All Questions
12
questions
0
votes
1
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153
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Is this the expansion for any known function?
The expansion is $$\sum_{n=1}^{\infty}\frac{x^n}{n!(n-1)!}\left[c(1+c)\dots((n-1)^2+c)\right]$$
So the first 3 terms are $cx$, $\dfrac{c(1+c)}{2}x^2$, $\dfrac{c(1+c)(4+c)}{12}x^3$.
- 35
3
votes
1
answer
187
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Derivation of the behavior of solutions to $\tan x = x$
This question is related to Chapter IV, Note IV.36 of Flajolet & Sedgewick's Analytic Combinatorics, and The question: Sum of the squares of the reciprocals of the fixed points of the tangent ...
- 80
4
votes
0
answers
402
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Conditions and correct interpretation of Borel summation
Hello to the community.
In my line of research (theoretical particle physics) it is customary to apply the strategy of Borel summation to infinite power series in order to find closed forms and/or ...
2
votes
0
answers
77
views
Large $x$ power series for $e^{-\alpha x}$
Let $\alpha>0$, then do there exist continuous functions $c_n\colon\mathbb R^+\to\mathbb R$ and $\gamma\colon\mathbb R^+\to\mathbb R$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$ and the ...
- 1,278
3
votes
2
answers
175
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Asymptotic expansion of $\sum_{n=1}^\infty\frac{H_n}{n!}z^n$ for $z\to\infty$
What is the asymptotic expansion of $\sum_{n=1}^\infty\frac{H_n}{n!}z^n$ for $z\to\infty$ where $H_n=\sum_{k=1}^n \frac{1}{k}$?
I thought of using the Euler-Mascheroni constant, the fact that $\gamma=\...
- 107
0
votes
0
answers
97
views
What is the asymptotic expansion of a function defined by a converging power series?
Given an analytic function $f(z)=\sum_{n=0}^{\infty} \frac{a_n}{n!}z^n$, how could one go about finding the asymptotic expansion of $f(z)$ for $z\to\infty$?
In particular given $g(z)=\sum_{n=1}^\infty ...
- 107
11
votes
0
answers
298
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Roots of partial sum of power series
Consider the power series
$$ \sqrt{1+z} = \sum_{k=0}^\infty \left( \begin{array}{c} \frac{1}{2} \\ k \end{array} \right) z^k. $$
I am interested in characterizing the roots of the partial sum
$$ s_n(z)...
- 1,436
4
votes
2
answers
3k
views
Asymptotics of Hypergeometric $_2F_1(a;b;c;z)$ for large $|z| \to \infty$?
I found this list of asymptotics of the Gauss Hypergeometric function $_2F_1(a;b;c;z)$ here on Wolfram's site for large $|z| \to \infty$
In particular there is a general formula for $|z| \to \infty$
$...
- 2,767
1
vote
0
answers
103
views
Analytic Continuation with Real Coefficients
I have an interesting problem that I cannot make progress on over several days. It is as follows:
Suppose $f(z) = \sum a_nz^n$ is analytic around $0$ such that $a_n \geq 0$ for all $n$, in ...
- 51
1
vote
0
answers
88
views
Analytic Function with Exponential Coefficients
I've been struggling with a weird complex analysis statement for a few days without making really any headway; I'd really appreciate a hint to get me past where I'm at. The theorem goes like this:
...
- 51
1
vote
1
answer
177
views
Technique for constructing an entire function satisfying a given growth condition
How can I construct an entire function whose growth rate at infinity satisfies
$$ \lim_{r \to \infty} \frac{\log M(r)}{\sqrt {r}} =1$$
where $M(r) = \max_{|z|=r} |f(z)|$?
Based on the above limit,...
- 1
4
votes
1
answer
792
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Trying to find more information about "Darboux's method/theorem" on coefficients of an analytic function
My supervisor briefly showed me a statement of something she called "Darboux's theorem," but I am having trouble finding more information about it on the internet. Here is what I have written down (...
- 91.1k