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The spherical harmonics form a complete set of the Hilbert space of square integrable functions on the sphere. However, looking at them, I can't see how they could ever be summed to equal a function which has a complex value for $\varphi =0$, because looking at the definition of the spherical harmonics, they're always real when $\varphi =0$:

$Y_\ell^m (\theta, \varphi ) = N \, e^{i m \varphi } \, P_\ell^m (\cos{\theta} )$

(where N is a real normalization factor and $P_\ell^m()$ are the also real-valued associated Legendre Polynomials)

Surely there are square integrable functions on the sphere that have a complex component at some point of the 0° meridian?

I'm sure I'm making some stupid mistake - so I would like to apologize for the question beforehand

yippy_yay
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    the coefficients are (allowed to be) complex. (also you work in $L^2$, so in principle questions amout values on a set of measure zero make little sense - but this is less important) – user8268 Nov 27 '14 at 10:42
  • Thanks. I knew I was overlooking something basic. – yippy_yay Nov 27 '14 at 10:45

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