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Referring to this article:

https://i.stack.imgur.com/sfQ1C.png

and

https://i.stack.imgur.com/LelKb.png

How is it that they get from equation 2 to equation 3?

Whenever I do it, I get only cosine terms, no sine terms and can't cancel the imaginary ones like they do.

This is taking into account Euler's formula and even/odd properties of the trig functions (i.e. m terms combine with their -m counterparts)

Is there some spherical harmonics property that I'm missing out?

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    $\begingroup$ How about giving an actual bibliographic reference, so that people can check the article themselves? No comment as to the appropriateness of MO for this question $\endgroup$
    – David Roberts ♦
    Jul 28, 2015 at 2:36
  • $\begingroup$ P Méléard, T Pott, H Bouvrais, JH Ipsen, Advantages of statistical analysis of giant vesicle flickering for bending elasticity measurements. The European physical journal. E, Soft matter 34 (2011) 116-129, citeseerx.ist.psu.edu/viewdoc/… $\endgroup$
    – David Roberts ♦
    Jul 28, 2015 at 2:40

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