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Questions tagged [entire-functions]

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0
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1answer
38 views

non-constant entire functions (Liouville) [on hold]

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
83 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
48 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
22 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
25 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 ...
1
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1answer
37 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
20 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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0answers
31 views

What can we say about $f$? [on hold]

Let $f$ be an entire function. Consider $A= \left \{z \in \Bbb C : f^{(n)} (z) = 0\ \text {for some positive integer}\ n \right \}.$ Then $1.$ if $A=\Bbb C$ then $f$ is a polynomial. ...
0
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1answer
22 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
66 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
41 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
26 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
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0answers
20 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
51 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
34 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
52 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
32 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
52 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
68 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
votes
1answer
53 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
29 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
63 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
52 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
44 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
44 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
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1answer
117 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
84 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
61 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
107 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
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1answer
72 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
61 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
44 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
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1answer
75 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
137 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
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2answers
61 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
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2answers
66 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
246 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
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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
39 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
73 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
82 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
71 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
35 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
62 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
45 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 ...