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Coefficient spherical harmonics

  1. Oct 31, 2012 #1
    $$
    Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi}
    $$

    For ##\ell = m = 1##, we have
    $$
    \sqrt{\frac{(2 + 1)(0)!}{4\pi(2)!}}P^1_{1}(\cos\theta)e^{i\varphi} = \frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi}\sin \theta
    $$

    But Mathematica is telling me the solution is
    $$
    -\frac{1}{2} e^{i\varphi} \sqrt{\frac{3}{2\pi}} \sin\theta
    $$

    What is going wrong?
     
  2. jcsd
  3. Oct 31, 2012 #2

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Have you checked the definitions/conventions for the associated Legendre functions ? The definitions for Mathematica you can find on the functions.wolfram.com website. Unfortunately, it's difficult to say that special functions theory is a unitary process with unique definitions.
     
  4. Oct 31, 2012 #3
    Some are defined with a (-1)^m which is weird my professor was using that definition since we were not in class.
     
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