Numerator

# of order, m=

b0:

Denominator

# of Order, n=

a0:

Plot Options

Number of points: (≤1000)

Range of time (t): 0 to

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.

## Tutorial

Example

The input in the tool is

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

Poles

s1 = -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

## Subject Review

### 1. Definitions

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.

### 2. Some Properties of Laplace Transforms

1. Superposition or linear combination

2. Time delay or shift in time

Given a function f(t), a corresponding time-shifted function fd(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 F1(s) = L{f1(t)} and F2(s) = L{f2(t)}

Here

f1*f2 = the convolution of the time functions f1 and f2, which is defined by

where f1(t-τ) and f2(t-τ) are time-shifted according to f1(t) and f2(t), respectively.

### 3. A Laplace Transform Table

Table 1 lists some basic functions and their Laplace transforms.

Table 1 A Laplace Transform Table

### 4. Inverse Laplace Transform via Partial Fraction Expansion

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=bn/an, 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

rj = the residue of F(s) at pole pj, 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 pj 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.

### References

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.