Status: Valid
Owner: [email protected]
Labels: Type-Enhancement Priority-Medium Integration
New issue 3901 by [email protected]: Improve integration of li(z)
http://code.google.com/p/sympy/issues/detail?id=3901
Sympy can integrate only very simple integrands containing li(z):
In [3]: li(x)
Out[3]: li(x)
In [4]: integrate(_, x)
Out[4]: x*li(x) - Ei(2*log(x))
In [5]: _.rewrite(li)
Out[5]: x*li(x) - li(x**2)
In [6]: integrate(_, x)
Out[6]: Integral(x*li(x) - li(x**2), x)
This is because we build support into heurisch.
Fricas (an Axiom flavor) can easily handle much more complicated integrands:
(11) -> integrate(li(x), x)
2
(11) - li(x ) + x li(x)
Type:
Union(Expression(Integer),...)
(12) -> integrate(%, x)
3 2 2
li(x ) - 2x li(x ) + x li(x)
(12) ----------------------------
2
Type:
Union(Expression(Integer),...)
(13) -> integrate(%, x)
4 3 2 2 3
- li(x ) + 3x li(x ) - 3x li(x ) + x li(x)
(13) ------------------------------------------
6
Type:
Union(Expression(Integer),...)
(14) -> integrate(%, x)
5 4 2 3 3 2 4
li(x ) - 4x li(x ) + 6x li(x ) - 4x li(x ) + x li(x)
(14) ----------------------------------------------------
24
Type:
Union(Expression(Integer),...)
(15) -> integrate(%, x)
6 5 2 4 3 3 4 2 5
- li(x ) + 5x li(x ) - 10x li(x ) + 10x li(x ) - 5x li(x ) + x
li(x)
(15)
--------------------------------------------------------------------
120
Type:
Union(Expression(Integer),...)
They have extended Risch to do the hard work.
A second issue: at the moment it is not clear how to add li(z) to our
Meijer-g algorithm. In the discussion of PR #1708 we found:
------------------------------------------------------------------------------------
Unfortunately I see no way to accomodate the extra additive terms without
rewriting some parts of meijerint.py :(. [Which ought to be done anyway...]
But here we have meijerg with logarithmic arguments anyway, so this really
is no help ... this is probably why I did not take care of logarithmic
integrals in the first place.
If you really want to integrate these using meijerint, you should extend
the heuristics to start out with a substitution z -> exp(z), and then
recognise li(exp(z)) as meijerg with a normal argument. This is likely
non-trivial.
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