Comment #8 on issue 3732 by [email protected]: Integration of Piecewise
function should also be continuous
http://code.google.com/p/sympy/issues/detail?id=3732
Aaron, as suspected by me, the current implementation of _eval_integral for
Piecewise
returns incorrect results for Integral transforms on piecewise functions.
For example
from sympy import *
x = Symbol('x')
k = Symbol('k')
fourier_transform(sin(pi*x)/(pi*x), x, k)
Piecewise((-k/|k| + 1, 4*Abs(k**2) > 1), (1, True))
This solution is incorrect as it returns the value 2 for k < -1/2 while it
should be zero. sinc(x) = sin(pi*x)/(pi*x) and rect(x) are Fourier
transform pairs( http://cnx.org/content/m32899/latest/ )
If you try the suggested implementation in the PR :
https://github.com/sympy/sympy/pull/1971, by uncommenting the code I have
commited, you get the correct answer.
from sympy import *
x = Symbol('x')
k = Symbol('k')
fourier_transform(sin(pi*x)/(pi*x), x, k)
Piecewise((0, 4*Abs(k**2) > 1), (1, True))
The current behavior should be reflected in any kind of integral transform,
even convolution of two Piecewise functions.
Any integral transform on a Piecewise function would likely be a Piecewise
function and the output should be continuous as expected. This is not the
currently, since the integration constant is set to zero after integrating
a piece of a piecewise function.
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