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Question 1

2votes

1answer

73views

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 ...

Maxime Jaccon

5,299

askedyesterday

Question 2

1vote

0answers

50views

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}}...

Ben

609

asked2 days ago

Question 3

0votes

0answers

73views

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 $-...

Jamal Hanus Jr

1

askedNov 25 at 1:12

Question 4

2votes

1answer

136views

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}\...

Alessandro Pini

305

askedNov 24 at 15:49

Question 5

6votes

0answers

98views

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!...

Eli Bartlett

2,506

askedNov 22 at 18:14

Question 6

1vote

0answers

78views

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 ...

Emma Anderson

33

askedNov 22 at 16:51

Question 7

4votes

1answer

268views

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})= \...

Xiaobao

79

askedNov 21 at 15:32

Question 8

3votes

3answers

267views

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 ...

Alessandro Pini

305

askedNov 21 at 9:54

Question 9

4votes

0answers

125views

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}...

Gro-Tsen

6,568

askedNov 16 at 11:24

Question 10

9votes

1answer

281views

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 ...

Maxime Jaccon

5,299

askedNov 15 at 5:02

Question 11

1vote

0answers

35views

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 Polynomialsusing the definition of Hermite polynomial ...

R. Bhattacharya

344

askedNov 13 at 12:06

Question 12

17votes

1answer

551views

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

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 ...

Maxime Jaccon

5,299

askedNov 11 at 17:23

Question 13

1vote

1answer

117views

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) - \...

Andre Lin

491

askedNov 10 at 20:27

Question 14

4votes

1answer

264views

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) ...

Treesight

61

askedNov 3 at 20:50

Question 15

3votes

1answer

64views

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,...

Z. Alfata

1,577

askedNov 3 at 17:26

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Questions tagged [special-functions]

Prove $B_{2k+1}(1/4) = \frac{-(2k+1) E_{2k}}{4^{2k+1}}$

Inverse Fourier transform for the square root of the Ohmic bath spectral function

Does the integral $\int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx$ have a known closed form? [duplicate]

analytic continuation of the series $\sum_{n=0}^{\infty}\frac{n^2}{\sqrt{a^2+n^2}}$

Does the polylogarithm $\mathrm{Li}_k(x)$ solve a first or second order ODE/ADE?

Definite integral of the two Bessel functions / (x-a)

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

Analytic sum of an alternating series$\sum\limits_{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}$

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 $L^{\alpha}_m(x)$

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