Tagged Questions
9
votes
3answers
98 views
Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$
I'm trying to figure out how to evaluate the following:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx
$$
I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
1
vote
0answers
18 views
Closed form for $k$-th moment
I would like to calculate this $k$-th moment:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(i^n\frac{\sin(\pi a x+\frac{n\pi}{2})}{\pi ax+\frac{n\pi}{2}}+(-i)^n\frac{\sin(\pi a ...
1
vote
1answer
34 views
$0$-th moment of product of gaussian and sinc function
I would like to calculate the following integrals:
$$\int_{-\infty}^{+\infty} \quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$
$$\int_{-\infty}^{+\infty} \quad ...
6
votes
2answers
79 views
Computation of $\int_0^{\pi} \frac{\sin^n \theta}{(1+x^2-2x \cdot \cos \theta)^{\frac{n}{2}}} \, d\theta$
Show that
$$\begin{align*} \forall x \in [-1,1]: \int_0^{\pi} \frac{\sin^n \theta}{(1+x^2-2x \cdot \cos \theta)^{\frac{n}{2}}} \, d\theta &= c_n \tag{1} \\ \int_0^{\pi} \frac{\sin^{n+2} ...
4
votes
1answer
38 views
$k$-th moment of product of gaussian and sinc
I would like to calculate the following integrals:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$
$$\int_{-\infty}^{+\infty} \quad ...
1
vote
2answers
50 views
Convolution of $1/(1+x^2)$ and $\exp(-x^2/(4t))$
Is there a closed form formula for the convolution of $1/(1+x^2)$ and $\exp(-x^2/(4t))$, where $t>0$, i.e. the integral
$$\int_{-\infty}^\infty ...
2
votes
1answer
41 views
Integral of product of Bessel functions of the first kind
I would like to solve the integral:
$$\int_0^{+\infty}\quad rJ_n(ar)J_n(br)\quad dr$$
Is there any closed form for it?
Thanks!
5
votes
1answer
87 views
A problematic integral: $\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$
Is there a special trick to calculate this integral?
$$\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$$
for $\lambda>0$.
13
votes
1answer
159 views
$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$
I need help with calculating this integral:
$$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$
where ...
5
votes
1answer
80 views
Interesting definite integral involving exp and trig
I'm trying to evaluate the following integrals:
$$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$
$$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$
for which I want to find ...
4
votes
2answers
96 views
About the Beta function : $\text{B}\left(\frac{4}{3},\frac{2}{3}\right)$.
Find the value of : $\text{B}\left(\frac{4}{3},\frac{2}{3}\right)$, where $\text{B}(x,y)$ is the Beta function.
Why do I need this ? Because I want to calculate : $$
\int\limits_{ - \infty }^\infty ...
4
votes
1answer
117 views
Integral of $\ln |\sin(x)|$
Does anyone have a real formula for the integral $$\int\ln |\sin(x)|\,dx ?$$
Neither Maple nor Mathematica give a real answer.
Using integration by parts and the series for $x\cot x$, I get $$x\ln ...
1
vote
1answer
61 views
Rewriting the integral $\mathrm{erf}(x) = \frac{1}{\sqrt{\pi}} \int_{-x}^x e^{-t^2} dt.$
I'm trying to implement an equation into a programming language which doesn't have functions for integrals. However as it's many years since I've had any math exercise I'm having some trouble ...
1
vote
2answers
53 views
Orthogonality of Legendre Functions
The Legendre Polynomials satisfy the following orthogonality condition:
The definite integral of $P(n,x) \cdot P(m,x)$ from $-1$ to $1$ equals $0$, if $m$ is not equal to $n$: $$\int_{-1}^1 P(n,x) ...
2
votes
0answers
57 views
Calculating the integral $\int_{0}^{\pi /6}\sqrt{1-\left(\frac{R_s\sin \theta }{C_L}\right)^2} d\theta$
I want to integrate $I=\int\limits_{0}^{\pi /6}{\sqrt{1-{{\left( \frac{{{R}_{s}}\sin \theta }{{{C}_{L}}} \right)}^{2}}}d \theta}$.
I get incomplete elliptic integral $E(z\mid m)$ in the calculation by ...
1
vote
0answers
72 views
Approximations of the incomplete elliptic integral of the second kind
For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
8
votes
2answers
100 views
Approximation of $\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$ [duplicate]
I am reading about the Riemann hypothesis, and the article mentioned the Li function:
$$\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$$
They said that this function can be approximated:
...
8
votes
1answer
152 views
Interesting log sine integrals $\int_0^{\pi/3}x \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{17\pi^4}{6480}$
Show that
$$\begin{aligned} \int_0^{\pi/3}x \log^2 \left(2\sin \frac{x}{2}
\right)dx &= \frac{17\pi^4}{6480} \\ \int_0^{\pi/3}\log^2
\left(2\sin\frac{x}{2} \right)dx &= ...
