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Questions tagged [entire-functions]

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2
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3answers
81 views

Show $\frac{f(z)}{\sin(\pi z)} = \sum_{-\infty}^{\infty} \frac{(-1)^nf(n)}{(z-n)}$

Let $f(z)$ be entire function satisfying $|f(x+iy)| \leq Ce^{a|y|}$ for $C > 0$ and $a \in (-\pi, \pi).$ Show $\frac{f(z)}{\sin(\pi z)} = \sum_{-\infty}^{\infty} \frac{(-1)^nf(n)}{(z-n)}$ All I ...
-1
votes
3answers
52 views

Is $f$ is constant ? Yes/No [duplicate]

Is the following statement is true/false let $f$ be entire function and if $\;\operatorname{Re}f(z) = <\operatorname{Im}f(z)$ then $f$ is constant . My attempts : No . I take $f(z) = z$...
0
votes
1answer
22 views

Regarding an entire function being affine

I have been reading this article A characterization of multiplicative linear functionals in Banach algebras and got stuck in the middle of the proof of theorem 1.2 on page 217. In the 3rd line from ...
0
votes
2answers
43 views

Prove/disprove the existence of an analytic map using Identity Theorem [closed]

I want to prove/disprove the following statement using Identity Theorem. Does there exists an analytic map $f: \Bbb C \to \Bbb C$ such that $f(z)=z$ for all $z$ such that $|z|=1$ and $f(z)=z^2$ ...
1
vote
1answer
41 views

Understanding infinite products convergence

I am reading about a convergence problem with infinite products and I am told: Any finite sequence $\{c_{n}\}$ in the complex plane has an associated polynomial p(z) that has zeroes precisely at ...
0
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1answer
38 views

Understanding entire functions

I am trying to understand the Weierstrass factorization theorum and I was told that the following is true for entire functions... Any finite sequence $\{c_{n}\}$ in the complex plane has an ...
2
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0answers
39 views

Reference Request: Parabolic Cylinder Functions (decaying at $+ \infty$), Order and Type in the Parameter

Background Reciprocal Gamma function as an entire function The reciprocal Gamma function, \begin{equation} \begin{split} \frac{1}{\Gamma(z)} &= e^{\gamma z} \prod_{n = 1}^{\infty} \left( 1 + \...
0
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3answers
45 views

Examples of $f:\mathbb{C} \rightarrow \mathbb{C}$ entire function with $|f(z)| \leq C|z|^2$ $\forall z \in \mathbb{C}$

Find the most general function $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f$ is entire and $\exists C > 0$ with $|f(z)| \leq C|z|^2$ $\forall z \in \mathbb{C}$. I'm really not sure ...
2
votes
3answers
87 views

If $f$ is an entire function that goes to infinity, $f$ is a polynomial

I am asked to show that if $f$ is entire with the property that $\lim_{z\to\infty} f(z)=\infty$, then $f$ must be a polynomial. However, I feel as if I am missing something. $e^z$ is entire (...
0
votes
2answers
36 views

is the real part of a holomorphic function holomorphic?

Say we have some entire function $f:\mathbb{C} \rightarrow \mathbb{C}$. Does this guarantee that the function $Re(f)$ will also be entire?
1
vote
3answers
60 views

Find all entire functions that satisfy following inequality [closed]

Find all entire functions that satisfy following inequality: $$ |f(z)| \leq |z| e^{\Re(z)} $$ for all $ z \in \Bbb C $
1
vote
1answer
22 views

Affine transformations that correspond to entire complex functions

I assume the following is a standard consideration and question, but I don't know how to prove it: There is a trivial one-to-one correspondence between affine transformations $f = (f_x,f_y)$ of the ...
1
vote
1answer
27 views

Find a and b values so a given function is harmonic

The Problem: Let $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ be an entire function. If $g(z)=au^2(x,y) - bv^2(x,y)$ find values for a and b so $g(z)$ is a harmonic function. My attempt to find a solution: Well ...
3
votes
1answer
39 views

Proving the injectivity of an entire function

This is an exercise from a book I was reading: Suppose that an entire function $f:\mathbb{C} \to \mathbb {C}$ satisfies $$f(z)\in \mathbb{R} \iff z\in \mathbb{R} .$$Prove that $f'$ does not ...
0
votes
1answer
62 views

Existence of an entire function

my Complex Analysis final is in a couple of days and i'm struggling with this question - "Is there an entire function $f$ that satisfies $|f(z)| = |z| + 1$ for every $z$ in the complex plane for ...
1
vote
2answers
40 views

Existence of analytic function having different values on $|z|=1$ and $|z|=2$

Does there exist an entire function satisying $f(z)=z$ for $|z|=1$ and $f(z)=z^2$ for $|z|=2$? I think no, but what argument would do here? Would it be maximum modulus or Liouville theorem? A ...
0
votes
1answer
21 views

holomorphic functions with special condition

i want to find all holomorphic funtions $ f: \mathbb{C} \rightarrow \mathbb{C}$ with $ f(0)=0$ and $ f(f(f(f(z))))=z \forall z \in \mathbb C$ I would say rotations like $f(z)= |z| e^{i(arg z + \frac{...
0
votes
1answer
35 views

bijective holomorphic entire functions [closed]

