Questions tagged [entire-functions]
This tag is for questions relating to the special properties of entire functions, functions which are holomorphic on the entire complex plane. Use with the tag (complex-analysis).
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Does the Gauss-Lucas theorem generalize to arbitrary entire functions with a finite number of zeros?
The Gauss-Lucas theorem states that if $P(z)$ is a non-constant polynomial with complex coefficients, then all the zeros of $P'(z)$ lie within the convex hull of the set of zeros of $P(z)$.
Does this ...
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sum of entire nonvanishing functions is constant implies functions are constant
Let $f,g$ be nonvanishing entire functions such that $f + g = 1$ for every $z\in \mathbb{C}$. Do $f$ and $g$ have to be constants themselves.
My attempt: $$f = 1-g \implies \frac{1}{f} = \frac{1}{1-g}$...
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Find the Hadamard factorization of $f(z)=\sinh(z)-e^z+1$
Find the Hadamard factorization of $f(z)=\sinh(z)-e^z+1$.
Here's my attempt.
First of all let us notice that $f$ is entire and that there are infinitely many zeros, namely $z_k=2k\pi i$ for all $k\in ...
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Procedure to solve $\mathcal{P}\int_{-\infty}^{+\infty} \frac{\cos (\alpha x)}{x^2-1} dx$
I recently solved a complex analysis exercise that required to solve this integral using residue theory, but I don't know where to check for the correctness of the result I got, so I thought of ...
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Prove or disprove that function $r\mapsto\frac{M(r,f)}{r^k}$ is increasing for transcendental entire function $f$ and fixed $k>0$.
Prove or disprove that function
$$r\mapsto\frac{M(r,f)}{r^k},\quad (\text{here}\ M(r,f)=\max_{|z|=r}|f(z)|)$$ is increasing for transcendental entire function $f$ and fixed $k>0$ when $r$ large ...
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Does there exist an entire function $f$ such that $f(z)+f(z^2)=z^3$ for all $z \in \mathbb{C}$?
Does there exist an entire function $f$ such that $f(z)+f(z^2)=z^3$ for all $z \in \mathbb{C}$?
This problem showed up on a qualifying exam; these exams enjoy including problems that utilize Liouville'...
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Holomorphic surjective map
Let $\Omega_1=\lbrace{z \in \mathbb{C} : |z| < 1\rbrace} $and $\Omega_2= \mathbb{C}$. Which of the following statements are true?
There exists a holomorphic surjective map $f:\Omega_1\to\Omega_2$.
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Entire function such that $f(\sin z)=\sin(f(z))$
Find all the entire functions $f$ such that
$$f(\sin z)=\sin(f(z)),\quad z\in\mathbb C.\tag {*}$$
The motivation to ask this question is an old post Find all real polynomials $p(x)$ that satisfy $\...
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How to handle questions of the form "Find all entire functions such that... "
I apologize if this is not the right forum to ask this question. Could you please recommend an online lecture, a textbook note to refer to understand how to answer questions of the form
Find all ...
- 379
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Growth order of $e^{p(z)}$ for a complex polynomial, $p(z)$
Let $p(z):\mathbb{C}\to\mathbb{C}$ be a degree $m$ polynomial, and consider $e^{p(z)}$. I'm wondering whether it's true that the order of $e^{p(z)}$ is equal to the degree of $p(z)$. My working ...
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An entire function such that $|\operatorname{Re}f(z)|>0.001 |\operatorname{Im}f(z)|$ must be constant.
I was going over some previous qualifying exams to prepare for my own, and came across the following problem:
Problem. Suppose that $f:\mathbb{C}\to\mathbb{C}$ is entire and has the property that for ...
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Entire function maps bounded sets to bounded sets
I was doing some problems in Complex Analysis… And I came across this.
Let $f: \mathbb{C} \to \mathbb{C}$ be entire. Then for any bounded set $B$, f ($B$) is bounded.
Now I know that if an entire ...
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Find all entire functions such that $|f(z+z')|\leq |f(z)| + |f(z')|$, for all $z,z'\in\mathbb{C}$
Find all entire functions such that $|f(z+z')|\leq |f(z)| + |f(z')|$, for all $z,z'\in\mathbb{C}$
In particular, let $z=z'$ yields $|f(2z)|\leq2|f(z)|$. This gives that $\frac{f(2z)}{f(z)}=c, $ for ...
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Show that any entire function satisfying the given conditions is a constant. [duplicate]
Let $f$ be an entire function. Consider the set
$$S=\bigg\{re^{i\theta}: r>0, \ \frac{\pi}{4}\leq \theta\leq \frac{7\pi}{4}\bigg\}\cup \{0\}.$$
It is given that $f$ is bounded on the set $S$.
...
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Entire function bounded in every horizontal line, and has limit along the positive real line
Let $f(z)$ be an entire function (holomorphic function on $\mathbb{C}$) satifying the following condition:
$$|f(z)|\leq \max (e^{\text{im}(z)},1 ),\ \forall z\in\mathbb{C}$$
$$\lim_{\mathbb{R}\ni t\...
