Get rid of sqrt in spherical harmonics

[egourgoulhon]

After #33117, Sage's spherical harmonics agree with those of SymPy, SciPy, Mathematica and Wikipedia. But for odd order m, they contain sqrt(sin(theta)^2) terms, which should be simplified to sin(theta), given that the colatitude angle theta lies in [0,pi] and hence fulfills sin(theta)>=0.
For instance, in Sage 9.6.beta5, we have

sage: theta, phi = var('theta phi')
sage: spherical_harmonic(1, 1, theta, phi)
-1/4*sqrt(3)*sqrt(2)*sqrt(sin(theta)^2)*e^(I*phi)/sqrt(pi)

whereas SymPy version has sin(theta) instead of sqrt(sin(theta)^2):

sage: from sympy import Ynm
sage: Ynm(1, 1, theta, phi).expand(func=True) 
-sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi))

CC: @slel @mjungmath @tscrim

Component: symbolics

Keywords: spherical harmonics

Author: Eric Gourgoulhon

Branch/Commit: 175cc9a

Reviewer: Matthias Koeppe, Frédéric Chapoton, Samuel Lelièvre

Issue created by migration from https://trac.sagemath.org/ticket/33501

[egourgoulhon]

Commit: d5ee9a5

[egourgoulhon]

Branch: public/functions/spherical_harmonic-33501

[egourgoulhon]

Author: Eric Gourgoulhon

[egourgoulhon]

comment:1

Here is the fix. Note that I have added some doctests to _evalf_, which, strickly speaking, do not pertain to the current ticket. Actually they should have been implemented in the preceding ticket, #33117, for they check the consistency between _eval_ (implementing the new formulas with the Condon-Shortley phase) and _evalf_ (where spherical harmonics are evaluated via mpmath).

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New commits:

d5ee9a5

Get rid of sqrt in spherical harmonics (#33501)

[mkoeppe]

Reviewer: Matthias Koeppe

[mkoeppe]

comment:2

LGTM

[fchapoton]

comment:3

missing double colons at the end of line here:

+        Check that :trac:`33501` is fixed:

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

5b2748e

Fix doctest in SphericalHarmonic._eval_ (#33501)