorem I know that if $f$ is a non-constant ...
1
vote
1answer
46 views

Holomorphic function $f: \Bbb C \rightarrow \Bbb C$ such that $|f(z)|\leq C|\negthinspace\cos(z)|$ for all $z \in \Bbb C$ [duplicate]

The Question: Let $f:\Bbb C \rightarrow \Bbb C$ be a holomorphic function (i.e. an entire function) such that $|f(z)|≤C|\negthinspace\cos(z)|$ for all $z \in \Bbb C$, where $C \in \Bbb R$ is a ...