12.1 Partial Fraction Expansion

The partial fraction expansion method is used extensively in integration and in finding the inverses of the Laplace, Fourier, and Z transforms. This method allows us to decompose a rational polynomial into smaller rational polynomials with simpler denominators, from which we can easily recognize their integrals or inverse transformations. In the subsequent discussion we will discuss the partial fraction expansion method and we will illustrate with several examples. We will also use the MATLAB residue(r,p,k) function which returns the residues (coefficients) r of a partial fraction expansion, the poles p and the direct terms k. There are no direct terms if the highest power of the numerator is less than that of the denominator.

Let

where N( s) and D( s) are polynomials and thus (12.1) can be expressed as

The coefficients a _k and b _k for k = 0, 1, 2, ..., n are real numbers and, for the present discussion, we have assumed that the highest power of N( s) is less than the highest power of D( s), i.e., m< n. In this case, F( s)...