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Math Help - Spherical Harmonics

  1. #1
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    Question Spherical Harmonics

    Let

    h(\theta, \phi)= 1 + sin\phi sin\theta

    By taking linear combinations of these spherical harmonics, find Anm for all n
    and m such that

    h(\theta, \phi)=\sum^{1}_{n=0}\sum^{n}_{m=-n} A_{nm}Y^{m}_{n}(\theta, \phi)

    Where Y^{m}_{n}(\theta, \phi) is the first few spherical harmonics.

    Any and All help is appreciated
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  2. #2
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    Re: Spherical Harmonics

    Expand it just like a Fourier Series.

    Y_{0}^{0} (\theta,\phi) =\frac{1}{2}\sqrt{\frac{1}{\pi}}

    Y_{0}^{0}h(\theta,\phi)=A_{00}Y_{0}^{0} (\theta,\phi)

    Now integrate both sides over a sphere of constant radius.

    \int_{0}^{\pi} \int_{0}^{2\pi} Y_{0}^{0}h(\theta,\phi)h(\theta,\phi)\sin(\theta)d  \theta d\phi = \int_{0}^{\pi} \int_{0}^{2\pi} A_{00}Y_{0}^{0} (\theta,\phi)\sin(\theta)d\theta d\phi

    \int_{0}^{\pi} \int_{0}^{2\pi}\frac{1}{2}\sqrt{\frac{1}{\pi}}(1+\  sin \phi \sin \theta)\sin(\theta)d\theta d\phi = \int_{0}^{\pi} \int_{0}^{2\pi} A_{00}\frac{1}{2}\sqrt{\frac{1}{\pi}}\sin(\theta)d  \theta d\phi

    Now just integrate and solve for A_{00}

    Then rinse and repeate with the next few spherical harmonics.
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