I want to find all entire bijective holomorphic functions $f:\mathbb{C}\rightarrow \mathbb{C}$. There are identity function -polynomials with odd degree Can I find an a way more abstract function ...
1
vote
1answer
22 views

Show that $f$ is a polynomial regarding a given question

This is in reference to this $f:\mathbb{C}\rightarrow\mathbb{C}$ entire function and $f(z)=u(x)+iv(y)$ then $f$ is a polynomial Since $f$ is entire it satisfies Cauchy Riemann so $u_x=v_y \& 0=...
0
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1answer
31 views

Showing the function is Entire

Suppose $f$ be a continuous function on $[0,1]$. Define $g(z)= \int_0^1 f(t)cos(tz) dt$. Prove that $g(z)$ is an entire function. I'm a beginner in complex analysis and read upto cauchy integral ...
13
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2answers
267 views

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 $...
0
votes
3answers
45 views

Show that entire function is constant.

$f$ is an entire function and $f(z) = i$ when $z = \left(1+ \frac kn \right)+i$ for every positive integer $k$. $n$ is fixed. How can we conclude that $f$ is constant? Is there any result about ...
2
votes
1answer
32 views

complex analysis : Growth

Let $f$ holomorphic on $C$. I'm looking for a counter exemple to : If $\sup_{|z|=r} |Re(f)| = O(r^{d})$ then $\sup_{|z|=r} |f| = O(r^{d})$ Actually, I'm wondering if I can find a entire function ...
0
votes
1answer
46 views

Is there is an entire function $f$ such that $f(1)=\pi$ and $f'(z)=|z|f(z)$ for all $z\in\mathbb C$?

Problem: Is there is an entire function $f$ such that $f(1)=\pi$ and $f'(z)=|z|f(z)$ for all $z\in\mathbb C$? Question: Is the following correct? On the unit circle line $\partial K_1(0)$ the above ...
8
votes
2answers
249 views

Why is $\cos \sqrt z$ entire but $\sin \sqrt{z}$ isn't?

I've been trying to formulate a way of comparing these two functions, in order to find out why the function $\sin \sqrt z$ is not entire, but I couldn't find a good way of doing that. What I tried so ...
1
vote
1answer
46 views

Entire function either constant or has a zero

Suppose that a Taylor series of an entire function $f$ converges to $f$ uniformly in $\mathbb{C}$. How do I show that either $f$ is a non-zero constant or $f$ has a zero? I was thinking about either: ...
0
votes
1answer
104 views

prove that $\;f(z) = g(z)\;$ for all $z\in \mathbb{C}$

The problem is: Let $f(z)$ and $g(z)$ be entire such that $r>0 ,\; f(z) = g(z)$ for all $|z| < r. $ Prove that $f(z) = g(z)$ for all $z \in \mathbb{C}$ Does that mean I should prove ...
0
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0answers
30 views

Cauchy estimates and the maximum principle

I am stuck at a step in a problem where I've been given hints. The hints confuse me, so I'm hoping for some help. Assuming that $f$ is entire and that $$|f(z)| \leq |z| + 1/|z|, $$ for $z \in \mathbb{...
2
votes
1answer
49 views

Complex Analysis : Prove f is constant if f is entire and $f(z)\notin \left[0,1 \right]$ [duplicate]

We are currently studying Liouville's Theorem and had a few questions which weren't too hard, but I'm kinda struggling with this one. $f$ is entire but $f(z)\notin \left[0,1 \right]$. I tried to play ...
0
votes
1answer
62 views

6 true or false questions in complex analysis

Let $f:\mathbb C\to\mathbb C$ be an entire function. Which of the following statements are true and which are false? If $f(z)\in\mathbb R$ for all $z\in\mathbb C$, then $f$ is constant. If $f(\frac{1}...
1
vote
1answer
40 views

Proving Liouville for entire functions using MVT for analytic functions

I am trying to prove Liouville's theorem: An entire bounded function is constant. I'm trying to use the Mean Value Theorem from my textbook. MVT: $\space\space $If $f$ is analytic in $D$ and $a \in D$...
0
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1answer
68 views

Canonical products and general form for entire functions

In Ahlfors book, chapter V - 2.3, we have a method to find a form to entire functions, that is: if $f$ is an entire function with a finite number of zeros $a_1, ..., a_N$, and a zero of multiplicity $...
0
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1answer
63 views

non-constant entire functions (Liouville) [closed]

I have to show by using Liouville's theorem, whether there are non-constant entire functions such that: $ f( \mathbb{C}) $ is in an half-plane. I found out, that this is not possible for non-constant ...
1
vote
1answer
95 views

If $Re(f)$ is a polynomial then $f$ is a polynomial, where$f$ is entire

If $f$ is entire, and $Re(f)$ is a polynomial in x,y I am trying to show that f is a polynomial in z. We can write $f(z) = u(z) + iv(z)$. where $Re(f) = u(z)$ I have solved problems similar to this ...
1
vote
1answer
68 views

