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Bessel functions

  1. May 15, 2014 #1
    1. The problem statement, all variables and given/known data
    Calculate:
    a) ##\frac{d}{dx}(xJ_1(x)-\int _0^xtJ_0(t)dt)##
    b) ##xJ_1(x)-\int _0^xtJ_0(t)dt##
    c) let ##\xi _{k0} ## be the ##k## zero of a function ##J_0##. Determine ##c_k## so that ##1=\sum _{k=1}^{\infty }c_kJ_0(\frac{x\xi _{k0}}{2})##.


    2. Relevant equations



    3. The attempt at a solution

    a) ##\frac{d}{dx}(xJ_1(x)-\int _0^xtJ_0(t)dt)=xJ_0(x)-xJ_0(x)=0##.

    b) What do I do with the integral? Should I calculate ##J_n(x)=\frac{1}{\pi }\int _0^{\pi }cos(tsin\varphi -n\varphi)d\varphi ## for n=0?

    c) Hmmm, no idea here :/
     
  2. jcsd
  3. May 15, 2014 #2

    Curious3141

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    Homework Helper

    I'm no expert in the theory of Bessel functions, but isn't the expression in part b) just the integral of the entire expression in a) wrt x? Integrating 0 gives you a constant. The constant can easily be found by subbing in a suitable value of x, right?

    c) exceeds my knowledge, someone else will have to help, sorry.
     
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