Updates:
Status: Fixed
Comment #9 on issue 2063 by [email protected]: Add complete elliptic
integrals
http://code.google.com/p/sympy/issues/detail?id=2063
Merged PR # 2165.
This affects only integrals with K(z) or E(z) inside the integrand, not
integrals which
would produce these functions as result (f.e. the defining integrals).
Sympy can now do among others the following integrals:
In [11]: integrate(z**b * elliptic_k(a*z**c) / z**d, z, meijerg=True)
Out[11]: pi*z*z**b*z**(-d)*gamma(b/c - d/c + 1/c)*hyper((1/2, 1/2, b/c -
d/c + 1/c), (1, b/c + 1 - d/c + 1/c),
a*z**c*exp_polar(2*I*pi))/(2*c*gamma(b/c + 1 - d/c + 1/c))
In [12]: pprint(_)
⎛ b d 1 │ ⎞
⎜1/2, 1/2, ─ - ─ + ─ │ ⎟
b -d ⎛b d 1⎞ ┌─ ⎜ c c c │ c 2⋅ⅈ⋅π⎟
π⋅z⋅z ⋅z ⋅Γ⎜─ - ─ + ─⎟⋅ ├─ ⎜ │ a⋅z ⋅ℯ ⎟
⎝c c c⎠ 3╵ 2 ⎜ b d 1 │ ⎟
⎜ 1, ─ + 1 - ─ + ─ │ ⎟
⎝ c c c │ ⎠
────────────────────────────────────────────────────────────────
⎛b d 1⎞
2⋅c⋅Γ⎜─ + 1 - ─ + ─⎟
⎝c c c⎠
In [13]: integrate(z**b * elliptic_e(a*z**c) / z**d, z, meijerg=True)
Out[13]: pi*z*z**b*z**(-d)*gamma(b/c - d/c + 1/c)*hyper((-1/2, 1/2, b/c -
d/c + 1/c), (1, b/c + 1 - d/c + 1/c),
a*z**c*exp_polar(2*I*pi))/(2*c*gamma(b/c + 1 - d/c + 1/c))
In [14]: pprint(_)
⎛ b d 1 │ ⎞
⎜-1/2, 1/2, ─ - ─ + ─ │ ⎟
b -d ⎛b d 1⎞ ┌─ ⎜ c c c │ c 2⋅ⅈ⋅π⎟
π⋅z⋅z ⋅z ⋅Γ⎜─ - ─ + ─⎟⋅ ├─ ⎜ │ a⋅z ⋅ℯ ⎟
⎝c c c⎠ 3╵ 2 ⎜ b d 1 │ ⎟
⎜ 1, ─ + 1 - ─ + ─ │ ⎟
⎝ c c c │ ⎠
─────────────────────────────────────────────────────────────────
⎛b d 1⎞
2⋅c⋅Γ⎜─ + 1 - ─ + ─⎟
⎝c c c⎠
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