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mukushla
Group Title
i need help with an integral\[\int x J_n^2(x) \ \text{d}x\]
 2 years ago
 2 years ago
mukushla Group Title
i need help with an integral\[\int x J_n^2(x) \ \text{d}x\]
 2 years ago
 2 years ago

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mukushla Group TitleBest ResponseYou've already chosen the best response.0
@mahmit2012
 2 years ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
\[J _{n}\] is just any arbitary function ?
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
bessel function of first kind
 2 years ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
i am out then
 2 years ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
\[x^2\frac{ d^2y }{ dx^2 }+x \frac{ dy }{ dx }+(x^2n^2)y=0\]
 2 years ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
so the bessel function is the solution to this curve
 2 years ago

Jonask Group TitleBest ResponseYou've already chosen the best response.0
can we find the intergral of these
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
emm...im just trying !! i dont know :(
 2 years ago

vf321 Group TitleBest ResponseYou've already chosen the best response.0
Well my good friend wikipedia tells us that we can expand the Bessel Function as follows with taylor series: http://upload.wikimedia.org/math/1/b/2/1b23400208b273377e8cdec7d82f0242.png
 2 years ago

vf321 Group TitleBest ResponseYou've already chosen the best response.0
So then the integral becomes: \[\int x\sum_{m=0}^\infty \frac{(1)^m}{m!(m+n)!}(\frac{1}{2}x)^{2m+n}\]for integer orders
 2 years ago

vf321 Group TitleBest ResponseYou've already chosen the best response.0
\[\int x\left(\sum_{m=0}^\infty \frac{(1)^m}{m!(m+n)!}(\frac{1}{2}x)^{2m+n}\right)^2 dx\]  forgot the dx
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
can u show me the steps with wolfram?
 2 years ago

vf321 Group TitleBest ResponseYou've already chosen the best response.0
No, I can't. But you do see the answer, right? My guess is that to get the answer you have to write \((\sum...)^2\) as a single sum using binomial expansion or something, then you can integrate it easily since it's just a polynomial. Then you reconstruct the sum as three separate sums and get the nice answer that alpha got: http://www.wolframalpha.com/input/?i=integrate+x+BesselJ[n%2C+x]^2+dx
 2 years ago

mahmit2012 Group TitleBest ResponseYou've already chosen the best response.2
dw:1350273811145:dw
 2 years ago

mahmit2012 Group TitleBest ResponseYou've already chosen the best response.2
dw:1350274006370:dw
 2 years ago

mahmit2012 Group TitleBest ResponseYou've already chosen the best response.2
dw:1350274137879:dw
 2 years ago

mahmit2012 Group TitleBest ResponseYou've already chosen the best response.2
dw:1350274335219:dw
 2 years ago
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