All Questions
12 questions
13
votes
2
answers
2k
views
$\mathrm{Bessel}^3$ Integral
I'm trying to calculate the following integral:
$\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$
($\mathrm{BesselJ}[n,x]$ is ...
- 133
5
votes
2
answers
1k
views
An integral involving three Bessel functions
I am looking for a closed form for the following integral
$$ I = \int_0^\infty \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx) $$
which can be thought of as a particular case of the more general integral
...
- 155
3
votes
2
answers
302
views
Definite integral of Bessel function of the first kind times $x^{3/2}$
I am looking for preferably a closed form (or series solution if not possible) for the following integral:
$$\int_0^a x^{3/2} J_\nu (bx) dx$$
where $\nu$ is an integer. This 1D integral appears when ...
- 83
3
votes
1
answer
733
views
Integral with 4 Bessel functions and an exponential
I would like to solve the following integral
$$
\int_0^\infty e^{-a k^2} J_{3/2}(b k) J_{3/2}(c k) J_{3/2}(f k) J_{1/2}(r k) k^{-3} dk,
$$
where $a,b,c,f,r > 0$, and $J_\nu(x)$ is the Bessel ...
- 61
3
votes
1
answer
260
views
Is it possible to get an equation with two exponentials and a bessel function in closed form?
Is it possible to get the equation below into closed form? I have tried using integration tables but I haven't found anything that matches. Are there any other methods to achieve a closed form ...
3
votes
1
answer
1k
views
Definite integral with modified Bessel functions, trigonometric function and a power
I require the following integral involving the modified Bessel functions of the first and second kinds of order one
$$I(a, b, c) = \int_0^{\infty} \frac{\sin(ax)}{x} I_1(bx) K_1(cx) \mathrm{d}x, \...
- 285
2
votes
0
answers
517
views
What am I missing in this highly oscillatory integral? [closed]
I want to numerically integrate this equation (in python):
$\int_{0}^{\infty}{\rm d}k f(k) J_v(r k)J_v(s k) $,
where f(k) is a non-smooth function, and $J_v$ are the Bessel function of the fist kind....
- 121
1
vote
1
answer
952
views
Integration of Bessel Function of the first kind
I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$. My statement as follows:
$$\int_{0}^\infty F(x)[Bx^3J_0(xy)+x^4J_1(xy)]dx=G(y)$$
where $B$ is a constant, ...
- 11
1
vote
1
answer
187
views
Definite intergal with two K-Bessel functions and x
I would like to calculate the definite integral with K-Bessel funcitons and a and b complex (n and k integers):
$$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(kx) \; dx$$
I could not find it in ...
- 1,199
1
vote
0
answers
302
views
Integration involving modified bessel function, exponential and power
I need to find the following integration.
$$
\int_0^a e^{-(N-1)x}(\sqrt{4x}K_{1}(\sqrt{4x}))^N
$$
where
$$
a>0, \quad N \geq 1
$$
Any help will be much appreciated.
BR
Frank
- 119
0
votes
2
answers
1k
views
Indefinite integration of multiplication of two Bessel function
I am trying to calculate this integral. I know it has an analytic expression when $a = 0$. But, is there any analytic expression for this case?
$$\int_{a}^{\infty}J_2(bx)J_1(cx)\,dx$$
Thanks in ...
- 101
0
votes
0
answers
81
views
Fourier transform of an exponential function with radical argument divided by a radical
I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...
- 1