Questions tagged [special-functions]

Question 1

I would like to prove that $$ \int_0^1 \text{Li}_2(\frac{1-x^2}{4}) \frac{2}{3+x^2}dx= \frac{\pi^3 \sqrt{3}}{486}$$ It's known that $\Im \text{Li}_3(e^{2πi/3})= \frac{2\pi^3}{81}$, but I struggle to ...

Question 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 ...

Question 3

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}...

Question 4

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 ...

Question 5

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 ...

Question 6

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 ...

Question 7

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) - \...

Question 8

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) ...

Question 9

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,...

Question 10

Let $z=x+i y$, $x\geq 1/2$. 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, ...

Question 11

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_{\...

Question 12

I'm trying to see if there are good closed form expressions for $$ \, _2F_1\left(-\frac{1}{2} (2 \nu +1),-\frac{1}{2} (2 \nu +1);\frac{1}{2};z\right) $$ where $\nu \in \{0,1,2,3,\ldots \}$. Using ...

Question 13

he Bessel functions of the first kind $J_n(x)$ are defined as the solutions to the Bessel differential equation: $$ x^2y''(x)+ xy'(x)+(x^2-n^2)y(x)=0.$$ The special case of $n=0$ gives $J_0(x)$ as the ...

Question 14

I am trying to evaluate $$\int_0^{\infty} \frac{1}{\sqrt{t}} \cos\left(\frac{t^3}{3} + \eta t \pm \frac{\pi}{4} \right) \, dt$$ where $\eta$ is some constant. The context of this problem comes from ...

Question 15

Can one express $$I =\int_0^1 (1-u) \left(\cot \pi u - \frac{1}{\pi u}\right) \log u\, du$$ in a closed form - say, in terms of known constants, or special values of special functions? This is an ...

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Questions tagged [special-functions]

Prove that $ \int_0^1 \text{Li}_2(\frac{1-x^2}{4}) \frac{2}{3+x^2}dx= \frac{\pi^3 \sqrt{3}}{486}$

Analytic sum of an alternating series involving $n/(n+\sqrt{a^2+n^2})^2$

Closed form for Dirichlet series whose coefficients are the Möbius function times a geometric series

Seeking generalizations of an Airy Integral

Question related to Hermite Polynomials

$\int_0^{\infty}\text{Ai}^4(x)dx = \ln(3)/24 \pi^2$

Evaluate $\int \frac{e^x [\operatorname{Ei}(x) \sin(\ln x) - \operatorname{li}(x) \cos(\ln x)]}{x \ln x} \, \mathrm {dx}$

Closed form for $\sum_{k=1}^{n-1} k \cot\left(\frac{\pi k}{n}\right)$ and its generalizations

Integral transform of $L^{\alpha}_m(x)$

Clean version of inequality for $\Gamma(z)$ - known?

Airy zeta as a polynomial

How to prove this special case of hypergeometric function identity

Bessel differential equation?

An Airy Integral

What does $\int_0^1 (1-u) \left(\cot \pi u - \frac{1}{\pi u}\right) \log u \,du $ equal

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