Questions tagged [entire-functions]
The entire-functions tag has no usage guidance.
2
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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\...
