architect
Oct2-07, 02:27 PM
Hi all,
I would like to get some advice regarding a difficult integration problem.
\int_{\varphi_{o}-\Delta\varphi}^{\varphi_{o}+\Delta\varphi}\int_{\v artheta_{o}-\Delta\vartheta}^{\vartheta_{o}+\Delta\vartheta}e^ {\kappa[sin\vartheta_{o}\sin\vartheta\sin\varthetacos(\var phi-\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_{\v artheta_{o}-\Delta\vartheta}^{\vartheta_{o}+\Delta\vartheta}e^ {\kappa\cos\gamma}sin\vartheta\partial\vartheta\pa rtial\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\varth etacos(\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_{\v artheta_{o}-\Delta\vartheta}^{\vartheta_{o}+\Delta\vartheta}\s um^{\infty}_{0}(2n+1)[\sqrt{\frac{\pi}{2\kappa}}I_{n+1/2}(\kappa)]P_{n}(cos\gamma)sin\vartheta\partial\vartheta\part ial\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})\Ups ilon{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
I would like to get some advice regarding a difficult integration problem.
\int_{\varphi_{o}-\Delta\varphi}^{\varphi_{o}+\Delta\varphi}\int_{\v artheta_{o}-\Delta\vartheta}^{\vartheta_{o}+\Delta\vartheta}e^ {\kappa[sin\vartheta_{o}\sin\vartheta\sin\varthetacos(\var phi-\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_{\v artheta_{o}-\Delta\vartheta}^{\vartheta_{o}+\Delta\vartheta}e^ {\kappa\cos\gamma}sin\vartheta\partial\vartheta\pa rtial\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\varth etacos(\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_{\v artheta_{o}-\Delta\vartheta}^{\vartheta_{o}+\Delta\vartheta}\s um^{\infty}_{0}(2n+1)[\sqrt{\frac{\pi}{2\kappa}}I_{n+1/2}(\kappa)]P_{n}(cos\gamma)sin\vartheta\partial\vartheta\part ial\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})\Ups ilon{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