In more detail, somewhere in functions/elementary/complexes.py I would
add principal_branch(z) which represents z with its argument converted
to range (-pi, pi], and argument_period(z) which then satisfies z =
principal_branch(z)*exp(I*2*pi*argument_period(z)). Then e.g.
uppergamma._rewrite_as_unbranched could return an expression involving
these functions. The integration code would use a sequence of operations
as follows to compute the branch of the result:
I went ahead today and did this [1] [2]. Here is what it looks like, for
the example of incomplete gamma functions (the only other example I have
currently implemented is meijerg, and that is shown in the first commit
message):
In [1]: lowergamma(y, x).rewrite('unbranched')
Out[1]:
2⋅ⅈ⋅π⋅y⋅argument_period(branched_argument(x))
ℯ ⋅γ(y, principal_branch(x))
In [6]: lowergamma(-3, exp(I*x)).rewrite('unbranched')
Out[6]:
ⅈ⋅π⋅argument_period(re(x)) ⎛ ⎛ ⅈ⋅x⎞⎞
- ────────────────────────── + γ⎝-3, principal_branch⎝ℯ ⎠⎠
In [7]: _6.subs(x, pi)
Out[7]: γ(-3, -1)
In [8]: _6.subs(x, -pi)
Out[8]:
ⅈ⋅π
γ(-3, -1) + ───
3
Notice that "-1" refers to the point exp(I*pi) by the branching
conventions of sympy, *not* exp(-I*pi).
I'm fairly convinced this is sufficient for my needs. I thought about
something along the lines of Polar(abs, arg) as suggested by Frederik,
but I feared getting it to work with all the rest of sympy would take
long (and I feel I'm already doing too many things that take long ^^).
The way it currently is my changes are very unintrusive, but admittedly
somewhat hacky.
[1]
https://github.com/ness01/sympy/commit/3b11e900e5849f6a0227eb0a4c531075cac52a4f
[2]
https://github.com/ness01/sympy/commit/9ac48a0250df01701d79125c54d7f1afdec4aa07
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