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Questions tagged [entire-functions]

This tag is for questions relating to the special properties of entire functions, functions which are holomorphic on the entire complex plane. Use with the tag (complex-analysis).

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Growth order of $e^{p(z)}$ for a complex polynomial, $p(z)$

Let $p(z):\mathbb{C}\to\mathbb{C}$ be a degree $m$ polynomial, and consider $e^{p(z)}$. I'm wondering whether it's true that the order of $e^{p(z)}$ is equal to the degree of $p(z)$. My working ...
Ty Perkins's user avatar
2 votes
2 answers
54 views

An entire function such that $|\operatorname{Re}f(z)|>0.001 |\operatorname{Im}f(z)|$ must be constant.

I was going over some previous qualifying exams to prepare for my own, and came across the following problem: Problem. Suppose that $f:\mathbb{C}\to\mathbb{C}$ is entire and has the property that for ...
martianpolarbear's user avatar
0 votes
0 answers
38 views

Entire function maps bounded sets to bounded sets

I was doing some problems in Complex Analysis… And I came across this. Let $f: \mathbb{C} \to \mathbb{C}$ be entire. Then for any bounded set $B$, f ($B$) is bounded. Now I know that if an entire ...
Maths wizard's user avatar
2 votes
1 answer
78 views

Find all entire functions such that $|f(z+z')|\leq |f(z)| + |f(z')|$, for all $z,z'\in\mathbb{C}$

Find all entire functions such that $|f(z+z')|\leq |f(z)| + |f(z')|$, for all $z,z'\in\mathbb{C}$ In particular, let $z=z'$ yields $|f(2z)|\leq2|f(z)|$. This gives that $\frac{f(2z)}{f(z)}=c, $ for ...
Derewsnanu's user avatar
0 votes
0 answers
45 views

Show that any entire function satisfying the given conditions is a constant. [duplicate]

Let $f$ be an entire function. Consider the set $$S=\bigg\{re^{i\theta}: r>0, \ \frac{\pi}{4}\leq \theta\leq \frac{7\pi}{4}\bigg\}\cup \{0\}.$$ It is given that $f$ is bounded on the set $S$. ...
PAMG's user avatar
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7 votes
2 answers
208 views

Entire function bounded in every horizontal line, and has limit along the positive real line

Let $f(z)$ be an entire function (holomorphic function on $\mathbb{C}$) satifying the following condition: $$|f(z)|\leq \max (e^{\text{im}(z)},1 ),\ \forall z\in\mathbb{C}$$ $$\lim_{\mathbb{R}\ni t\...
likeeatingoctopus's user avatar
0 votes
0 answers
90 views

If $f(z)=e^z$ then $f(A)=?$...

Let A denote the set of all complex numbers lying within the square having vertices $0,1,\pi i,\pi i +1$. let $f(z)=e^z$ then $f(A)=$ (1) $ \{ \omega : 1\leq \omega \le e \}$ (2) $\{\omega : |\omega| \...
math student's user avatar
  • 1,181
1 vote
1 answer
77 views

Let $f,g$ entire functions both $\neq 0$ such that $|f|^2\leq |g|^3$, what can we say about their zeroes?

$f,g$ entire, both $\neq 0$ such that $|f|^2\leq |g|^3$ My reasoning is the following: Both can have zeroes If $g$ has a zero at $z=a$, then $f$ must also have it, but we could find a neighborhood ...
Mateo's user avatar
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1 vote
1 answer
95 views

Existence or non-existence of an entire funtion

Does there exist an entire function such that it transforms the real axis in the imaginary axis, and the imaginary axis into the parabola $y^2=x$? I have been trying to solve this question for my ...
flgflg71's user avatar
2 votes
3 answers
134 views

Modulus of function is function of modulus

Is there a way to characterize all non-constant entire functions such that $|f(z)| = f(|z|)$ ? The monomials work, but I can't think of any other functions. So far, I have managed to show that $f$ is ...
ed268's user avatar
  • 69
0 votes
1 answer
82 views

