Updates:
Cc: [email protected]
Comment #6 on issue 3056 by [email protected]: Closed-form results of
sums of meijerg functions
http://code.google.com/p/sympy/issues/detail?id=3056
Nice to see! When did that start working?
Direct transform:
In [18]: fourier_transform(1/(t**2+a**2), t, w)
Out[18]: -pi*(sinh(2*pi*a*w) - cosh(2*pi*a*w))/a
And hyper expansion:
In [19]: meijerg(((1/S(2),), ()), ((1/S(2), 0, 1/S(2)), ()),
pi**2*a**2*w**2*exp_polar(-I*pi)) + meijerg(((1/S(2),), ()), ((1/S(2), 0,
1/S(2)), ()), pi**2*a**2*w**2*exp_polar(I*pi))
Out[19]: meijerg(((1/2,), ()), ((1/2, 0, 1/2), ()),
pi**2*a**2*w**2*exp_polar(-I*pi)) + meijerg(((1/2,), ()), ((1/2, 0, 1/2),
()), pi**2*a**2*w**2*exp_polar(I*pi))
In [20]: pprint(_)
╭─╮3, 1 ⎛ 1/2 │ 2 2 2 -ⅈ⋅π⎞ ╭─╮3, 1 ⎛ 1/2 │ 2
2 2 ⅈ⋅π⎞
│╶┐ ⎜ │ π ⋅a ⋅w ⋅ℯ ⎟ + │╶┐ ⎜ │ π ⋅a
⋅w ⋅ℯ ⎟
╰─╯1, 3 ⎝1/2, 0, 1/2 │ ⎠ ╰─╯1, 3 ⎝1/2, 0, 1/2
│ ⎠
In [21]: hyperexpand(_)
Out[21]: -sqrt(pi)*(2*(-I*Shi(2*pi*a*w) - pi/2)*cosh(2*pi*a*w) +
2*I*(Chi(2*pi*a*w) - I*pi/2)*sinh(2*pi*a*w)) - sqrt(pi)*(2*(I*Shi(2*pi*a*w)
- pi/2)*cosh(2*pi*a*w) - 2*I*(Chi(2*pi*a*w) + I*pi/2)*sinh(2*pi*a*w))
In [22]: simplify(_)
Out[22]: -2*pi**(3/2)*(sinh(2*pi*a*w) - cosh(2*pi*a*w))
In [23]: pprint(_)
3/2
-2⋅π ⋅(sinh(2⋅π⋅a⋅w) - cosh(2⋅π⋅a⋅w))
But the results are not equal!
I'll go through the notebooks and see what all now works.
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