Hi! Are x, mu and v constant? What do you mean by "in terms Bessel functions"? I think, calculating this integral numerically should not be a problem.
Hi Vikash, yes x, mu and v constant. We think in terms of Bessel functions can be found . Gaussian solutions may be .
Well, I am not sure on how to arrive at the analytical solution which has Bessel terms in it. But I guess, a numerical evaluation should not be difficult using Simpson's or Gaussian quadrature rule.
I am trying to get an anaytical solution by using the mathematica package, but as already suggested by Vikash and Eberto you can solve it by numerical integration.
Thank you Agustin. But I do not get the solution I want. Analytical solutions needed me.
Where do you get that sum? It looks like some expansion of a fundamental solution of the Helmholtz equation in some coordinate system? spheroidal?
I obtained from the differential equation. I was not found similar result in the literature. Dear Howard;
Is there a source that you can recommend related Helmholtz equation?
Can you give the differential equation and also the derivation which gives you the sum?
Hello, dear Kazakov;
Thank you so much. You're right. It can give specific results. I found the following results for some specific values. But I'm looking for a more general solution.
Dear colleague, may be, I have not expressed main idea clearly enough. If you have solution in closed form, like definite integral, this is the best possible expression. Further "improvements" of your relation are unnecessary. I do not know initial statement of your problem, but definite integral is the best description of the answer.
Abdulla, you sum representation is quite nice. I've been able to rederive it. I also have found another integral representation in terms of modified Bessel function of zero order. I was wrong about the Helmholtz equation. The kernel of your integral is simply the inverse distance in prolate spheroidal coordinates, and then you integrate multipled by e^{-x\mu'}. Using the DLMF http://dlmf.nist.gov/10.47.E7 for i_n^{(1)}(x), I find the sum to be 2\sum_{n=0}^\infty (-1)^n (2n+1) P_n(\mu) Q_n(v) i_n^{(1)}(x). To demonstrate how useful the sum representation is, look at the asymptotic result as x\to\infty. In that case i_n^{(1)}(x) \sim e^{x}/(2 x). In this case there is no x-dependence and your integral f(x,\mu,v)\sim e^x/(x(v+x)) using Heine's formula, \sum_{n=0}^\infty (2n+1) P_n(x) Q_n(z) = 1/(z-x) and using the Parity of Legendre polynomials. The integral representation I derived is quite useful as well. I'll show it you tomorrow.
Thank you dear Howard; you're right, not like the Helmholtz differential equation. This integral is not like integrals in the literature.
Thank you dear Kazakov.
Basic aim to solve this definite integral. I think I have the solution for some special cases. But I can not get an overall solution. I looked at the book you recommended earlier. But I could not find useful transformation.
Since the expression in the denominator, under the square root, is quadratic in the integration variable, the integral becomes a sum of error functions, integrals of
exp(-As2) over a finite interval, with A some constant, function of the other constants. (The limits of integration are deduced by the condition that the square root be real.) So error functions, rather than Bessel functions, are the most useful representation. Of course it's possible to relate the two. For with real roots one can simplify the denominator using trigonometric functions or hyperbolic functions, depending on the sign of the quadratic ted, and one finds the representation in terms of (modified) Bessel functions.
If they're equal then it's of the form e-Bs/|s-a| with given limits of integration and a, B constants.
If the roots are complex, then the expression under the root can be simplified using hyperbolic functions and one is led to the expression involving modified Bessel functions.
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