If $f$ is entire and nowhere zero, then there exists an entire function $g$ such that $f(z)=g(z)^2$

Problem: Show that if $f$ is entire and nowhere zero, then there exists an entire function $g$ such that $f(z) = g(z)^2$ for all $z\in \mathbb{C}$. Preliminary: To get some practice in, I tried to ...
mathlover314's user avatar
0 votes
0 answers
38 views

Entire function with bounded sequence of distinct real numbers

Problem: Suppose that $f$ is an entire function and that there is a bounded sequence of distinct real numbers $\{ a_n \}_{n\in \mathbb{N}}$ such that $f(a_n)\in \mathbb{R}$ for all $n\in \mathbb{N}$. ...
mathlover314's user avatar
0 votes
1 answer
47 views

Prove that order of function certain order and type properties

Currently reading the Levin book on Entire functions, Taking my time practicing by trying to prove the following statements, I have wasted a couple of days fighting with these folks: $p_{fg} \le max(...
Awesome Man's user avatar
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0 answers
68 views

Order zero entire function.

The order of an entire function $f$ is defined by the smallest $\alpha$ in $[0,\infty]$ such that $f(z)\ll e^{|z|^{\alpha+\epsilon}}$ for all $\epsilon>0$. An equivalent definition is, if $\limsup_{...
Factorial_zero's user avatar
0 votes
1 answer
55 views

How do I determine order of growth of periodic function? (sin)

I am self-reading Levin's book, on Entire functions. There is a task that needs to be verified by the reader, and I cannot reproduce it. It states: "Verify that $sin(Az)$ is of order $p = 1$ and ...
Awesome Man's user avatar
0 votes
0 answers
21 views

Fourier transform of entire function of exponential type $\nu$ equal to zero outside $\Delta_\nu$

Let $\Delta_\nu=\{|x_j|<\nu,j=1,...,n\}$ definition of entire function of exponential type $\nu$. Let $n\in\mathbb{N},~\nu>0 $. A function $g: \mathbb{C}^n \to \mathbb{C} $ is an entire function ...
DJ fade's user avatar
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1 vote
1 answer
86 views

Showing that an entire function is constant

Let $f$ be an entire function on $\mathbb{C}$ such that (a) $f$ has no zeros, and (b) $f^{-1}(1)$ is finite. How can I prove that $f$ is constant? I was thinking of using Little Picard, but this gets ...
user0134's user avatar
  • 404
0 votes
0 answers
128 views

$g(z) = \sum_{n=0}^{\infty} \frac{g(n)}{(n+1)!} z^n$ with $0 \leq g(n)$?

Im looking for functions $g(z)$ such that $$g(z) = \sum_{n=0}^{\infty} \frac{g(n)}{(n+1)!} z^n = g(0) + \frac{g(1)}{2} z + \frac{g(2)}{6} z^2 + \frac{g(3)}{24} z^3 + ...$$ and $g(n)$ are all positive ...
mick's user avatar
  • 16k
1 vote
0 answers
75 views

For arithmetical periodic function $f$, if $\sum_{r=1}^k f(r)=0$, then $S=\sum_{n=1}^\infty \frac{f(n)}{n^{s}}$ converges

[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 12, question 1(b)] Let $f(n)$ be an arithmetical function which is periodic mod $k$. If $$\sum_{r=1}^k f(r)=0$$ then prove that the ...
Sayan Dutta's user avatar
  • 8,911
1 vote
1 answer
88 views

A question on the coefficients of the Taylor's series of an entire function

Let $f:\mathbb{C} \to \mathbb{C}$ be defined by $$f(z)=(1-z)e^{\big( z+ \frac{z^2}{2} \big)}=1+ \sum_{n=1}^{\infty} a_nz^n.$$ Then, which of the following is FALSE? $f'(z)=-z^2e^{\big( z+ \frac{z^2}{...
MathRookie2204's user avatar
3 votes
1 answer
66 views

Extending a holomorphic function on $\mathbb{C}\backslash K$ to an entire function

Consider a compact set $K\subset \mathbb{R}$ with positive measure (i.e. $\mu(K)>0$), and for $z\in\mathbb{C}\backslash K$, define the holomorphic function $f$ on $\mathbb{C}\backslash K$ by \begin{...
IIIsomorphiii's user avatar
4 votes
1 answer
131 views

Is an entire function with this property necessarily a polynomial?

