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Questions tagged [entire-functions]

The tag has no usage guidance.

1
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1answer
19 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
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1answer
24 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
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1answer
37 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 ...
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1answer
60 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 ...
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2answers
39 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
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1answer
20 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{...
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1answer
34 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 ...
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1answer
21 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=...
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1answer
28 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 ...
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2answers
258 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 $...
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3answers
42 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
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1answer
31 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
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1answer
44 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 ...
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2answers
228 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
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1answer
42 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: ...
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1answer
102 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 ...
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0answers
23 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
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1answer
46 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 ...
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1answer
52 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}...
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1answer
37 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 $a \in D$...
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1answer
26 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
46 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
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1answer
91 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
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1answer
56 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 ...
0
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1answer
29 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 ...
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2answers
26 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 ...
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1answer
41 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
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1answer
23 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 ...
0
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1answer
25 views

Determination of entire functions given with a removable singularity. [closed]

Determine all entire functions $f(z)$ such that $0$ is a removable singularity of $f\big(\frac{1}{z}\big)$. I have no idea how to start with.
3
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1answer
69 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
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1answer
50 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
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1answer
30 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 ...
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0answers
26 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
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1answer
36 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
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3answers
55 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
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1answer
39 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
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1answer
73 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
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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
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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\...
2
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1answer
55 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 ...
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0answers
31 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
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0answers
66 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
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1answer
62 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
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1answer
47 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
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1answer
45 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
34 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
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1answer
124 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
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1answer
100 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
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3answers
68 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 $...