# Source code for diofant.series.residues

from ..core import Integer, Mul, sympify

[docs]def residue(expr, x, x0):
"""
Finds the residue of expr at the point x=x0.

The residue is defined as the coefficient of 1/(x - x_0)
in the power series expansion around x=x_0.

This notion is essential for the Residue Theorem.

Examples
========

>>> residue(1/x, x, 0)
1
>>> residue(1/x**2, x, 0)
0
>>> residue(2/sin(x), x, 0)
2

References
==========

* https://en.wikipedia.org/wiki/Residue_%28complex_analysis%29
* https://en.wikipedia.org/wiki/Residue_theorem

"""
# The current implementation uses series expansion to
# calculate it. A more general implementation is explained in
# the section 5.6 of the Bronstein's book {M. Bronstein:
# Symbolic Integration I, Springer Verlag (2005)}. For purely
# rational functions, the algorithm is much easier. See
# sections 2.4, 2.5, and 2.7 (this section actually gives an
# algorithm for computing any Laurent series coefficient for
# a rational function). The theory in section 2.4 will help to
# understand why the resultant works in the general algorithm.
# For the definition of a resultant, see section 1.4 (and any
# previous sections for more review).

from ..simplify import collect
from .order import Order

expr = sympify(expr)
if x0 != 0:
expr = expr.subs({x: x + x0})
s, n = Order(1, x), 1
while s.has(Order) and s.getn() <= 0:
s = expr.nseries(x, n=n)
n *= 2
s = collect(s.removeO(), x)