Given an entire function $f(z)$, define $\displaystyle m(r):=\inf_{\vert z\vert=r}\vert f(z)\vert$. If $f$ satisfies $$\limsup_{r\to+\infty} m(r)=+\infty,$$ can we assert that $f$ is a polynomial? ...
Hamming Zhao's user avatar
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0 answers
38 views

Proving an entire function has at least one zero provided that it maps the open disk $D(0,R)$ to $D(0,M)$

Suppose $f$ is an entire function such that $f(D(0,R)) \subset D(0,M)$ for some $R,M > 0$. Is it possible to prove that f must have at least one root in the open disk $D(0,R)$? I really struggle at ...
oti nane's user avatar
3 votes
2 answers
165 views

Non-polynomial entire function with finitely many zeros tends to zero on circles

Suppose $f$ is entire with finitely many zeros. Assume $f$ is not a polynomial. Let $m(r) = \inf_{|z| = r} |f(z)|$. I want to show that $$\lim_{r \rightarrow \infty} m(r) =0.$$ I want to note that a ...
Dalop's user avatar
  • 715
0 votes
0 answers
23 views

Non-constant entire function without taking any real values [duplicate]

Does there exist a non-constant entire function $f$ such that $f(\mathbb{C})\subseteq \mathbb{C}\smallsetminus \mathbb{R}?$ Can we show that such a function doesn't exist by combining the two facts ...
nkh99's user avatar
  • 459
2 votes
1 answer
79 views

If f is entire, find all g entire such that $|f(z)-g(z)|\leq |f(z)+g(z)|$ for every $z\in \mathbb{C}$

I have the following problem: If f is entire, find all g entire such that $|f(z)-g(z)|\leq |f(z)+g(z)|$ for every $z\in \mathbb{C}$ My initial thought was to write $f(x+iy)=u_1(x,y)+iv_1(x,y)$ and $g(...
obitobi_tobias's user avatar
11 votes
5 answers
887 views

Find all entire functions that satisfy the following equality

For $n\geq 2$ I need to find all entire functions $f:\mathbb{C} \rightarrow \mathbb{C}$ Such that $f(z^n)=f(z)^n$ for all complex numbers $z$. I've tried expanding it's series at 0 and have found some ...
Confused Integrator's user avatar
0 votes
0 answers
54 views

Entire function with real part having limit zero at infinty

Describe all the entire functions,$f(z)$ with $Re(f(z))$ tending to $0$ as $z\to \infty$. My approach: $lim_{z\to \infty}Re(f(z))=0$ implies $lim_{z\to \infty}e^{Re(f(z))}=1$. Thus for some $N\in \...
nkh99's user avatar
  • 459
0 votes
0 answers
43 views

A question related to entire function

$f(z)$ be an entire function such that $|zf(z)-1+e^z| \leq 1+|z| \forall z \in \mathbb{C}$ then what can we say about such function value at $z=0$ and its derivative at $z=0$? I know the basic ...
Maths's user avatar
  • 182
1 vote
2 answers
94 views

Image of imaginary axis under $f$ is either real axis or imaginary axis

Let $f$ be a nonzero entire function such that $f(0) = 0$ and $f(\mathbb{R}) \subseteq \mathbb{R}$. Show that if the image of the imaginary axis under $f$ is contained in a line, then that line must ...
R_Squared's user avatar
  • 273
0 votes
0 answers
44 views

On the topology of the space of entire functions.

