sympy/sympy — pull request #7422 by raoulb (2014-04-21 02:24:29+00:00)
Title: Expansions for harmonic numbers at general rational arguments n + p/q

In [2]: harmonic(2/S(3))
Out[2]: harmonic(2/3)

In [3]: expand_func(_)
(0, 2, 3)
Out[3]: -log(6) + 2*Sum(log(sin(_k*pi/3))*cos(4*_k*pi/3), (_k, 1, 1)) + sqrt(3)*pi/6 + 3*Sum(1/(3*_k + 2), (_k, 0, 0))

In [4]: pprint(_)
1 0
____ ____
╲ ╲
╲ ⎛ ⎛k⋅π⎞⎞ ⎛4⋅k⋅π⎞ ___ ╲ 1
╲ log⎜sin⎜───⎟⎟⋅cos⎜─────⎟ ╲╱ 3 ⋅π ╲ ───────
-log(6) + 2⋅ ╱ ⎝ ⎝ 3 ⎠⎠ ⎝ 3 ⎠ + ─────── + 3⋅ ╱ 3⋅k + 2
╱ 6 ╱
╱ ╱
‾‾‾‾ ‾‾‾‾
k = 1 k = 0

In [5]: _.doit()
Out[5]: -log(6) - log(sqrt(3)/2) + sqrt(3)*pi/6 + 3/2

In [6]: simplify(_)
Out[6]: log(sqrt(3)/9) + sqrt(3)*pi/6 + 3/2

In [7]: _.n()
Out[7]: 0.758981249114944

In [8]: harmonic(2/S(3))
Out[8]: harmonic(2/3)

In [9]: _.rewrite(polygamma)
Out[9]: -3*log(3)/2 + sqrt(3)*pi/6 + 3/2

In [10]: _.n()
Out[10]: 0.758981249114944