#3188 spherical_harmonic(n,m,x,y) incorrect

[dan hayes]

spherical_harmonic(n,m,x,y) incorrect
For example for spherical_harmonic(1,1,x,y) gives exactly the negative of the correct expression
as does also spherical_harmonic(2,1,x,y)

For example see many texts including Merzbacher :
page 248 equation (11.78) and the list (11.82) in
instrumentation.tamu.edu/~ting/other/QM_Merzbacher.pdf

[dan hayes]

a test of trial runs indicated it is the wrong sign for m odd only. If so could correct it by multiplying by (-1)^m. Though have not done an exhaustive search.

[David Billinghurst]

I agree that spherical_harmonic(n,m,x,y) differs in sign from Merzbacher for odd m, so that at least we have a documentation error. Apparently there are alternate normalizations of this function that differ by (-1)^m. (http://dlmf.nist.gov/14.30)
.
According to Arfen and Weber, Mathematical methods for physicists, 7th ed, (2015) section 15.5

The definition we introduced for the associated Legendre functions leads to specific signs
for the Y^m_l that are sometimes identified as the Condon-Shortley phase, after the authors
of a classic text on atomic spectroscopy. This sign convention has been found to simplify
various calculations, particularly in the quantum theory of angular momentum. One of
the effects of this phase factor is to introduce an alternation of sign with m among the
positive-m spherical harmonics.

Affen and Weber gives Table 15.4 Spherical Harmonics (Condon-Shortley Phase) on p 760. At first glance this agrees with Merzbacher.

We should check the conventions used by mathematica and maple.