I have a doubt. When we have a Banach space $X$, by definition, each element $f$ in $X$ has a norm $\left\|f\right\|_X$. On the other hand, I understand that the space of entire functions $\mathcal{O}(...
eraldcoil's user avatar
  • 3,574
1 vote
0 answers
38 views

exponential type and distribution of zeros

Let $H\doteqdot\{z\in\mathbb{C}\colon \operatorname{Re}z>0\}$ denotes the half plane and $f\colon H\rightarrow\mathbb{C}$ be a holomorphic function, with the property that $$f(n)=0\qquad\forall n\...
SprtWhitebeard's user avatar
2 votes
0 answers
51 views

Bounds on asymptotic behavior of Taylor coefficients of entire functions?

It is relatively straightforward to show that when an analytic function has a simple pole, then the coefficients $a_n$ of the Taylor series asymptotically follow $$ \left|a_n\right| \sim O\left(z_0^{-...
sasquires's user avatar
  • 1,650
0 votes
0 answers
21 views

A problem about entire function and polynomial [duplicate]

Let $p(z)$ be a polynomial not identically zero on $\Bbb C$ with degree n. Suppose that $|p(z)|\le|z|^n$ for any z satisfies $|z|\le1$. Then there exists a complex number $c\in \Bbb C$ such that $p(z)$...
QIRUN CONG's user avatar
4 votes
3 answers
422 views

Prove an entire function is constant on complex plane

Let $f(z)$, $F(z)$ be two analytic functions on $\Bbb C$ satisfies $f(z)=F(\overline{f(z)})$. Here $\overline{f(z)}$ is the complex conjugate of $f(z)$. Prove that $f(z)$ is constant on $\Bbb C$. I ...
QIRUN CONG's user avatar
2 votes
0 answers
49 views

Explicit example of an entire function with simple zeros at precisely the square roots of the positive half integers

I'm looking for an entire function with the property that $f(\sqrt{n+1/2}) = 0$ for $n=0,1,2,\dots$, all of which are simple zeros and $f$ has no other zeros. I know that such functions exist and can ...
Tree Wizard's user avatar
2 votes
1 answer
104 views

Given $\frac{H(z)}{\prod_{n=1}^\infty \left(1-\left(\frac{z}{\lambda_n}\right)^2\right)} = \phi(z),$ determine if $\phi(z)$ is an entire function.

Let us consider $\lambda_n=n-\frac14$ and define $$\frac{H(z)}{\prod_{n=1}^\infty \left(1-\left(\frac{z}{\lambda_n}\right)^2\right)}=\phi(z) $$ where $H(z)$ is an entire function. I would like to ...
Mark's user avatar
  • 7,861
4 votes
1 answer
260 views

Conditions to calculate an integral through a series expansion

Let $f,g:\mathbb{R}\longrightarrow\mathbb{R_+}$ be Lebesgue integrable functions. We can show that if $g$ has compact support and $f$ has a Maclaurin series that converges absolutely in the support of ...
P.S. Dester's user avatar
0 votes
0 answers
32 views

Entire functions that have a zero in the disk $\{z \colon |z| < r\}$ or constant functions.

Question: Let $f\colon \mathbb{C} \to \mathbb{C}$ be an entire function. Suppose that there are real constants $r, R > 0$ so that $|f(z)| > R$ for all $z \in \mathbb{C}$ with $|z| > r$. Show ...
L-JS's user avatar
  • 715
0 votes
1 answer
83 views

An entire function satisfying $f(z)=f(z+\xi t)$ for all $t\in\mathbb{R}$ and some $\xi\in\mathbb{C}$

Let $f$ be entire function satisfying $f(z)=f(z+\xi t)$ for all $t\in\mathbb{R}$, some $\xi\in\mathbb{C}\neq 0$ and all $z\in\mathbb{C}$. I would like to show that $f$ is constant. Since $f(z+\xi t)-f(...
Dispersion's user avatar
  • 5,531
1 vote
1 answer
121 views

