Complex Partial Fraction Decompositions of Rational Functions

This work is aimed to obtain the complex partial fraction decompositions of rational functions. We express the coefficients of complex partial fraction decomposition of arbitrary rational functions in terms of the coefficients of their real partial fraction decomposition. This type of decompositions is then used to generalize the high order derivatives of such rational functions. Moreover, different applications are selected to demonstrate the applicability of introduced algorithms.


Introduction
Rational functions are widely used in many branches of mathematics such as numerical analysis i.e Pade Approximations, mathematical analysis as well as mathematical modelling and they appear in mathematical representations of many problems in science and engineering [1][2][3][4][5][6][7]. Unfortunately, working with rational functions except polynomials is generally not very easy. From the point of view, some methods can be used to facilitate the computations. One of the famous and simplest methods is partial fraction decomposition method (PFD) for suitable applications: Consider a polinomial function ( ) Q x with real coefficients and recall that there exist some integers ≥ are also pairwise different.
Assume that P(x) is another polynomial such that deg(P)<deg (Q). In this case, the real partial fraction decompositions of the rational function where β γ ∈  , , .
ir js js a One important problem of these PFDs is to determine the corresponding coefficients. To determine the coefficients of these PFDs, some methods/algorithms can be applied with respect to the given rational functions. For example, long division (routine calculation), algorithms introduced in [1,2] can be used for suitable applications. These algorithms especially focus for determining the real PFDs for given rational functions. On the other hand, real PFDs may fail in some applications; for example a failure of real PFD is mentioned in [3]. From the point of view, complex PFD can alternatively be used and this yields simplest form for representation of rational function (1). In this paper, we aim to expose the full complex PFDs of rational function (1). We first give some complex partial fraction decompositions and then emphasize the relationship between the coefficients of real PFD and complex PFD. Then, some applications of these complex PFDs will be discussed.

Complex Partial Fraction Decompositions
Complex PFDs can effectively be used in the parts of suitable applications. For example, rational algorithm, which is introduced in [3] for computing the coefficients of formal power series of a rational functions, requires complex section we provide the relations between the coefficients of real PFD and the coefficients of their complex PFD. Note that to find corresponding real PFDs, the algorithms discussed in [1,2] can be used for suitable applications.

Theorem 1
ω ω ω x z x z x z x z x z x z x z x z x z x z R R (2) We first observe that By assigning successively the values 2; 3; : : : ; j to k, one gets: which shows that The proof is done by induction on q. If q = 1 or q = 2 one gets the obvious equalities

. R R and R R aR
Assume the equality for q-1 and observe that one gets successively: Now the new algorithms for finding the desired complex coefficients of a certain type of rational function with respect to corresponding PFDs are given by the following proposition and corollary:     be also pairwise different. If is a polinomial with real coefficients whose degree satisfies the inequality deg The relations between the coefficients of the real partial fraction decomposition and the coefficients of the complex partial fraction decomposition are:

Example 1
Consider the rational function For the first step we need to find corresponding PFD of given function. The given rational function is decomposed in [3] by "Two Brick Method" as Then, according to the Proposition (1) the complex PFDs of decomposed functions are obtained respectively as,

Example 2
Apply complex PFD of a more complicated rational function Using proposition 2, the desired complex PFDs with corresponding complex coefficients can be given as

Differentiation via Complex Partial Fraction Decompositions
Higher order derivatives of functions can be used in many applications. Hovewer, representing the higher order derivatives of many functions explicitly cannot be easy and in general they are computed recursively. From the point of view, higher order derivatives of a certain class of rational functions can easily be computed through complex PFDs and the k th order derivatives can be computed directly. In this section we will give the general form of higher order derivatives of rational function (1). Finally, the results then will be used in some certain applications.
In order to find the high order derivatives of rational functions in (1), we suppose to find the high order derivatives of rational functions of types β + − − + While computing high order derivatives of the first type of rational functions is direct, computing the high order derivatives of the other type of rational functions needs some more computations which will be given in this section.
If ∈  z is a fixed complex number, then the function with k an integer, is arbitrarily many times differentiable and  Corollary 6: The n th order derivatives of the function R and R, respectively, are given by the following formulae.
ω ω 2 ( ( )) ( 2 ( ) ) q q n n n q n q n q n q k q n n q k k k n q n n k k q even n q Re z x z Proof: We just need to use the formula   where f is the rational function given by ( ) Re z x z s Proof: It follows immediately by induction on n.
is the rational function as in corollary (3), then its nth order derivative is given by Practically, it may be difficult to represent a k explicitly for all k for a certain class of functions. From the point of view, if the desired power series deals with a rational function, the complex PFDs can be applied to represent the coefficients explicitly by the following theorem:  If the function f is deal with the rational function (1); the complex PFD algorithms can be used to defined its Schwarzian derivative as well as higher order Schwarzian derivatives in a closed form. Let us consider a particular type of rational functions β α Then the higher order derivatives of the rational function R(x) according to complex PFDs is Thus Schwarzian derivative of the rational function R(x) can be easily represented in the closed form as