12
votes
1answer
187 views
Interesting Integral $\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx$
I am asking this question out of curiosity.
$$\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx = \frac{ \left(2\cos \frac{n}{2} \right)^{\alpha ...
4
votes
1answer
77 views
the solution for an integral including exponential integral function
I have the following integral
$$\int_c^\infty{x^{a-1} e^{\ p \ x} \ \mathrm{Ei}(-p\ x) \ \mathrm{d}x}.$$
I'd like you to help me to evaluate it or giving me a hint to proceed.
2
votes
1answer
119 views
Verification of integral over $\exp(\cos x + \sin x)$
I found the following integral in a paper I was reading:
\begin{equation}
\frac{1}{2\pi} \int\limits_{-\pi}^{\pi} \exp\left(a \cos x + b \sin x\right) dx = I_0\left(\sqrt{a^2+b^2}\right),
...
8
votes
2answers
155 views
Integral $\int\limits_0^\infty \prod\limits_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx$
Does anybody know how to prove this identity?
$$\int_0^\infty \prod_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx=\frac{\sqrt{\pi}}{2}\frac{\Gamma ...
0
votes
2answers
248 views
7
votes
2answers
165 views
Evaluating $\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} dx$
What is the value of the following integral?
$$\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} \,dx$$
Here $\Gamma(x)$ is Euler's gamma function.
EDIT: Can we improve the upper bound strictly smaller than $1$?
...
2
votes
0answers
69 views
A more general exponential integral
More generalized on the previous question: A improper integral with Glaisher-Kinkelin constant
Show that :
...
1
vote
1answer
61 views
Proof of $\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$?
Can anyone show a proof of $$\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$$
where $0<a<1$, and $$\Gamma(a)=\int_0^\infty t^{a-1}e^{-t}\,dt.$$ Thank you.
3
votes
2answers
180 views
Calculate an integral involving Hermite polynomials
I have to calculate the integral
$$\frac{1}{\sqrt{2^nn!}\sqrt{2^ll!}}\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx}H_l(x)\;\mathrm{d}x$$
where $H_n(x)$ is the $n^{th}$ Hermite ...
5
votes
1answer
103 views
Integral representation of cosecant function
According to Wolfram website http://functions.wolfram.com/ElementaryFunctions/Csc/introductions/Csc/05/,
There exists a "well-known" integral representation for the cosecant function, i.e. ...
3
votes
1answer
108 views
Integrals of Hermite polynomials over $(-\infty, 0)$
Does there exist a simple expression for integrals of the form,
$I = \int_{-\infty}^0 H_n(u) H_m(u)\, \mathrm{e}^{-u^2}\,du$,
where $m$ and $n$ are nonnegative integers and $H_n$ is the $n$'th ...
4
votes
0answers
149 views
How to find the inverse Fourier Transform of the product of two bessel functions of the first kind and a complex exponential function?
I am attempting to find a closed form or symbolic expression of the inverse Fourier transform of the product of two Bessel functions of the first kind and a complex exponential, e.g.
$P(t) = IFT_w \{ ...
3
votes
2answers
76 views
Definite Integral Arising from a Double Integral
I gave an integral to a student. She reported back to me that she could not do it. I've tried a couple of approaches and have failed. I imagine it is fairly easy. It's a double integral. And, no ...
2
votes
2answers
286 views
Integral of Hermite polynomial multiplied by $\exp(-x^2/2)$
What is the value of $\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}H_n(x)dx$ where $H_n(x)$ is the $n^{\small\mbox{th}}$ Hermite Polynomial (physicist's convention)?
6
votes
1answer
180 views
How to evaluate the following integral using hypergeometric function?
May I know how this integral was evaluated using hypergeometric function?
$$\int \sin^n x\ dx$$
Wolframalpha showed this result but with no steps
Thanks in advance.
4
votes
1answer
107 views
Definite integral with $\mathrm{Si}$ in integrand
Does the function
$$f(t) = \int_0^{\sqrt{3}} (x^2-1) \;\mathrm{Si}((x^2-1)\,t)\; \mathrm{d}x$$
have a representation in terms of elementary functions of $t$ for real, positive $t$? Here,
...
3
votes
3answers
98 views
Can this function be expressed in terms of other well-known functions?
Consider the function $$f(a) = \int^1_0 \frac {t-1}{t^a-1}dt$$
Can this function be expressed in terms of 'well-known' functions for integer values of $a$? I know that it can be relatively simply ...
8
votes
1answer
134 views
Evaluation of an integral involving the Lambert W function
Wikipedia claims that
$$\int_0^\infty W\left(\frac{1}{x^2}\right) \,\text dx=\sqrt{2\pi}$$
and a numerical computation seems to confirm this.
