Solving Integration Problem: Advice Needed

[architect]

Hi all,

I would like to get some advice regarding a difficult integration problem.

$$\int_{\varphi_{o}-\Delta\varphi}^{\varphi_{o}+\Delta\varphi}\int_{\vartheta_{o}-\Delta\vartheta}^{\vartheta_{o}+\Delta\vartheta}e^{\kappa[sin\vartheta_{o}\sin\vartheta\sin\varthetacos(\varphi-\varphi_{o})+cos\vartheta_{o}cos\vartheta]}sin\vartheta\partial\vartheta\partial\varphi$$ Equation 1

Let me briefly explain what I have attempted so far.

I know that I can re-write Equation 1 as

$$\int_{\varphi_{o}-\Delta\varphi}^{\varphi_{o}+\Delta\varphi}\int_{\vartheta_{o}-\Delta\vartheta}^{\vartheta_{o}+\Delta\vartheta}e^{\kappa\cos\gamma}sin\vartheta\partial\vartheta\partial\varphi$$ Equation 2

This is because $$\gamma$$ can be thought of as the angular displacement from $$(\theta,\varphi) to (\theta_{o},\varphi_{o})$$. We can therefore write

$$\cos\gamma=sin\vartheta_{o}\sin\vartheta\sin\varthetacos(\varphi-\varphi_{o})+cos\vartheta_{o}cos\vartheta$$

Since I could not see a straight forward solution I thought of re-writing the above integral in terms of an infinite series represantation and then integrate the series(which would be simpler). I noticed that in Abramowitz and Stegun 's's book (Handbook of Mathematical Functions) there is an equation which might help me achieve my goal. This is equation 10.2.36 and can be found in the following website.

$$e^{\kappa\cos\gamma}=\sum^{\infty}_{0}(2n+1)[\sqrt{\frac{\pi}{2\kappa}}I_{n+1/2}(\kappa)]P_{n}(cos\gamma)$$ Equation 3 or 10.2.36 in Abramowitz&Stegun

where we have the modified bessel function and a Legendre polynomial. Therefore my new integrand will be:

$$\int_{\varphi_{o}-\Delta\varphi}^{\varphi_{o}+\Delta\varphi}\int_{\vartheta_{o}-\Delta\vartheta}^{\vartheta_{o}+\Delta\vartheta}\sum^{\infty}_{0}(2n+1)[\sqrt{\frac{\pi}{2\kappa}}I_{n+1/2}(\kappa)]P_{n}(cos\gamma)sin\vartheta\partial\vartheta\partial\varphi$$ Equation 4

Now I think I must do something about the modified bessel function and the Legendre polynomial.

Do you think I can substitute Equation 10.2.5 from Abramowitz&Stegun (http://www.math.sfu.ca/~cbm/aands/page_443.htm) for the modified Bessel Funtion?

How would I re-write the Legendre polynomial in this case? Can I use the Addition Theorem of Spherical Harmonics?Or I will complicate things even further?If I use the Addition Theorem then I would get:

$$P_{n}(\cos\gamma)=\sum^{n}_{m=-n}\Upsilon^{*}_{nm}(\vartheta_{o},\varphi_{o})\Upsilon{nm}(\vartheta,\varphi)$$ Will this be correct?

Please advise if you think that the procedure I have followed so far is incorrect!

Thanks & Regards

Alex

 

[AiRAVATA]

I think that would be correct. The problem can be justifying the exchange of the double integral with the sum.

Is the first integral in it's original form or have you already transformed it? Maybe there is a simpler way to express it?

---EDIT---
I've found this two identities in N.N. Lebedev's book:

$$P_n(\cos \theta)=\frac{1}{\pi}\int_0^\pi (\cos \theta +i\sin \theta \cos \varphi)^n d\varphi,\qquad 0<\theta<\pi.$$

$$e^{ika \cos \varphi}=J_0(ka)+2\sum_{n=1}^\infty (-1)^nJ_n(ka) \cos n\varphi,$$

where $J_n$ are Bessel functions of order $n$.

Maybe this can help.

 

[architect]

Hi,

Many Thanks for your reply. Yes the first double integral is in its original form (Equation 1). The expression that you wrote for the Legendre polynomial from Lebedev's book will it not make it more complicated(because of the integral)?

Thanks again

Regards

Alex

 

[AiRAVATA]

Could be, the only way to find out is to get your hands dirty!

I posted both formulas in order to give you another choice for your calculations. The thing with special functions is that some representations make the work easier, while others make it impossible. Let me know how it went.