# Integration FB5 problem

1. Feb 12, 2008

### architect

Hi,

I have a problem in integrating the FB5 distribution. The FB5's probability density function is given by $$f(x)=\frac{1}{c(\kappa, \beta)}\exp[\kappa \gamma_1 x +\beta[(\gamma_2 x)^2 - (\gamma_3 x)^2]]$$. We know that $$\gamma_1$$ represents the mean direction of the distribution and that $$\gamma_2$$ and $$\gamma_3$$ represent the major and minor axes respectively. The three columns $$\gamma_1$$, $$\gamma_2$$ and $$\gamma_3$$ 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 $$\gamma_1 x$$ can be expressed as:$$\gamma_1 x= \cos(\theta) \cos(\theta_o) + \sin(\theta) \sin(\theta_o) \cos(\phi - \phi_o)$$ since it is the vector dot product of two 3d vectors. In the above $$\gamma_1 = [\sin(\theta_o) \cos(\phi_o) \sin(\theta_o) \sin(\phi_o) \cos(\theta_o) ]$$ and $$x= [\sin(\theta) \cos(\phi) \sin(\theta) \sin(\phi) \cos(\theta) ]$$.

How can we express the $$\gamma_2 x$$ and $$\gamma_3 x$$ in a similar way?

Do you think that the following form would be correct? :$$\gamma_2 x= \cos(\theta) \cos(\theta major) + \sin(\theta) \sin(\theta major) \cos(\phi - \phi major)$$...same for $$\gamma_3 x$$ ???? . 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 $$\gamma_2 x$$ and $$\gamma_3 x$$ as stated above. Please advise!!!!!!!!

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

Many Thanks

BR

Alex.