How can this result be proven?
-1
votes
1answer
150 views
Integral of the square of the Bessel Function
Has anyone ever solved the integral of square of Bessel and Neumann functions? This is for standing wave analysis on a cylindrical object.
I need to integrate $(AJ(kr) + BY(kr))^2$ from $r=a$ to ...
3
votes
2answers
57 views
Advice on an integral involving the error function
I'd like to calculate the following integral:
$$\int^{\infty}_{0} \mathrm{erf}\left(\frac{\alpha}{\sqrt{1+x}} - \frac{\sqrt{1+x}}{\beta}\right) \exp\left(-\frac{x}{\gamma}\right)\, dx,$$
where ...
5
votes
1answer
128 views
Error Function limit
$$\prod_{n=1}^{\infty}{\frac{2}{\sqrt{\pi}}\int_0^n e^{-x^{2}} \mathrm{d}x} \approx 0.83874 $$
Is it a known constant? I couldn't find anything about it. Do you know ways to calculate the value ...
0
votes
1answer
203 views
Solve in terms of the Gamma function
Show:
\begin{align*}
\int\limits_0^1\sqrt{\frac{1-x^2}{1+x^2}}\,\mathrm d x
&=\frac{\sqrt \pi}{4}\left(\frac{\Gamma ...
2
votes
1answer
111 views
Definite integral involving Fresnel integrals
I am seeking to evaluate
$\int_0^{\infty} f(x)/x^2 \, dx$
with
$f(x)=1-\sqrt{\pi/6} \left(\cos (x) C\left(\sqrt{\frac{6 x}{\pi }} \right)+S\left(\sqrt{\frac{6 x}{\pi }} \right) \sin
...
1
vote
1answer
119 views
How to compute $\int\frac{x^7}{\sin(x)} dx$ efficiently?
How to compute $\int\frac{x^7}{\sin(x)} dx$ efficiently ?
We need $Polylog$ for this.
4
votes
1answer
179 views
Integral related to the modified Bessel function
I would like to solve the integral
$$F_n(\kappa,\theta,\phi)=\int_{-\pi}^{\pi}{\rm e}^{\kappa\cos(x-\theta)}\cos(n\, x-\phi)\,{\rm d}x$$
that appears related to the identity
...
0
votes
2answers
394 views
what are the properties of gamma function? [closed]
In mathematics, the gamma function (represented by the capital Greek letter $\Gamma$) is an extension of the factorial function,
example:
$\Gamma(x)$, $\Gamma(ix)$.
What are the physical properties ...
0
votes
1answer
114 views
Problem with ratios of integrals
I have the following integral from a paper I'm reading:
$$f(z)=\frac{\displaystyle\int_0^{\pi/2}\,\tan \alpha\, J_0(z \sin\alpha)\, d\alpha}{\displaystyle \int_0^{\pi/2}\tan\alpha\,d\alpha}$$
...
17
votes
4answers
573 views
Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$?
I would like to evaluate the sum
$$
\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)
$$
Where $\operatorname{Si}$ is the sine integral, defined as:
$$\operatorname{Si}(x) := ...
1
vote
1answer
122 views
Calculate $I(\alpha, x,y)=\int\limits_0^1 {{v^{\alpha - 1}}{{(1 - vx)}^{\alpha - 1}}{e^{vy}}dv,\,\,\,0 < \alpha ,x,y < 1}.$
I want to calculate this integral with singularity:
$$I(\alpha, x,y)=\int\limits_0^1 {{v^{\alpha - 1}}{{(1 - vx)}^{\alpha - 1}}{e^{vy}}dv,\,\,\,0 < \alpha ,x,y < 1}. $$
I hope to obtain a ...
3
votes
2answers
85 views
A question of the norm calculation of Hermite function.
Define the Hermite function $H_n (x)$ by $$H_n (x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} $$ then prove that $$ \int_{\mathbb R} |H_n (x) |^2 e^{-x^2} dx = 2^n n! \sqrt{\pi}$$
10
votes
1answer
407 views
Evaluation of $\sum_{x=0}^\infty e^{-x^2}$
Most of us are aware of the classic Gaussian Integral
$$\int_0^\infty e^{-x^2}\, dx=\frac{\sqrt{\pi}}{2}$$
I would be interested in evaluating the similar sum
$$\sum_{x=0}^\infty e^{-x^2}$$
Now, ...
6
votes
4answers
624 views
Calculate integrals involving gamma function
What are the usual ways to follow in order to solve the integrals given below?
$$\begin{align*}
I&=\int_0^1 \ln\Gamma(x)\,dx\\
J&=\int_0^1 x\ln\Gamma(x)\,dx
\end{align*}$$
