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I can't believe we haven't done this before, nor that someone hasn't requested it. See this ask.sagemath post.
sage: f(x)=1/x
sage: f.maxima_methods().residue(x,0)
1
f(x)=1/(x^2+1)
f.maxima_methods().residue(x,i)
-1/2*I
This should be a little easier to do than that.
CC: @jasongrout @burcin @wdjoyner
Component: calculus
Issue created by migration from https://trac.sagemath.org/ticket/11210