Questions tagged [special-functions]
This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).
4,884 questions
0
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Maximisation of functions of the form $f(x) = \sqrt{1 - x^2} + (ax+b)x$
I am studying a function arising in the analysis of robust aggregation rules in distributed learning, but the question is purely analytical. The function I am facing depends on parameters $a, b > 0$...
- 936
2
votes
1
answer
78
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Prove $B_{2k+1}(1/4) = \frac{-(2k+1) E_{2k}}{4^{2k+1}}$
Question. Is there a simpler way to prove $$B_{2k+1}(1/4) = \frac{-(2k+1) E_{2k}}{4^{2k+1}}$$ where $B_n(x)$ is the $n$-th Bernoulli polynomial and $E_n$ is the $n$-th Euler number?
I have verified ...
- 5,309
2
votes
0
answers
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Inverse Fourier transform for the square root of the Ohmic bath spectral function
I think this is a bit hopeless but let me ask just in case. Consider the real and positive function:
$$
\hat{f}(\omega) = \sqrt{\frac{\omega}{1-e^{-\frac{\omega}{T}}}} e^{- \frac{\omega^2}{4\Lambda^2}}...
- 619
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Does the integral $\int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx$ have a known closed form? [duplicate]
I am studying the definite integral
$$
I = \int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx .
$$
The integral does converge:
as $x \to 0$, $\ln(1+x) \sim x$ and $\ln(1-x) \sim -x$, so the ratio tends to $-...
2
votes
1
answer
139
views
analytic continuation of the series $\sum_{n=0}^{\infty}\frac{n^2}{\sqrt{a^2+n^2}}$
I would like to understand how to make sense of the following divergent series, or at least to identify the appropriate analytic continuation of its general term:
\begin{align}
\sum_{n=0}^{\infty}\...
- 305
6
votes
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answers
98
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Does the polylogarithm $\mathrm{Li}_k(x)$ solve a first or second order ODE/ADE?
I have been working with the Polylogarithm on several problems and I think it might help if I knew an algebraic/ordinary differential equation (ADE/ODE) which it satisfies (and just for the sake of it!...
- 2,506
1
vote
0
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79
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Definite integral of the two Bessel functions / (x-a)
In my work, an integral of the following type arose:
$$\int_0^\infty dx J_l(a x) J_l(b x) \frac{1}{x - c + i 0}.$$
Here $J$ is the Bessel function of the first kind. Assume that $a$, $b$, and $c$ are ...
4
votes
1
answer
271
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Prove that $\int_0^1\operatorname{Li}_2\left(\frac{1-x^2}{4}\right)\frac{2}{3+x^2}\,\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}$
I would like to prove that
$$\int_0^1 \operatorname{Li}_2 \left(\frac{1-x^2}{4}\right) \frac{2}{3+x^2} \,\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}.$$
It’s known that $\Im \operatorname{Li}_3(e^{2πi/3})= \...
- 79
3
votes
3
answers
268
views
Analytic sum of an alternating series$\sum\limits_{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}$
I recently came across the following series with a positive real number $a$:
\begin{align}
S(a) = \sum _{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}
\end{align}
Does anyone know if ...
- 305
4
votes
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126
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Closed form for Dirichlet series whose coefficients are the Möbius function times a geometric series
By definition,
$$
\sum_{n=1}^{+\infty}\frac{1}{n^s} = \zeta(s)
\tag{*}
$$
when the real part of $s$ is large enough ($>1$). I am also aware that
$$
\sum_{n=1}^{+\infty}\frac{\mu(n)}{n^s} = \frac{1}...
- 6,568
9
votes
1
answer
281
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Seeking generalizations of an Airy Integral
Some crude numerical experiments led me to stumble upon the amusing result that $$\int_{0}^{\infty} \frac{1}{\operatorname{Bi}(t)^2}\, dt = \frac{\pi}{\sqrt{3}}$$
where $\text{Bi}(x)$ is an Airy ...
- 5,309
1
vote
0
answers
35
views
Question related to Hermite Polynomials
I try to express the following:
$$\text{e}^{(ax^2+bx+c)}(-\hbar\frac{\partial}{\partial x})^n\text{e}^{-(ax^2+bx+c)}$$
in terms of the Hermite Polynomials
using the definition of Hermite polynomial ...
- 344
17
votes
1
answer
554
views
$\int_0^{\infty}\text{Ai}^4(x)dx = \ln(3)/24 \pi^2$
How can I prove that
$$\Omega = \int_{0}^{\infty} \text{Ai}^4(x) \, dx = \frac{\ln(3)}{24 \pi^2}$$
where $\text{Ai}(x)$ is the Airy-function.
Using the Fourier integral representation of the Airy ...
- 5,309
1
vote
1
answer
117
views
Evaluate $\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \operatorname{li}(x) \cos(\ln x)]}{x \ln x} \, \mathrm {dx}$
Evaluate: $$\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \operatorname{li}(x) \cos(\ln x)]}{x \ln x} \, \mathrm {dx}$$
My approach:
$$\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \...
- 491
4
votes
1
answer
271
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Closed form for $\sum_{k=1}^{n-1} k \cot\left(\frac{\pi k}{n}\right)$ and its generalizations
I’ve been looking at the sum
$$
S(n) = \sum_{k=1}^{n-1} k \cot\left(\frac{\pi k}{n}\right),
$$
and after some manipulations, I arrived at the following explicit (though somewhat complicated) ...
- 61