Prove that a function is a polynomial of degree at most one given $v(x,y) \geq x$

The question is Let $f(z) = u(x,y) + iv(x,y)$ be an entire function satisfying $v(x, y) ≥ x$ for all $z = x + iy$. Then show that f(z) is a polynomial of degree at most one. I know that I have to ...
SHIVODIT's user avatar
0 votes
0 answers
51 views

Want an example of entire function of order $\sqrt 2$ [duplicate]

I want an example of entire function with order $\sqrt 2$. If $f$ is a entire function of finite order $\rho$ then $\rho=\lim_{R \to \infty} \sup_{r \geq R} \frac{\log \log M(f,r)}{\log(r)}$, where $M(...
MAS's user avatar
  • 10.7k
2 votes
0 answers
187 views

An entire function $f(z)$ that is real if and only if $z$ is real

I would like to prove that if $f(z)$ is an entire function that is real if and only if $z$ is real, then $f'(z)\neq 0$ for all real $z$. I first wrote $$f(z)=u(x,y)+iv(x,y)$$ with $z=x+iy$ and ...
Dispersion's user avatar
  • 5,531
0 votes
1 answer
67 views

Polya's representation (Boas Book)

I am studying the Borel transform. If $f(z)$ is an entire function $f(z)=\sum_{n=0}^{\infty}a_nz^n$, then, the Borel transform is $F(z)=\sum_{n=0}^{\infty}\frac{n! a_n}{z^{n+1}}$. Question: why $\...
eraldcoil's user avatar
  • 3,574
1 vote
1 answer
233 views

Continuity of maximum modulus function $M(r)=\max_{|z|=r}|f(z)|$

I am looking to prove that the maximum modulus function $$M(r)=\max_{|z|=r}|f(z)|$$ is continuous on $[0, \infty)$ for $f$ an entire function. My idea was to use the representation of $f$ as a power ...
Dispersion's user avatar
  • 5,531
0 votes
1 answer
50 views

All $f\in H(\mathbb{C})$ with $Re(f(z))=u(z)=u(x+iy)=x^2-y^2+2x+1$

In my complex analysis course I'm supposed to compute all entire functions with the given requirement for the Real part. How do I work on this? I hav no clue how to work this out. Can someone please ...
MilesDefis's user avatar
1 vote
0 answers
65 views

Question of the order of growth

In Stein and Shakarchi's book on Complex analysis, there's a theorem (p.138) that states: Theorem: Let $f$ be an entire function that has an order of growth $\leq \rho$. If $z_1,z_2,\dots$ denotes ...
Richard D's user avatar
  • 157
0 votes
0 answers
50 views

Extended Liouville to Real Part of Entire Functions

I want to prove if $f$ is entire function and $u(x,y)$ is real part of $f$ satisfying \begin{align*} |u(x,y)|\leq C|z|^n, z=x+iy \end{align*} then $f$ is polynomial with degree at most $n$. If $f(z)=\...
Laurence PW's user avatar
1 vote
1 answer
154 views

Find all the entire functions satisfying $|f(z)|\leq-\ln(|z|^2)+|z|^2$, $f(0)=i$.

Problem. Find all the entire functions $f:\mathbb{C}\rightarrow\mathbb{C}$ such that $$|f(z)|\leq-\ln(|z|^2)+|z|^2, f(0)=i$$ Since $-\ln(|z|^2)+|z|^2\leq2|z|^2,$ we have$|f(z)|\leq2|z|^2$. hence $f$ ...
topst's user avatar
  • 149
0 votes
2 answers
34 views

Entire function taking real values on two parallel lines is periodic [duplicate]

Given an entire function $f$, I have that $f(ix)$ and $f(1+ix)$ are real for all $x \in \mathbb{R}$. I want to show that $f(z) = f(z+2)$ for all $z \in \mathbb{Z}$. I've thought about considering the ...
Dalop's user avatar
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