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Questions tagged [entire-functions]

The tag has no usage guidance.

1
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1answer
41 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
69 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
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2answers
112 views

Proof of Casorati-Weierstrass [on hold]

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
46 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
54 views

Removable Singularity with Entire Function

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}(z-0)...
12
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1answer
239 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 ...
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0answers
36 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
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1answer
35 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, ...
1
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2answers
43 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
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1answer
76 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
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1answer
31 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
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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
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1answer
57 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
24 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
54 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
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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
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0answers
42 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 ...
0
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1answer
22 views

existence of non constant entire function that is bounded outside an annulus

Is there a function $f$ which is not constant, analytic in $\mathbb{C}\setminus\{0\}$ and $$\forall z\in\{w\in\mathbb{C}: 0<|w|<\frac{1}{100} \lor |w|>100\}:|f(z)|<800$$ Probably not. ...
1
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1answer
71 views

Analytic continuation on disconnected sets

Let $f(z,s)$ be a complex-valued function in two complex variables, entire in $s$, and let $$F(s)=\int_\Omega f(x+iy,s)\,dx\,dy$$ where $\Omega$ is an open subset of $\mathbb{C}$. Assume that the ...
4
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1answer
52 views

entire function that maps real line to itself is linear

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function be such that $\mathbb{R}=f^{-1}(\mathbb{R})$. Show that $f$ is linear. i.e. $$\exists\ a,b\in\mathbb{R}:f(z)=az+b$$ Hint I think that $f$ must ...
2
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2answers
71 views

If $f(z)$ is entire function such that $|f(z)f'(z)| \leq 1$ then $f(z)$ is constant

If $f(z)$ is entire function such that $|f(z)f'(z)| \leq 1$ then $f(z)$ is constant. Choose $g(z) = \frac {(f(z))^2}{2}$, $g'(z) = f(z)f'(z) \implies |g'(z)| \leq 1$(given condition) So $g'(z)$ is ...
0
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1answer
34 views

Find all entire functions with real part greater than 1 and imaginary part less than -1

I'm stuck on the following question: Find all entire functions $f$ such that $Re(f) >1$ and $Im(f)<-1$. Unfortunately this doesn't seem like a problem that can be quickly solved by ...
2
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1answer
29 views

Liouville's theorem non-entire function

I have the following problem which I don't understand. Find all entire functions $f$ such that $$|f(z)|\ge \frac{1}{1+|z|^{2017}}=g(z), \; \forall z \in \Bbb{C}$$ The answer says that $f$ has to ...
0
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2answers
50 views

What conclusion can we make if $f$ is a holomorphic function?

Suppose $f$ is holomorphic in an open neighbourhood of $z_0 \in \Bbb C$. Given that the series $$\sum\limits_{n=1}^{\infty} f^{(n)} (z_0)$$ converges absolutely, we can conclude that $(1)$$\ \ \ \...
2
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0answers
49 views

Finding all entire functions $f(z)$ for which $|f(z)|\leq M.|\sin z|$

I've seen the other way for this question (Entire functions $f$ for which there exists a positive constant $M$ such that $|f(z)|\le M|\cos z|$), I just want to make sure that the same argument can be ...
1
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1answer
52 views

a question about specific entire function

Find all the entire functions $f(z)$ for which there exists positive constants $M, \:R$ and a positive integer $n$ such that $$|f(z)|\geq M.|z|^n\:\:\text{whenever}\:\:|z|>R.$$ What if the ...
1
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0answers
22 views

Possible existence of entire function

Is there an entire function $f$ such that $f(z)=1/z$ for all z in the plane punctured at $0$? I am viewing it as a problem to analytically extend the function $1/z$ in the entire plane which is not ...
1
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1answer
69 views

Showing $f(z)$ is a rational function

I could prove that if $g(z)$ is bounded and analytic except at a finite number of points in the complex plane $\mathbb{C}$, $g(z)$ must be constant. First I thought I can use the above to prove the ...
3
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1answer
201 views

Entire function mapping a parallelogram onto another one is a degree 1 polynomial

I am reading Bak and Newman's Complex Analysis and I can't figure out how to do the following exercise: Prove that an entire function which maps a parallelogram onto another parallelogram, and maps ...
4
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1answer
54 views

Entire function having the property [duplicate]

Let $f$ be an entire function. Consider $A=\{z \in \Bbb{C} : f^{(n)}(z)=0\; \text{for some}\; n \in \Bbb{N}\}$. Then how to prove if $A=\Bbb{C}$, then $f$ is a polynomial ? This is same as proving ...
1
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1answer
54 views

Examples of different type of entire functions

I try to answer following question. I would like to find a general approach for part b, since I think any entire function with $n $ roots is either polynomial of degree $n$ or is a product of ...
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0answers
53 views

Uniformly continuous entire functions

Find all entire functions that are uniformly continuous on the complex plane. I think the answer must be the the linear polynomials, but nor sure, since neither of polynomials of degree greater than ...
0
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1answer
67 views

Entire function $f$ with infinitely many zeros [closed]

Let $f$ be an entire function, non constant with the property: The set $\{w : f(w)=0\}$ has ininite elements. Show that for all c there exist a sequence $\{z_n\}$ such that $z_n\to\infty$ and $f(z_n)\...
0
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1answer
63 views

Essential singularity at infinity of exponential function.

