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).
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Does the integral $\int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)},円dx$ have a known closed form? [duplicate]
I am studying the definite integral
$$
I = \int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)},円dx .
$$
The integral does converge:
as $x \to 0,ドル $\ln(1+x) \sim x$ and $\ln(1-x) \sim -x,ドル so the ratio tends to $-...
2
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1
answer
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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}\...
6
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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!...
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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,ドル $b,ドル and $c$ are ...
4
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1
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Prove that $\int_0^1\operatorname{Li}_2\left(\frac{1-x^2}{4}\right)\frac{2}{3+x^2},円\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}$ [closed]
I would like to prove that
$$\int_0^1 \operatorname{Li}_2 \left(\frac{1-x^2}{4}\right) \frac{2}{3+x^2} ,円\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}.$$
It’s known that $\Im \operatorname{Li}_3(e^{2πi/3})= \...
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3
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259
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Analytic sum of an alternating series$\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 ...
4
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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
$$
\sum_{n=1}^{+\infty}\frac{\mu(n)}{n^s} = \frac{1}...
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Seeking generalizations of an Airy Integral
Some crude numerical experiments led me to stumble upon the amusing result that $$\int_{0}^{\infty} \frac{1}{\operatorname{Bi}(t)^2},円 dt = \frac{\pi}{\sqrt{3}}$$
where $\text{Bi}(x)$ is an Airy ...
1
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0
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Question related to Hermite Polynomials
I try to express the following:
$$\text{e}^{(ax^2+bx+c)}(-\hbar\frac{\partial}{\partial x})^n\text{e}^{-(ax^2+bx+c)}$$
in terms of the Hermite Polynomials
using the definition of Hermite polynomial ...
16
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1
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$\int_0^{\infty}\text{Ai}^4(x)dx = \ln(3)/24 \pi^2$
How can I prove that
$$\Omega = \int_{0}^{\infty} \text{Ai}^4(x) ,円 dx = \frac{\ln(3)}{24 \pi^2}$$
where $\text{Ai}(x)$ is the Airy-function.
Using the Fourier integral representation of the Airy ...
1
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1
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Evaluate $\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \operatorname{li}(x) \cos(\ln x)]}{x \ln x} ,円 \mathrm {dx}$
Evaluate: $$\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \operatorname{li}(x) \cos(\ln x)]}{x \ln x} ,円 \mathrm {dx}$$
My approach:
$$\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \...
4
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1
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261
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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) ...
3
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1
answer
63
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Integral transform of $L^{\alpha}_m(x)$
According to p.244 in "Magnus, W., Oberhettinger, F., Soni, R.: Formulas and Theorems for the Special Functions of Mathematical Physics, Grundlehren der mathematischen Wissenschaften 52, Springer,...
7
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1
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204
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Clean version of inequality for $\Gamma(z)$ - known?
Let $z=x+i y,ドル $x\geq 1/2$. Is the following inequality true?
$$|\Gamma(z)|\leq (2\pi)^{1/2} |z|^{x-1/2} e^{-\pi |y|/2}$$
If you allow a fudge factor such as $e^{\frac{1}{6|z|}}$ on the right side, ...
3
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1
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Airy zeta as a polynomial
The Airy zeta function is defined as a sum over the zeroes of the $\mathrm{Ai}$ airy function.
$$\zeta _{\mathrm {Ai} }(s)=\sum _{i=1}^{\infty }{\frac {1}{|a_{i}|^{s}}}$$
Integer values for $\zeta_{\...