In complex analysis, a partial fraction expansion is a way of writing a meromorphic function f ( z ) {\displaystyle f(z)} as an infinite sum of rational functions and polynomials. When f ( z ) {\displaystyle f(z)} is a rational function, this reduces to the usual method of partial fractions.
Motivation 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 1 ( a z + b ) k + p ( z ) {\textstyle {\frac {1}{(az+b)^{k}}}+p(z)} , where a {\displaystyle a} and b {\displaystyle b} are complex, k {\displaystyle k} is an integer, and p ( z ) {\displaystyle p(z)} is a polynomial. 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 f ( z ) {\displaystyle f(z)} for which | f ( z ) | {\displaystyle |f(z)|} goes to 0 as z {\displaystyle z} goes to infinity at least as quickly as | 1 z | {\textstyle |{\frac {1}{z}}|} has an expansion with no polynomial terms.
Calculation Let f ( z ) {\displaystyle f(z)} be a function meromorphic in the finite complex plane with poles at λ 1 , λ 2 , . . . {\displaystyle \lambda _{1},\lambda _{2},...} and let ( Γ 1 , Γ 2 , . . . ) {\displaystyle (\Gamma _{1},\Gamma _{2},...)} be a sequence of simple closed curves such that:
The origin lies inside each curve Γ k {\displaystyle \Gamma _{k}}
No curve passes through a pole of f {\displaystyle f}
Γ k {\displaystyle \Gamma _{k}} lies inside Γ k + 1 {\displaystyle \Gamma _{k+1}} for all k {\displaystyle k}
lim k → ∞ d ( Γ k ) = ∞ {\displaystyle \lim _{k\rightarrow \infty }d(\Gamma _{k})=\infty } , where d ( Γ k ) {\displaystyle d(\Gamma _{k})} gives the distance from the curve to the origin one more condition of compatibility with the poles λ k {\displaystyle \lambda _{k}} , described at the end of this section Suppose also that there exists an integer p {\displaystyle p} such that
lim k → ∞ ∮ Γ k | f ( z ) z p + 1 | | d z | < ∞ {\displaystyle \lim _{k\rightarrow \infty }\oint _{\Gamma _{k}}\left|{\frac {f(z)}{z^{p+1}}}\right||dz|<\infty }
Writing PP ( f ( z ) ; z = λ k ) {\displaystyle \operatorname {PP} (f(z);z=\lambda _{k})} for the principal part of the Laurent expansion of f {\displaystyle f} about the point λ k {\displaystyle \lambda _{k}} , we have
f ( z ) = ∑ k = 0 ∞ PP ( f ( z ) ; z = λ k ) , {\displaystyle f(z)=\sum _{k=0}^{\infty }\operatorname {PP} (f(z);z=\lambda _{k}),}
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