Comment #11 on issue 2624 by ness...@googlemail.com: Sympy 0.7.1 can't
integrate Gaussians
http://code.google.com/p/sympy/issues/detail?id=2624
I should mention that, of course, we can do all the steps of the paper by
hand:
In [6]: mellin_transform((2-x)**alpha*Heaviside(2-x), x, s)
Out[6]:
⎛ α + s ⎞
⎜2 ⋅Γ(s)⋅Γ(α + 1) ⎟
⎜────────────────────, (0, ∞), True⎟
⎝ Γ(α + s + 1) ⎠
In [7]: mellin_transform(sin(alpha/x), x, s)
Out[7]:
⎛ -s - 1 ⎽⎽⎽ s ⎛ s 1⎞ ⎞
⎜2 ⋅╲╱ π ⋅α ⋅Γ⎜- ─ + ─⎟ ⎟
⎜ ⎝ 2 2⎠ ⎟
⎜───────────────────────────, (-1, 1), True⎟
⎜ ⎛s ⎞ ⎟
⎜ Γ⎜─ + 1⎟ ⎟
⎝ ⎝2 ⎠ ⎠
In [8]: inverse_mellin_transform(_6[0], s, x, _6[1], as_meijerg=True)
Out[8]:
α ╭─╮0, 1 ⎛1 │ 2⎞
2 ⋅Γ(α + 1)⋅│╶┐ ⎜ │ ─⎟
╰─╯1, 1 ⎝ -α │ x⎠
In [9]: inverse_mellin_transform(_7[0], s, x, _7[1], as_meijerg=True)
Out[9]:
⎛ │ 2 ⎞
⎽⎽⎽ ╭─╮1, 0 ⎜ │ α ⎟
╲╱ π ⋅│╶┐ ⎜ │ ────⎟
╰─╯0, 2 ⎜1/2 0 │ 2⎟
⎝ │ 4⋅x ⎠
In [10]: integrate(_8*_9, (x, 0, oo))
Out[10]:
⎛ α 1 α │ 2⎞
⎽⎽⎽ ╭─╮3, 0 ⎜ ─ + ─, ─ + 1 │ α ⎟
╲╱ π ⋅α⋅Γ(α + 1)⋅│╶┐ ⎜ 2 2 2 │ ──⎟
╰─╯2, 4 ⎜ │ 16⎟
⎝0, 0, 1/2 -1/2 │ ⎠
───────────────────────────────────────────────────────
4
The only differences are that (1) meijerint_definite uses a table of mellin
transforms instead of the mellin_transform() function (at least initially)
- but this table is of course where mellin_transform() gets its results
from in the end, and (2) that the algorithm adds in the Heaviside function
for you (which I did by hand in the very first line).
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