The tool MA-05 determines the poles and partial fraction expansion and inverse Laplace transform of a given rational function F(s) and obtain the inverse Laplace transform of F(s), where s is the Laplace transform parameter. The tool also plots the inverse Laplace transform of F(s) in a preselected time region.

In addition, a tutorial on how to use the tool is provided and a subject review on Laplace transform and inverse Laplace transform is presented.

**Example**

The input in the tool is

After clicking "Run", the results will be shown as follows in another page.

Poles

s_{1} = -0.25

Partial Fraction Expansion of the Function

F(s) = k + (2.3125)/(s-(-0.25))

where k = 0.75

Inverse Laplace Transform

f(t) = 0.75 * δ(t) + (2.3125) * e^{-0.25*t}

where δ(t) being the delta function

For a real-valued, piecewise continuous function f(t) specified for t≥0, its Laplace transform, denoted by F(s), is defined by

Furthermore, can be expressed by the inverse Laplace transform of F(s):

In the above equations

The f(t) and F(s) forms a Laplace transform pair.

1. Superposition or linear combination

2. Time delay or shift in time

Given a function f(t), a corresponding time-shifted function f_{d}(t-T) is

where T is a positive time delay parameter. The Laplace transform of the time-shifted function is

3. Differentiation

4. Integration

5. Initial value theorem

For a Laplace transform pair f(t) and F(s)

if the indicated limits exist.

6. Final value theorem

Let and F(s) be a Laplace transform pair. If sF(s) is analytic on the imaginary and in the right-half of the s-plane (namely, all the poles of sF(s) have negative real parts), then

7. Convolution theorem

For F_{1}(s) = L{f_{1}(t)} and F_{2}(s) = L{f_{2}(t)}

Here

f_{1}*f_{2} = the convolution of the time functions f_{1} and f_{2}, which is defined by

where f_{1}(t-τ) and f_{2}(t-τ) are time-shifted according to f_{1}(t) and f_{2}(t), respectively.

Table 1 lists some basic functions and their Laplace transforms.

**Table 1** A Laplace Transform Table

Consider a rational function of the form

where all the coefficients are real. When m=n, by long division, Eq. (1) can be written as

where the direct term coefficient k=b_{n}/a_{n}, and

For this reason, without loss of generality, we only need to consider the case of m < n for partial fraction expansion.

The denominator D(s) has n roots. These roots are also called the poles of function F(s). In what follows, partial fraction expansion and inverse Laplace transform are presented in two cases of the poles of F(s).

**Case 1. All poles of F(s) are distinct**

The partial fraction expansion of F(s) is

where

r_{j} = the residue of F(s) at pole p_{j}, and is given by

By Eq. (2), the inverse Laplace transform is

**Case 2. Poles of F(s) are repeated**

Assume that F(s) has l different poles:

where

The partial fraction expansion of F(s) is of the form

where the residues about the pole p_{j} are given by

with

It follows from Eq. (5) that

The tool MA-05 gives the expressions (3) and (8) in real format; see Reference 2.

1. M. R. Spiegel, 1965, Schaum's Outlines: Laplace Transforms, 1st edition, McGraw-Hill.

2. Yang, B., 2005, Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes, Elsevier Science.