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

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

mukushla
 3 years ago
Best ResponseYou've already chosen the best response.0bessel function of first kind

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

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

Jonask
 3 years ago
Best ResponseYou've already chosen the best response.0can we find the intergral of these

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

vf321
 3 years ago
Best ResponseYou've already chosen the best response.0Well 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

vf321
 3 years ago
Best ResponseYou've already chosen the best response.0So 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

vf321
 3 years ago
Best 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

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

vf321
 3 years ago
Best ResponseYou've already chosen the best response.0No, 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

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

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

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

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