Show that if $f(z)$ is a non-constant entire function, then $e^{f(z)}$ has an essential singularity at $z=\infty$. This is my approach: By Liouville's theorem I know that if $f$ is a non-constant ...
1
vote
1answer
45 views

Holomorphic function $f: \Bbb C \rightarrow \Bbb C$ such that $|f(z)|\leq C|\negthinspace\cos(z)|$ for all $z \in \Bbb C$ [duplicate]

The Question: Let $f:\Bbb C \rightarrow \Bbb C$ be a holomorphic function (i.e. an entire function) such that $|f(z)|≤C|\negthinspace\cos(z)|$ for all $z \in \Bbb C$, where $C \in \Bbb R$ is a ...
1
vote
1answer
32 views

Examples of entire functions with none, one and infinite zeros.

I know that an entire function is one which can be differentiated on the entire complex plane, and I believe that a zero of an entire function is the z where f(z)=0. I was tring to think of example ...
1
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1answer
36 views

A nonconstant complex function satisfying $f(z+w)=f(z)f(w)$ and differentiable at the origin is entire

Suppose we are given a non-constant function $f: \mathbb{C} \rightarrow \mathbb{C}$ which satisfies the property $f(z+w)=f(z)f(w)$ for all $z, w \in \mathbb{C}$ and is differentiable at the origin, i....
2
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1answer
55 views

$P(z)$ be a monic polynomial with complex co-efficients of degree $n$ such that $|P(z)| \le 1, \forall |z| \le 1$ ; then is $P(z)=z^n$?

Let $P(z)$ be a monic polynomial with complex co-efficients of degree $n$. Since by Cauchy integral formula bound, $n!=P^{(n)} (0) \le n! Sup_{|z|=1} |P(z)|$ , so $Sup_{|z|=1} |P(z)| \ge 1$ . So by ...
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0answers
35 views

Determine all entire functions which are injective [duplicate]

Determine all entire functions which are injective. This is it. Now, I know all entire functions that are injective take a specific form under certain conditions, but the formulation seems a bit ...
1
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2answers
31 views

Show that there cannot be an entire function that satisfy $|z+(\cos (z)-1)f(z)|\leq 7$

Show that there cannot be an entire function $f(z)$ that satisfy $|z+(\cos (z)-1)f(z)|\leq 7$. I thought about showing somehow that $f(z)$ is bounded and by Liouville's theorem it is constant, then ...
2
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1answer
40 views

Question on entire function

Question: let $f$ be an entire function on $\mathbb{C}$ and let $g(z)=\overline{f(\bar{z})}$ then which of the following is/are correct? (a) if $f(z) ∈\mathbb{R}$ for all $z∈\mathbb{R}$ then $f=g$ (...
1
vote
1answer
82 views

Can zeros of the Fourier transform of an asymmetric kernel all be real?

Can zeros of the Fourier transform of an asymmetric kernel all be real ? Be specific, in the following, if $f(t)$ is a real function, NOT an EVEN function, $z$ is complex, ${\displaystyle {\hat {f}}...
3
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0answers
61 views

entire function that misses $[0,1]$ is constant [duplicate]

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function. Suppose that $\forall z\in\mathbb{C}:f(z)\not\in[0,1]$. Prove that $f$ is a constant function. I've heard about Picard's little theorem, but ...
0
votes
2answers
49 views

Corollary of Liouville's Theorem

I need someone to verify my proof. I had Liouville's Theorem presented to me as If $f$ is entire and bounded, then $f$ is constant. I am then asked to prove If $f$ is entire and there exists $...
0
votes
1answer
30 views

Check $r\mapsto M_r=\max\{|f(z)|~:~|z|=r\}$ is an increasing function

Let $f$ be a non constant entire function and $M_r=\max\{|f(z)|~:~|z|=r\}$. I need to check whether the mapping $r\mapsto M_r$, is increasing. I am trying to prove it by the open mapping theorem, but ...
1
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0answers
44 views

A nonvanishing entire function of finite order is the logarithm of a polynomial

Let $f$ be a nonvanishing entire function of finite order. It follows immediately from Hadamard's factorization theorem that $f(z)=e^{P(z)}$ for some polynomial $P(z)$ whose degree is at most the ...
2
votes
1answer
61 views

Show that if $f$ is entire and $(\forall z \in \mathbb{C})$ $|f(z^2)| \leq |f(z)|$, then $f$ is constant

Show that, if $f$ is entire and $(\forall z \in \mathbb{C})$ $|f(z^2)| \leq |f(z)|$, then $f$ is constant How should I approach this problem? Should I use the power series of $f(z)$ or Liouville's ...
1
vote
1answer
38 views

Express entire functions $f(z)$ by exact formula

An entire function $f(z)$ has the properties that $f(0)=10,f'(0)=0$, and $f''(1+ \frac{1}{n})=2- \frac{3}{n}$ for each $n=1,2,3,...$ Express $f(z)$ by an exact formula and prove that it is the unique ...
2
votes
2answers
97 views

If $f$ is entire and for some $r>0$ we have $f(z)=Cz^n$ for a constant $C \in\mathbb C$ on $|z|=r$ then does it follow that $f(z)=Cz^n$ for all $z$?

If $f$ is entire and for some $r>0$ we have $f(z)=Cz^n$ for a constant $C \in\mathbb C$ on $|z|=r$ then does it follow that $f(z)=Cz^n$ for all $z$? Since a circle is not open the identity ...
2
votes
1answer
33 views

On a pseudo periodic entire function being an exponential function

Let $p_1,p_2\in \mathbb C$ be linearly independent over $\mathbb R$. Let $f: \mathbb C \to \mathbb C$ be an entire function such that $f(0)=f'(0)=1$ and $f(z+p_1)=af(z) $ and $f(z+p_2)=bf(z)$ , $\...