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Math Help - Bessel function integration

  1. #1
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    Bessel function integration

    I cannot do the following integrals, its an assignment we should do, someone please help

    1) ∫ x^(-2) * J2(x) dx
    where J2(x) is Bessel function of order 2

    2) ∫ x* (Jυ(λx))^2 dx
    this is a definite integral with
    upper index b
    lower index a
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  2. #2
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    Quote Originally Posted by Aggressive Duck View Post
    I cannot do the following integrals, its an assignment we should do, someone please help

    1) ∫ x^(-2) * J2(x) dx
    where J2(x) is Bessel function of order 2

    2) ∫ x* (Jυ(λx))^2 dx
    this is a definite integral with
    upper index b
    lower index a
    Do youi have limits of integration for the first. The second (a la Maple) is

    \frac{1}{2} b^2 \left( J^2_n(b) - J_{n+1}(b) J_{n-1}(b) \right) - \frac{1}{2} a^2 \left( J^2_n(a) - J_{n+1}(a) J_{n-1}(a) \right)
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  3. #3
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    Thank you

    No the first has no limits. Note: It should be solved in terms of Jo and J1

    Would you please just provide the steps of the second one, or is it a standard form? Because I just can't deliver it like that. We were told to use Bessel equation in the form
    t^2 * Jn(t) = n^2 * Jn(t) - tJ'n(t) - t^2Jn''(t)
    Last edited by Aggressive Duck; January 25th 2009 at 10:18 AM. Reason: Forgot something
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  4. #4
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    Quote Originally Posted by Aggressive Duck View Post
    Thank you

    No the first has no limits. Note: It should be solved in terms of Jo and J1

    Would you please just provide the steps of the second one, or is it a standard form? Because I just can't deliver it like that. We were told to use Bessel equation in the form
    t^2 * Jn(t) = n^2 * Jn(t) - tJ'n(t) - t^2Jn''(t)
    OK. Let me show that

    \int x J_n^2\, dx = \frac{1}{2} x^2 \left(J_n^2 - J_{n+1} J_{n-1} \right) and then your answer is just evaluating at the upper and low limits (note, I've set  \lambda = 1 and all constants of integration to zero)

    First, integration by parts u = J_n^2,\;\;\;dv = x\,dx

    \int x J_n^2\, dx = <br />
\frac{1}{2} x^2 J_n^2 - \int x^2 J_n J_n '\, dx \;\;\;\;(*)<br />

    Next, consider the differential equation for J_n

    x^2 J''_n + x J'_n + x^2 J_n - n^2 J_n = 0

    Multiply this by J'_n and integrating (again, no constant of integration)

    \int x^2 J'_n J''_n\, dx + \int x J'^2_n\,dx + \int x^2 J_n J'_n\,dx - \int n^2 J_n J'_n\, dx = 0

    Integrating the first term by parts gives

    \frac{1}{2} x^2 J'^2_n - \int x J'^2_n\,dx + \int x J'^2_n\,dx + \int x^2 J_n J'_n\,dx - \int n^2 J_n J'_n\, dx = 0

    and we see cancellation. The last term also integrates

    \frac{1}{2} x^2 J'^2_n + \int x^2 J_n J'_n\,dx - \frac{1}{2} n^2 J^2_n = 0

    Thus,

    \int x^2 J_n J'_n\,dx = \frac{1}{2} n^2 J^2_n- \frac{1}{2} x^2 J'^2_n

    or

    \int x^2 J_n J'_n\,dx = - \frac{1}{2}\left(x J'_n - n J_n \right)\left(x J'_n + n J_n \right)

    Now two properties of the Bessel J function Bessel Function

    J'_n = J_{n-1} - \frac{n}{x} J_n\;\;\;J'_n = \frac{n}{x} J_n - J_{n+1}

    Using these gives

    \int x^2 J_n J'_n\,dx = \frac{1}{2}\left(x J_{n+1} \right)\left(x J_{n-1}\right) = \frac{1}{2}x^2 J_{n+1} J_{n-1}

    and substitution into (*) gives the result.
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