Status: New
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Labels: Type-Defect Priority-Medium
New issue 1792 by kirzhanov: SymPy does not integrate a real power of
sine/cosine
http://code.google.com/p/sympy/issues/detail?id=1792
I want to get a general explicit expression of an integral of this function:
delta
cos (theta)
where delta is any positive real, but not integer value. It seems to me
that this problem has a known solution, but SymPy
does no offer it.
A minimal program to demonstrate an issue is discussed below:
1 #!/usr/bin/python
2
3 from sympy import *
4
These are my variables:
5 t=Symbol('theta')
6 d=Symbol('delta')
This is the function which will be integrated:
7 myfunction=cos(t)**(d)
First, let's try d=1 (or any other integer). Such integrals are very
well-known, the integration result can be found either on wikipedia
[http://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions]
or in a handbook (for example, Dwight H.B. Tables of integrals. 1957)
8 print integrate(myfunction.subs({d:1}),(t,0,pi/2))
In this case SciPy gives an expected result: "1".
Now let's try to integrate expression cos(theta)**delta, were delta is not
integer. In the next line value 1.5 is given to
delta variable, but you can try any other real non-integer value:
9 print integrate(myfunction.subs({d:1.5}),(t,0,pi/2))
The result is "-Cos_2k_integrate(0.75, 0) + Cos_2k_integrate(0.75, pi/2)".
It seems that I cannot do anything with this result
- neither get an algebraic expression (which can contain special
functions), nor calculate numerical value.
It seems that such integrals can be expressed via hypergeometric special
function
[http://en.wikipedia.org/wiki/Euler_hypergeometric_integral]. The
antiderivative, given by Mathematica 7 to function
delta
cos (theta)
is (Latex notation is used):
-\frac{\cos^{2+\delta}\theta \sin\theta}{(2 + \delta) |\sin\theta|
}\;_2\mathbf{F}_1\left( \frac{2 +
delta}{2},\frac{1}{2},\frac{4+\delta}{2},\cos^2\theta \right)
Is it possible to implement calculation of antiderivatives for such
functions?
Cheers,
Dima
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