# Search results

1. ### How to integrate -exp((s^2 - t^2)/2)*(f'(t) - t*f(t)) dt

Apply the product rule for differentiation to the solution. This gives you the integrand. If you recognize this then you just need to do the reverse to get the solution.
2. ### Show integral is equal to Bessel function

You can expand the exponential term on the rhs as a product of two series: \exp\left(\frac{\sqrt{x^2-1}}{2}\left(t-\frac{1}{t}\right)\right)=\sum_{n=0}^{\infty}\frac{(\sqrt{x^2-1})^n}{2^nn!}t^n\sum_{k=0}^{\infty}\frac{(\sqrt{x^2-1})^k}{2^kk!}(-t)^{-k} and so it becomes...
3. ### Is there any website that goes through the derivation of all the integral formulae?

This website is an attempt to compile proofs of all integrals listed in the Gradshteyn and Ryzhik integral tables: http://129.81.170.14/~vhm/Table.html" [Broken] There are still lots of gaps to be filled in though!
4. ### Is there any way to solve this?

you can use the identity \sin N\theta = \sum_{k=0}^N \binom{N}{k} \cos^k \theta\,\sin^{N-k} \theta\,\sin\left(\frac{1}{2}(N-k)\pi\right) to obtain a polynomial in \sin \theta which can then be solved for \theta . However, you will probably need to solve it numerically for N>3.
5. ### Multivariable Dirac Delta Functions

Are you sure that the only nonzero point is (0,\pi)? Depending on the values of a and b (e.g. if one of them is negative) the function g(x,y)=a(cos(x)-1)+b(cos(y)+1) could have a number of roots between the limits of integration. Given the following identity for a single variable Dirac Delta...
6. ### Hermite Polynomial identity

Does anyone know how to prove the following identity: \Sigma_{k=0}^{n}\left(\stackrel{n}{k}\right) H_{k}(x)H_{n-k}(y)=2^{n/2}H_{n}(2^{-1/2}(x+y)) where H_{i}(z)represents the Hermite polynomial?
7. ### Multidim. Gaussian integral with linear term

Yes, I think you can assume that the angle is dependent on only one of the angles in a spherical polar co-ordinate system by assuming that the vector \vec{a} lies along one of the axes in the system similar to the approach in the following integral...
8. ### Solution to an exponential integral

Hi, I am trying to find an analytic solution to the following double integral...
9. ### Exponential integral with trigonometric argument

Yea, I got the hypergeometric functions from mathematica as well but I really don't think it improves the result! Thanks for your help!
10. ### Multidim. Gaussian integral with linear term

Woops! yea, of course you are right. I forgot the r^{d-1} term! I think your answer looks correct now.
11. ### Exponential integral with trigonometric argument

Hi Jason, That is very nice, thanks! Do you know of any methods/tricks that I could try to reduce it from a double to a single summation? Thanks for your help.
12. ### Multidim. Gaussian integral with linear term

Hi Orbb, I think the formula you are looking for is \frac{\pi^{\frac{d}{2}}}{\alpha}\frac{1}{\Gamma(\frac{d}{2})} It is straightforward to get this result in spherical co-ordinates and I don’t think you can find any easier way to do it. Assume |\vec{y}| = r =...
13. ### Exponential integral with trigonometric argument

Hi JasonRF, Thanks for all your help, it looks like you put in a lot of work! I am going to take sometime to go through it and see how it looks - I am still clinging to the hope that a neater analytical solution might pop out! Thanks again.
14. ### Exponential integral with trigonometric argument

Hi, Does anyone know of an analytic solution for the integral \int_{0}^{\pi}\sin\theta\exp\left(a\sin^{2}\theta+b\sin\theta\right)d\theta Thanks.
15. ### Series of a Bessel function

Hello, There is a book called "Integral representation and the computation of combinatorial sums" by G. P. Egorychev that might be useful. The general idea is to convert each term of the series to a contour integral and then using some theorems from several complex variables to manipulate the...
16. ### Proof of a definite multiple integral relation

Hi Mathman, I am trying to extend the method you suggested to solving integrals of the form \int_0^{\pi}\int_0^{2\pi}\sin\theta f\left(\alpha\sin\theta\cos\phi+\beta\sin\theta\sin\phi+\gamma\cos\theta\right)g\left(a\sin\theta\cos\phi+b\sin\theta\sin\phi+c\cos\theta\right)...
17. ### Proof of a definite multiple integral relation

Thanks for your help Mathman, it makes sense to me now! I had begun to work on it from the point of view of the surface area of a sphere but hadn't managed to get it. Thank you!
18. ### Proof of a definite multiple integral relation

Mathman, Thanks for your help. I have followed your advice on first \phi and then \theta and got the following. Starting from \alpha\cos\theta+\beta\cos\phi\sin\theta+\gamma\sin\theta\sin\phi=\alpha\cos\theta+\sin\theta\left(\beta\cos\phi+\gamma\sin\phi\right)\nonumber...
19. ### Proof of a definite multiple integral relation

Hello, The following is identity no. 4.624 in Gradshteyn & Ryzhik's Table of Integrals, Series and Products: \int_0^{\pi}\int_0^{2\pi}f\left(\alpha\cos\theta+\beta\sin\theta\cos\phi+\gamma\sin\theta\sin\phi\right)\sin\theta d\theta d\phi=2\pi\int_0^{\pi}f\left(R\cos...
20. ### Integrating products of Bessel functions

Hi guys, Does anyone have any ideas about an analytical solution for the following integral? \int_{0}^{2\pi}J_{m}\left(z_{1}\cos\theta\right)J_{n}\left(z_{2}\sin\theta\right)d\theta J_{m}\left(\right) is a Bessel function of the first kind of order m. Thanks.
21. ### Analytic solution to an exponential integral

Excellent, thanks Count Iblis!
22. ### Analytic solution to an exponential integral

Thanks Count Iblis, I think the result does involve a Bessel function. Related to this, does anyone know how the result \int_{0}^{2\pi}\exp(x\cos\theta + y\sin\theta)d\theta = 2\pi I_{0}(\sqrt{x^2+y^2}) is obtained? I_{0} is the modified Bessel function of the first kind.
23. ### Analytic solution to an exponential integral

Hello, I am trying to find an analytic solution to the following: \int_{-1}^{1}\exp(-p\sqrt{1-x^{2}}-qx)dx where p,q > 0. Does anyone have any ideas? Thanks.