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Showing That the Modified Bessel Function of the First Kind is a Solution

  1. Nov 22, 2008 #1

    I am in the process of showing that the modified Bessel function, I_v(x), is a solution to the modified Bessel equation,


    I have differentiated the MBF twice and plugged it in to show that the left hand side is in fact 0.

    After a good amount of work, I've come to the following left hand side:


    Where sigma=v.

    Is that right? The math seems straight forward, and I only did one change of index that looks correct to me. I'm skeptical about the end result though.
  2. jcsd
  3. Nov 22, 2008 #2
    Sorry, the actual left hand side I have are those two terms inside the sum multiplied by (x/2)^2k+v.

    All inside the sum.
  4. Nov 25, 2008 #3
    Not sure either. But the second term is undefined when k=0 , i.e. (-1)! = Γ(0) is undefined. Or do we assume the second term to be zero?
  5. Nov 25, 2008 #4
    Thank you for the response.

    I just realized I made a mistake by changing the index, resulting in that denomintor. I eventually obtained a series that converged to 0.

    Thank you again.
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