5.1 The Inverse Laplace Transform Integral

The Inverse Laplace Transform Integral was stated in the previous chapter; it is repeated here for convenience.

This integral is difficult to evaluate because it requires contour integration using complex variables theory. Fortunately, for most engineering problems we can refer to Tables of Properties, and Common Laplace transform pairs to lookup the Inverse Laplace transform.

5.2 Partial Fraction Expansion

Quite often the Laplace transform expressions are not in recognizable form, but in most cases appear in a rational form of s, that is,

where N( s) and D( s) are polynomials, and thus (5.2) can be expressed as

The coefficients a _k and b _k are real numbers for k = 1, 2, ..., n, and if the highest power m of N( s) is less than the highest power n of D( s), i.e., m < n, F( s) is said to be expressed as a proper rational Junction. If m ? n, F( s) is an improper rational function.

In a proper rational function, the roots of N( s) in (5.3) are found by...