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Integration FB5 problem

  1. Feb 12, 2008 #1

    I have a problem in integrating the FB5 distribution. The FB5's probability density function is given by [tex]f(x)=\frac{1}{c(\kappa, \beta)}\exp[\kappa \gamma_1 x +\beta[(\gamma_2 x)^2 - (\gamma_3 x)^2]][/tex]. We know that [tex]\gamma_1[/tex] represents the mean direction of the distribution and that [tex]\gamma_2[/tex] and [tex]\gamma_3[/tex] represent the major and minor axes respectively. The three columns [tex]\gamma_1[/tex], [tex]\gamma_2[/tex] and [tex]\gamma_3[/tex] form an orthogonal 3x3 matrix.

    I would like to integrate this PDF and therefore I need to express its exponent in polar coordinates. I know that [tex]\gamma_1 x[/tex] can be expressed as:[tex]\gamma_1 x= \cos(\theta) \cos(\theta_o) + \sin(\theta) \sin(\theta_o) \cos(\phi - \phi_o)[/tex] since it is the vector dot product of two 3d vectors. In the above [tex]\gamma_1 = [\sin(\theta_o) \cos(\phi_o) \sin(\theta_o) \sin(\phi_o) \cos(\theta_o) ][/tex] and [tex]x= [\sin(\theta) \cos(\phi) \sin(\theta) \sin(\phi) \cos(\theta) ][/tex].

    How can we express the [tex]\gamma_2 x[/tex] and [tex]\gamma_3 x[/tex] in a similar way?

    Do you think that the following form would be correct? :[tex]\gamma_2 x= \cos(\theta) \cos(\theta major) + \sin(\theta) \sin(\theta major) \cos(\phi - \phi major)[/tex]...same for [tex]\gamma_3 x[/tex] ???? . However, I have read that one needs to define only one angle to specify the major and minor axes. This is very confusing to me. As a result, I am not sure if I can express [tex]\gamma_2 x[/tex] and [tex]\gamma_3 x[/tex] as stated above. Please advise!!!!!!!!

    If can someone think of any other way to numerically integrate this please let me know.

    Many Thanks


  2. jcsd
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