# 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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### 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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It is relatively straightforward to show that when an analytic function has a simple pole, then the coefficients $a_n$ of the Taylor series asymptotically follow $$\left|a_n\right| \sim O\left(z_0^{-... • 1,650 0 votes 0 answers 21 views ### A problem about entire function and polynomial [duplicate] Let p(z) be a polynomial not identically zero on \Bbb C with degree n. Suppose that |p(z)|\le|z|^n for any z satisfies |z|\le1. Then there exists a complex number c\in \Bbb C such that p(z)... • 121 4 votes 3 answers 422 views ### Prove an entire function is constant on complex plane Let f(z), F(z) be two analytic functions on \Bbb C satisfies f(z)=F(\overline{f(z)}). Here \overline{f(z)} is the complex conjugate of f(z). Prove that f(z) is constant on \Bbb C. I ... • 121 2 votes 0 answers 49 views ### Explicit example of an entire function with simple zeros at precisely the square roots of the positive half integers I'm looking for an entire function with the property that f(\sqrt{n+1/2}) = 0 for n=0,1,2,\dots, all of which are simple zeros and f has no other zeros. I know that such functions exist and can ... 2 votes 1 answer 104 views ### Given \frac{H(z)}{\prod_{n=1}^\infty \left(1-\left(\frac{z}{\lambda_n}\right)^2\right)} = \phi(z), determine if \phi(z) is an entire function. Let us consider \lambda_n=n-\frac14 and define$$\frac{H(z)}{\prod_{n=1}^\infty \left(1-\left(\frac{z}{\lambda_n}\right)^2\right)}=\phi(z) $$where H(z) is an entire function. I would like to ... • 7,861 4 votes 1 answer 260 views ### Conditions to calculate an integral through a series expansion Let f,g:\mathbb{R}\longrightarrow\mathbb{R_+} be Lebesgue integrable functions. We can show that if g has compact support and f has a Maclaurin series that converges absolutely in the support of ... • 983 0 votes 0 answers 32 views ### Entire functions that have a zero in the disk \{z \colon |z| < r\} or constant functions. Question: Let f\colon \mathbb{C} \to \mathbb{C} be an entire function. Suppose that there are real constants r, R > 0 so that |f(z)| > R for all z \in \mathbb{C} with |z| > r. Show ... • 715 0 votes 1 answer 83 views ### An entire function satisfying f(z)=f(z+\xi t) for all t\in\mathbb{R} and some \xi\in\mathbb{C} Let f be entire function satisfying f(z)=f(z+\xi t) for all t\in\mathbb{R}, some \xi\in\mathbb{C}\neq 0 and all z\in\mathbb{C}. I would like to show that f is constant. Since f(z+\xi t)-f(... • 5,531 1 vote 1 answer 121 views ### Prove that a function is a polynomial of degree at most one given v(x,y) \geq x The question is Let f(z) = u(x,y) + iv(x,y) be an entire function satisfying v(x, y) ≥ x for all z = x + iy. Then show that f(z) is a polynomial of degree at most one. I know that I have to ... • 25 0 votes 0 answers 51 views ### Want an example of entire function of order \sqrt 2 [duplicate] I want an example of entire function with order \sqrt 2. If f is a entire function of finite order \rho then \rho=\lim_{R \to \infty} \sup_{r \geq R} \frac{\log \log M(f,r)}{\log(r)}, where M(... • 10.7k 2 votes 0 answers 187 views ### An entire function f(z) that is real if and only if z is real I would like to prove that if f(z) is an entire function that is real if and only if z is real, then f'(z)\neq 0 for all real z. I first wrote$$f(z)=u(x,y)+iv(x,y)$$with z=x+iy and ... • 5,531 0 votes 1 answer 67 views ### Polya's representation (Boas Book) I am studying the Borel transform. If f(z) is an entire function f(z)=\sum_{n=0}^{\infty}a_nz^n, then, the Borel transform is F(z)=\sum_{n=0}^{\infty}\frac{n! a_n}{z^{n+1}}. Question: why \... • 3,574 1 vote 1 answer 233 views ### Continuity of maximum modulus function M(r)=\max_{|z|=r}|f(z)| I am looking to prove that the maximum modulus function$$M(r)=\max_{|z|=r}|f(z)|is continuous on [0, \infty) for f an entire function. My idea was to use the representation of f as a power ... • 5,531 0 votes 1 answer 50 views ### All f\in H(\mathbb{C}) with Re(f(z))=u(z)=u(x+iy)=x^2-y^2+2x+1 In my complex analysis course I'm supposed to compute all entire functions with the given requirement for the Real part. How do I work on this? I hav no clue how to work this out. Can someone please ... • 177 1 vote 0 answers 65 views ### Question of the order of growth In Stein and Shakarchi's book on Complex analysis, there's a theorem (p.138) that states: Theorem: Let f be an entire function that has an order of growth \leq \rho. If z_1,z_2,\dots denotes ... • 157 0 votes 0 answers 50 views ### Extended Liouville to Real Part of Entire Functions I want to prove if f is entire function and u(x,y) is real part of f satisfying \begin{align*} |u(x,y)|\leq C|z|^n, z=x+iy \end{align*} then f is polynomial with degree at most n. If f(z)=\... • 859 1 vote 1 answer 154 views ### Find all the entire functions satisfying |f(z)|\leq-\ln(|z|^2)+|z|^2, f(0)=i. Problem. Find all the entire functions f:\mathbb{C}\rightarrow\mathbb{C} such that|f(z)|\leq-\ln(|z|^2)+|z|^2, f(0)=i Since $-\ln(|z|^2)+|z|^2\leq2|z|^2,$ we have$|f(z)|\leq2|z|^2$. hence $f$ ...
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Given an entire function $f$, I have that $f(ix)$ and $f(1+ix)$ are real for all $x \in \mathbb{R}$. I want to show that $f(z) = f(z+2)$ for all $z \in \mathbb{Z}$. I've thought about considering the ...