Liouville's theorem for non constant functions

I have to show by using Liouville's theorem, whether there are non-constant entire functions such that: $$ |f^k(z)| \leq 1 \forall z \in \mathbb{C}$$ and fixed $k$. for $k=0$: There is such a ...
1
vote
1answer
32 views

Line integral depending on a parameter is entire

Suppose you have a continuous function: $$\phi:[0,1]\rightarrow \mathbb{C}$$ define the complex function: $$f(z)=\int_0^1\phi(t)e^{itz}dt$$ prove that it is entire and calculate it's Taylor ...
0
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2answers
28 views

Sequence of entire function that converges uniformly over on sets with empty interior

I have to prove that the sequence of entire functions: $$f_n(z)=\frac 1n \sin(nz)$$ converges uniformly over $\mathbb{R}$ (and this I managed to verify) but doesn't on every set with non-empty ...
0
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1answer
43 views

Identity theorem for a holomorphic funtion defined near zero

I have to show, whether there is a holomorphic funtion $f$ defined in an open neighborhood of zero, such that: $$ f\left(\frac{1}{n}\right)=(-1)^n \frac{1}{n^3}$$ for all positive integer $n$. My ...
0
votes
1answer
28 views

Entire function invariant by translation is constant [duplicate]

I need to apply Liouville theorem ("entire bounded complex functions are constant") to prove that an entire function satisfying: $$f(z)=f(z+1)=f(z+i)$$ for all complex numbers $z$ is constant. I'm ...
3
votes
1answer
70 views

$f$ is an entire function s.t $|f(z)|=1$ $\forall z \in \Bbb R$. prove that $f$ has no zeros in $\Bbb C$

My attempt: I was trying to apply identity theorem that if the zeros of the function do have any limit point and it will be a zero function but setting $g(z)=f(z)-1$ will not help me. Can anyone help ...
0
votes
1answer
59 views

Entire function bounded on a set

"Consider the set $T = \{ \alpha ∈ C|∃ a, b ∈ Z :\alpha = a + bi \}$ Let $g$ be an entire function which satisfies that $g(z + \alpha) = g(z)$ for all $z ∈ \mathbb{C}$ and all $\alpha ∈ T$. Prove ...
1
vote
1answer
39 views

Application of Hadamard's Formula to a function that never vanish

Suppose $f$ is an entire and never vanishes, and that none of higher derivatives of $f$ ever vanish. Prove that if $f$ is also of finite order, then $f(z) = e^{az+b}$ for some constants $a,b$. I have ...
0
votes
0answers
37 views

Prove that Jacobi theta function is of order $2$ as a function of $z$.

Let $$\Theta(z|\tau) = \sum_{n = -\infty}^{\infty}e^{\pi in^{2}\tau}e^{2\pi inz}$$ with $\tau = s + it$ where $t > 0$. Show that $\Theta$ is of order $2$ as function of $z$. [Hint: $-n^{2}...
0
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2answers
53 views

Show $ \int_{0}^{1}\frac{\sin (ax)}{x}dx $ is an entire function.

I am struggling in evaluating the following integral: $$\int_{0}^{1}\frac{\sin (ax)}{x}dx$$ I know that if the integral is from $0$ to infinity, it will be a constant of $\pi/2$ which is analytic, ...
0
votes
1answer
38 views

Does such entire function exist?

Does there exist a nonconstant entire function with $f^{(n)}(0)=3^n$ for $n$ even and $f^{(n)}(0)=(n-1)!$ for $n$ odd. My attempt: If such $f$ exists, then $f(z)=\sum_{n=0}^{\infty}a_n\,z^n$ where $...
0
votes
3answers
64 views

If $f$ is an entire function with $|\,f(z)|\le|\operatorname{Re}(z)|$, then $\,f\equiv 0$.

If $f$ is an entire function so that $|\,f(z)|\le|\operatorname{Re}(z)|$ for all $\mathbb{C}$, then $f\equiv0$ on $\mathbb{C}$. There are various ways showing the above property. For example, using ...
1
vote
1answer
47 views

Use maximum modules principle to prove that if $\sup_{|z| = R}|f(z)|\leq AR^{k} + B$ with $f$ entire, then $f$ is a polynomial

Use Cauchy inequality or maximum modules principle to prove that if $f$ is entire function that satisfies $$\sup_{|z| = R}|f(z)|\leq AR^{k} + B$$ for all $R >0$, for some $k \in \mathbb{Z}$ and ...
1
vote
1answer
79 views

Take $f$ and entire function such that $f(z) = f(z+1)=f(z+\sqrt{2})$, then it is constant [duplicate]

I am working through some practice problems and I have this one which is stumping me: Take $f$ and entire function such that $f(z) = f(z+1)=f(z+\sqrt{2})$, then it is constant. I was thinking, given ...
2
votes
1answer
71 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
62 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\...