Results 1 to 2 of 2

Math Help - Annoying Integral involving Bessel Functions

  1. #1
    Member
    Joined
    May 2008
    Posts
    106

    Annoying Integral involving Bessel Functions

    Hi I'm having trouble with this integral
    <br />
\int_{0}^{\infty} \frac{J_0(kR)}{(1+(kR_d)^2)^{3/2}} dk<br />

    I'm supposed to evaulate it using
    <br />
\int_{0}^{\infty} J_{\nu}(xy) \frac{dx}{(x^2+a^2)^{1/2}} = I_{\nu/2} (ay/2) K_{\nu/2} (ay/2)<br />

    Where standard notation has been used for the bessel functions, any hints on how to transform it to the correct forn would be much appreciated, I can't really see how to get this to work
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,887
    Thanks
    326
    Awards
    1
    Quote Originally Posted by thelostchild View Post
    Hi I'm having trouble with this integral
    <br />
\int_{0}^{\infty} \frac{J_0(kR)}{(1+(kR_d)^2)^{3/2}} dk<br />

    I'm supposed to evaulate it using
    <br />
\int_{0}^{\infty} J_{\nu}(xy) \frac{dx}{(x^2+a^2)^{1/2}} = I_{\nu/2} (ay/2) K_{\nu/2} (ay/2)<br />

    Where standard notation has been used for the bessel functions, any hints on how to transform it to the correct forn would be much appreciated, I can't really see how to get this to work
    Hint:
    \displaystyle \int_0^{\infty} \frac{J_0(kR)}{(1 + (kR)^2 )^{3/2}}dk = ~...~= \frac{1}{y}\int_0^{\infty}J_0(m) \cdot m(1 + m^2)^{-3/2}dm
    (after letting y = R and m = xy.)

    Now integrate by parts:
    \displaystyle \int p~dq = pq - \int q ~dp
    using \displaystyle p = J_0(m) and \displaystyle dq = m(1 + m^2)^{-3/2}

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. integral involving the Bessel function
    Posted in the Calculus Forum
    Replies: 0
    Last Post: January 15th 2012, 08:03 AM
  2. Replies: 3
    Last Post: April 18th 2011, 10:36 AM
  3. Proof involving an integral of a product of 2 functions
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 12th 2010, 03:29 AM
  4. Integral involving Bessel Function
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 26th 2009, 10:33 AM
  5. integral expression for bessel functions...
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 17th 2009, 02:07 PM

Search Tags


/mathhelpforum @mathhelpforum