In complex analysis, a partial fraction expansion is a way of writing a meromorphic function
as an infinite sum of rational functions and polynomials.
is a rational function, this reduces to the usual method of partial fractions.
By using polynomial long division and the partial fraction technique from algebra, any rational function can be written as a sum of terms of the form
Just as polynomial factorization can be generalized to the Weierstrass factorization theorem, there is an analogy to partial fraction expansions for certain meromorphic functions.
A proper rational function (one for which the degree of the denominator is greater than the degree of the numerator) has a partial fraction expansion with no polynomial terms.
Similarly, a meromorphic function
goes to infinity at least as quickly as
has an expansion with no polynomial terms.
be a function meromorphic in the finite complex plane with poles at
be a sequence of simple closed curves such that: Suppose also that there exists an integer
for the principal part of the Laurent expansion of
itself does not have a pole at 0, the residues of
about the origin to get so that the polynomial terms contributed are exactly the regular part of the Laurent series up to
can be pulled out of the residue calculations: The simplest meromorphic functions with an infinite number of poles are the non-entire trigonometric functions.
is meromorphic with poles at
, which are easily seen to satisfy the necessary conditions.
On the horizontal sides of
is continuous, decreasing, and bounded below by 1, so it follows that on the horizontal sides of
, and the partial fraction expansion of
looks like The principal parts and residues are easy enough to calculate, as all the poles of
are simple and have residue -1: We can ignore
are analytic at 0, so there is no contribution to the sum, and ordering the poles
, etc., gives Because the partial fraction expansion often yields sums of
, it can be useful in finding a way to write a function as an infinite product; integrating both sides gives a sum of logarithms, and exponentiating gives the desired product: Applying some logarithm rules, which finally gives The partial fraction expansion for a function can also be used to find a Laurent series for it by simply replacing the rational functions in the sum with their Laurent series, which are often not difficult to write in closed form.
This can also lead to interesting identities if a Laurent series is already known.
Recall that We can expand the summand using a geometric series: Substituting back, which shows that the coefficients
in the Laurent (Taylor) series of
Conversely, we can compare this formula to the Taylor expansion for
to calculate the infinite sums: