1
vote
0answers
49 views

L2 norm of a M-Whittaker function

Let $M_{\kappa,\mu}(z)$ be the Whittaker function, as defined here http://en.wikipedia.org/wiki/Whittaker_function. Does any one know the evaluation of the following integral? ...
1
vote
2answers
118 views

Evaluate an integral or Fourier coefficients

Consider an integral $$ \int_0^\pi \frac{\cos(kx)}{\cosh(ax)}\ dx $$ there $k\in \mathbb{Z}, a\in \mathbb{R}.$ Of course that is Fourier coefficient for the function $f(x)=\frac{1}{\cosh(ax)}.$ ...
0
votes
0answers
74 views

Integral of Bessel function of 1st kind with complex exponential

Does someone know the solution (simple closed form) of one of theses integrals: $$\int_0^t J_l(s) e^{-iA(t-s)}ds$$ $$\int_0^t \frac{J_l(s)}{s} e^{-iA(t-s)}ds$$ with $l>0$, $t>0$, $\Re(A)>0$, ...
8
votes
2answers
288 views

On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation ...
0
votes
0answers
63 views

What function is “$U_{\nu}(\cdot, \cdot)$”?

I was searching in the Prudnikov (vol. 2) how to solve an integral and I finally found it. However, I didn't recognized a function that appears in the answer. Integral 1.8.2.4: $$ \int_0^x x^{\nu+1} ...
4
votes
1answer
309 views

Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...
1
vote
0answers
246 views

complex contour integral calculation after Möbius transformation

Good day to everyone. In my scientific research I've got stuck with a contour integration problem. I would like to evaluate the following integral: $$I=\int_0^{\infty } \frac{e^{\frac{\alpha -\mathrm ...
4
votes
1answer
226 views

How to get an expression for this integral(Numerically/Analytically)

I have the following problem: I need to evaluate the integral $$\int_{\cos(\alpha)}^{1} P_l(t)P_{l'}(t) dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the l-th ...
4
votes
1answer
388 views

Identity involving Fresnel integrals

In the paper E. Mehlum, Appell and the apple (nonlinear splines in space), Technical Report No. 1676 (1981), Central institute for industrial research, Oslo (reproduced in the book Mathematical ...
2
votes
1answer
285 views

Integral of Modified Bessel Function of the Second Type

Given the identity $$ \int^\infty_0 K_v\left(\alpha\sqrt{x^2+z^2}\right) \frac{x^{2\mu+1}}{\left(\sqrt{x^2+z^2}\right)^v}\:\mathrm{d}x = \frac{2^\mu \Gamma(\mu+1)}{\alpha^{\mu+1}z^{v-\mu-1}} ...
2
votes
1answer
92 views

Dertivative of a Special Function with respect to Order

The marcum Q-function is defined by $$ Q_m(a,b) = \int^\infty_b x \left(\frac{x}{a}\right)^{m-1} \exp\left(-\frac{x^2+a^2}{2}\right) I_m\left(a x\right) \:\mathrm{d} x,$$ where $m\in\mathbb{N}$ , ...
0
votes
1answer
230 views

derivative of a special function in integral form

What is the derivative of $Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right)$ with respect to $x$, i.e, $$\frac{\partial}{\partial x}Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right), \quad ...
5
votes
1answer
438 views

Contour Integral with Gamma functions and 2F1

Given the following contour integral $$\frac{1}{2\pi j}\int^{c+j\infty}_{c-j\infty} \frac{\Gamma(-1+a+s)\Gamma(b+s)}{\Gamma(3+a-s)}\cos(-1+a+s)\, {}_2F_1\Big(-1-a+s,-1+a+s;\frac{1}{2};z\Big) y^s\: ...
5
votes
2answers
1k views

How to do integrals involving two Bessel functions and another function?

I often encounter the integrals in the following form: $\int_0^\infty{\rm Bessel}(ax)\cdot{\rm Bessel}(bx)\cdot f(cx)dx$, where Bessel can be $J$, $N$, $H^{(1)}$, $H^{(2)}$, $I$, or $K$; and $f(x)$ ...
1
vote
2answers
435 views

High dimensional beta integral (a typo in Stein's book “singular integrals”)

Hello, When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake: $$ \int_{R^n} |x-y|^{-n+\alpha} ...
1
vote
1answer
211 views

evaluating an integral related to the volume of Hessenberg orthogonal matrices

Consider the following integral, $$ {1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi} \sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right) \sin^{2}\left(\theta_{2} \over 2\right)\,} \,{\rm ...