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Spherical Harmonic Question

  1. May 16, 2008 #1
    Spherical Harmonic Question!!!

    Hi,

    I have a spherical harmonic which is a function of [tex]\theta[/tex] and [tex]\phi[/tex], written as [tex]Y_n_m(\theta,\phi)[/tex]. I recently found an equation which is written as follows:

    [tex]Y_n_m(\frac{x_2 - x_1}{\left\|x_2 - x_1\right\|})[/tex]

    I am trying to understand its meaning in order to include it in my simulations. Note that in my case [tex]\theta = \pi/2[/tex] so the vectors become 2-dimensional.

    I know that in order to evaluate a Spherical Harmonic we need a value for [tex]\theta[/tex] and a value for [tex]\phi[/tex]. As I can understand from the equation above, the [tex]\frac{x_2 - x_1}{\left\|x_2 - x_1\right\|}[/tex] term will produce a new normalized vector, that is the difference between [tex]x_2 - x_1[/tex]. In my case the points/vectors are located on a circle (not inside the circle) of some radius R.

    The author of the book mentions that the magnitude is [tex]\left\|x_2 - x_1\right\| = 0.1,0.2,...2[/tex], in steps of 0.1. My problem is that I want to evaluate the Spherical Harmonic for all the values of [tex]\left\|x_2 - x_1\right\| = 0.1,0.2,...2[/tex] but I cannot see how to do it since the input to the spherical harmonic is the angle [tex](\theta=\pi/2, \phi)[/tex]. Knowing that [tex]\left\|x_2 - x_1\right\| = 0.1,0.2,...2[/tex] does not tell me anything about [tex]x_2 - x_1[/tex] or DOES IT???

    I would appreciate someone's help

    Thanks

    Regards

    Alex
     
  2. jcsd
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