Status: Valid
Owner: ----
Labels: Type-Defect Priority-Medium
New issue 3362 by [email protected]: Integration of sin(x^n)/x^n and
sinh(x^n)/x^n
http://code.google.com/p/sympy/issues/detail?id=3362
Integration of sin(x^n)/x^n and sinh(x^n)/x^n works in principle however
the resulting pFq function should be simplified further to incomplete gamma
functions.
In [9]: integrate(sin(x**n)/x**n, x, meijerg=True)
Out[9]: x*gamma(1/(2*n))*hyper((1/(2*n),), (3/2, 1 + 1/(2*n)),
-x**(2*n)/4)/(2*n*gamma(1 + 1/(2*n)))
In [10]: combsimp(_)
Out[10]: x*hyper((1/(2*n),), (3/2, 1 + 1/(2*n)), -x**(2*n)/4)
In case n=1 we get the well known Si(x) function:
In [12]: _.subs(n,1)
Out[12]: x*hyper((1/2,), (3/2, 3/2), -x**2/4)
In [14]: hyperexpand(_)
Out[14]: Si(x)
In [27]: integrate(sinh(x**n)/x**n, x, meijerg=True)
Out[27]: x*gamma(1/(2*n))*hyper((1/(2*n),), (3/2, 1 + 1/(2*n)),
x**(2*n)/4)/(2*n*gamma(1 + 1/(2*n)))
In [28]: combsimp(_)
Out[28]: x*hyper((1/(2*n),), (3/2, 1 + 1/(2*n)), x**(2*n)/4)
In [29]: hyperexpand(_)
Out[29]: x*hyper((1/(2*n),), (3/2, 1 + 1/(2*n)), x**(2*n)/4)
In [30]: _.subs(n,1)
Out[30]: x*hyper((1/2,), (3/2, 3/2), x**2/4)
In [31]: hyperexpand(_)
Out[31]: Shi(x)
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