residue: mathematically wrong output

[behackl]

The complex function f(s) = 1/(1 - 2^(-s)) has poles of residue 1/log(2) at s = 2*k*pi*I/log(2) for all integers k.

Currently sage recognize these poles just at s=0:

sage: f(s).residue(s==0)
1/log(2)
sage: f(s).residue(s==2*pi*I/log(2))
0

In essence, this happens because the series-method does not recognize the pole. The priority is critical because mathematically wrong output is produced.

CC: @rwst @cheuberg

Component: symbolics

Author: Benjamin Hackl

Branch/Commit: 68fea61

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/20084

[behackl]

comment:1

The most elegant solution would of course be the automatic simplification of expressions like 2^(something/log(2)) --> exp(something), such that

sage: 2^(2*pi*I/log(2))
1

However, I'm not sure if that can be achieved so easily.

A second suggestion would be something like

diff --git a/src/sage/symbolic/expression.pyx b/src/sage/symbolic/expression.pyx
index 3a5c864..175bcad 100644
--- a/src/sage/symbolic/expression.pyx
+++ b/src/sage/symbolic/expression.pyx
@@ -4076,7 +4076,7 @@ cdef class Expression(CommutativeRingElement):
             a = 0
         if a == infinity:
             return (-self.subs({x: 1/x}) / x**2).residue(x == 0)
-        return self.subs({x: x+a}).series(x == 0, 0).coefficient(x, -1)
+        return self.subs({x: x+a}).canonicalize_radical().series(x == 0, 0).coefficient(x, -1)
 
     def taylor(self, *args):
         r"""

where there are several ways to refine this approach:

• introduce an additional keyword that could disable this additional simplification such that performance does not suffer necessarily,
• only try to simplify for complex arguments of a

and so on.

Thoughts?

[behackl]

Branch: u/behackl/symbolics/residue/exp-complex-poles

[behackl]

comment:2

This is just the quick workaround from above.

---

New commits:

45d16e2

simplify substituted expression before series expansion

[behackl]

Author: Benjamin Hackl

[behackl]

Commit: 45d16e2

[williamstein]

comment:3

Please add an example doctest to illustrate what you're fixing.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

8719364

add a doctest

[sagetrac-git]

Changed commit from 45d16e2 to 8719364