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Prove a sum identity for bessel function

  1. Dec 16, 2007 #1
    Hi This is one of the problems for my take home final exam on differential equations.
    I have been looking for a solution for this problem intensely for the last two days. This problem comes from Calculus vol 2 by Apostol section 6.24 ex 7. here it is

    1. The problem statement, all variables and given/known data
    Use the identities in exercise 6 to show that:

    [tex]J^{2}_{0}+2\sum^{\infty}_{n=1}J^{2}_{n}=1[/tex]

    and

    [tex]\sum^{\infty}_{n=0}(2n+1)J^{2}_{n}J^{2}_{n+1}=\frac{1}{2}x[/tex]




    2. Relevant equations

    The identities that I already proved in exercise 6 are:


    [tex]\frac{1}{2}\frac{d}{dx}(J^{2}_{\alpha}+J^{2}_{\alpha+1})=\frac{\alpha}{x}J^{2}_{\alpha}-\frac{\alpha+1}{x}J^{2}_{\alpha+1}[/tex]


    and

    [tex]\frac{d}{dx}(xJ^{2}_{\alpha}J^{2}_{\alpha+1})=x(J^{2}_{\alpha}-J^{2}_{\alpha+1})[/tex]


    3. The attempt at a solution

    As I mentioned in the introduction I have looked for possible solutions and background information for a while but I'm still stuck.

    The best source for information on this problem is this book but I honestly don't understand whats going on. I have tried A treatise on the theory of bessel functions but I also don't undertstand how hansen changes the from
    [tex]J^{2}_{0}+2\sum^{\infty}_{n=1}J^{2}_{n}[/tex]
    to
    [tex]\left(e^{\frac{1}{2}z\frac{t-1}{t}}\right)\left(e^{\frac{1}{2}z\frac{-t+1}{t}}\right)[/tex]

    I think the problem is that I just don't have enough knowledge of power series to know how to handle this problem. I just don't know how to approach the problem. I though t that maybe:


    [tex]2\sum^{\infty}_{n=1}\frac{J^{2}_{0}}{2}+J^{2}_{n}[/tex](is this even right?)

    and then try to change it into something that looks like a recurrence relation

    other approach I tried was to expand the series and try recurrence relations there. but I got stuck there too.


    Thanks very much for any help or links to relevant information.
     
    Last edited: Dec 16, 2007
  2. jcsd
  3. Dec 16, 2007 #2

    cristo

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    Staff Emeritus
    Science Advisor

    Note that we write sums as [tex]\sum_{n=1}^{\infty}[/tex] and not with the limits the other way around. Also there's a typo in your first equation; I presume it should read J_n^2.

    I'm not too sure about helping you pass a final, though; it doesn't seem right to me.
     
  4. Dec 16, 2007 #3
    Thanks for the corrections. I made an error and copied and pasted it over and over.

    I understand, but just so you know, I have been working very hard for the last three days to try and solve this problem. I simply don't have enough background knowledge to solve this. I'm not a genius, but I compensate for it by working very hard. I also dont have a very good mathematical background since I wasted my time in my school years, joined the army and I'm now 29 trying to learn what I should have learned then, but I'm trying to make up for it by working hard and trying to fill the great gaps in my knowledge.

    If I don't solve this problem I might get a B in the course, as I have all A's so far. It's not bad, except that a B will lower my GPA to 3.89 taking away my full scholarship with it. I have to admit that is my primary concern. But I have a second concern, missing out on techniques that might come useful later in my computer science career.

    That said, I understand that grades should be earned, and I respect that, it's just that form my point of view I'm earning it by time spent working on it, knowledge gained by reading about it and taking the time for asking for help here. I still have to understand the solution, and while I might never be as creative as Euler or Bessel and might understand just enough to make it clearer to other by teaching or applying it to computer science.

    Thanks for the corrections and honest response, I hope I have given a satisfactory reason to get help, since it is the most sincere I can give.
     
  5. Dec 16, 2007 #4

    Defennder

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    Homework Helper

    I don't mean to digress from the topic, but what exactly is a "take home final exam"?
     
  6. Dec 16, 2007 #5
    it means that I don't sit in the classroom to take the test, instead it's homework that I have to turn in for the final grade.
     
  7. Dec 16, 2007 #6
    IF anyone that have any idea reads this, I will be happy with a quick pointer in the right direction, like reading material or general subject for the proposed solution. Also excuse my English, it is not my first language.
     
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