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).
683 questions
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Bounded Lindelöf indicator implies finite order?
Let $f : \mathbb C \rightarrow \mathbb C$ be an entire function. Assume that its (Lindelöf, for the first order) indicator
$$
h(\theta) := \limsup_{r \rightarrow \infty} \frac{\log |f(re^{i\theta})|}{...
- 2,124
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43
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Limit of $g^{(k)}(x)/g^{(k+1)}(x)$ as $x \to \infty$ for entire functions with positive coefficients
Let
$
g(z) = \sum_{n=0}^\infty a_n z^n, \qquad a_n > 0,
$
be an entire function. Fix an integer $k \ge 0$. I am interested in the limit
$
L_k := \lim_{x \to +\infty} \frac{g^{(k)}(x)}{g^{(k+1)}(x)},...
- 1
4
votes
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answer
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views
Asymptotics of $\sum_n^\infty \left(\log\left|1 - \frac z{a_n}\right| + \Re\left(\frac z{a_n}\right)\right)$
I would like to find an asymptotic for
$$
\sum_{n=2}^\infty \left(\log\left|1 - \frac z{a_n}\right| + \Re\left(\frac z{a_n}\right)\right), \tag{$*$}
$$
as $z\to \infty e^{i\theta}$, where $a_n = -n/\...
- 2,228
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1
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84
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Find an entire function knowing its real part and using Cauchy-Riemann equations.
Find for which $a,b,c\in \mathbb{R}$ exists an entire function $f$ such that $\Re(f(z))=ax^2+bxy+c$ and $f(0)=i$. Then, write it in the variable $z$.
Let $z=x+iy$, I can rewrite $f(z)$ as follows:
$$...
- 1,953
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votes
4
answers
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Does $f(x) = \sum_{k=1}^{\infty} (-1)^{k+1} \sin(x/k)$ have infinitely many real zeros?
I am investigating the properties of the function $f(x)$ defined for $x \in \mathbb{C}$ by the series:
$$f(x) = \sum_{k=1}^{\infty} (-1)^{k+1} \sin\left(\frac{x}{k}\right)$$
This function was the ...
- 1,695
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vote
1
answer
148
views
Determine all of the entire functions such that $\left|f\left(f(z)\right)\right|\leq 1 \space \forall z\in \mathbb{C}$
This is taken from an old exam of my complex analysis course.
If $f$ is entire then $f\circ f$ is entire. Moreover, the requested condition implies that $f\circ f$ is an entire bounded function. So, ...
- 1,953
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85
views
Does there exist an entire function $f$ such that $f(1/n) = \frac{1}{2^n}$ for all $n \in \mathbb{N}$? [duplicate]
Does there exist an entire function $f$ such that $f(1/n) = \frac{1}{2^n}$ for all $n \in \mathbb{N}$?
I feel like such a function shouldn't exist, but I am struggling to prove it.
Attempt 1: Assume ...
- 170
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Under what oscillation conditions on $f(t)$ does its Laplace transform avoid singularities on the abscissa of convergence?
I’m interested in the following general phenomenon:
Let
$$
F(s) = \mathcal{L}\{f\}(s) = \int_{0}^{\infty}f(t)\,e^{-st}\,dt
$$
have abscissa of convergence $\sigma_c $, i.e. the integral ...
- 1,141
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60
views
Non-constant entire functions that grow less rapidly than $f(z)=z$ [duplicate]
Do there exist non-constant entire functions whose maximum modulus increases less rapidly than the function f(z)=z? Give an example if you can. Obviously the constant functions have no growth and $f(z)...
13
votes
1
answer
440
views
Prove the existence of $2$-periodic point of $\exp(z)$.
To prove that $\exp(z)$ has at least one fixed point, consider the function
$$
f(z) = \exp(z) - z,
$$
which satisfies $f(z+2\pi i)=f(z)-2\pi i$. By Picard's little theorem, we conclude that $f$ is ...
5
votes
2
answers
507
views
Entire function that grows faster than any iteration of exponentials
Define the class $\mathcal{F}$ of entire functions satisfying, for some integer $n$: $$\limsup_{r\rightarrow \infty}\frac{\log _{n}M(r)}{\log r}<\infty,$$where $M(r)$ is the maximum of the function ...
- 290
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vote
1
answer
89
views
Characterize all entire functions such that $|f(z)| < (1+|z|)^{\frac{4}{3}}$.
I've been thinking about this question for a while and I don't know how I can finish the exercise.
So, what I found is that if $f(z)$ is an entire function and $|f(z)| < (1+|z|)^{\frac{4}{3}}$ for ...
- 587
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votes
1
answer
204
views
Does there exist an entire function with exactly two zeros whose derivative is nonvanishing?
The derivative of a polynomial $f$ over $\mathbb{C}$ has one less zero than $f$. Including various entire transcendental $f$, we can show that the pair of counts of zeros $(\#Z(f), \# Z(f'))$ ranges ...
- 659
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votes
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62
views
Find (finite) order of an entire function.
I was trying to to find a function $f\in H(\mathbb{C})$ with a certain finite non-zero order $1/\alpha$. In particular I found an answer that does just this. I was able to understand why
$$g_\alpha(z) ...
- 1,410
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Discuss existence (and eventually uniqueness) of an $f \in H(\mathbb{C})$
I would like to discuss the existance (and eventually uniqueness) of an $f \in H(\mathbb{C})$ such that :
(i) $f(0) = 0 \iff z = \sqrt{k} \, e^{i \pi/5}$ for $k \in \mathbb{N}$ and $k \ge 2$;
(ii) $...
- 1,410