1 Laplace transforms

1.1 Introduction

The Laplace transform, named after its inventor Pierre-Simon Laplace, is an integral transform. Laplace transform converts a function of a real variable (often time) to a function of a complex variable (complex frequency).

Laplace transforms are useful to solve linear ordinary differential equations and analyze frequency response and stability analysis. Process control feedback loops and their response properties and stability can thus be conveniently analyzed in the Laplace domain.

Also, depending on the boundary conditions of your problem, it can be judicious to use a Laplace transform to solve the diffusion equation, heat transfer equation, and Navier-Stokes. So, Laplace transforms will show up in many core engineering curricula, for example, mass transport, heat transport, fluid transport, and process controls. For the basic concepts of Laplace transform and its applications, one can refer to [2-6, 8, 9, 13, 14].

1.2 Definition of Laplace transforms

As we have explained in Introduction, the integral transform of function ft in the interval a=x=b is given by,

where -8=a<b=8

Let ft be a function defined for t=0 0=t=8 and kernel Kt,s=e-st and then eq. (1.1) will be called Laplace transform under certain conditions to be explained later.

Thus, the Laplace transform of function ft which is defined for t=0 (t is real variable), is formally defined as follows:

where s is the transform variable, which is a complex number.

Figure 1.1: The Laplace transforms as a mapping.

Therefore, the Laplace transform converts time domain functions and operations into frequency domain ftFs t?R, s?CasshowninFigure1.1.

1.3 Laplace transform of some elementary functions using definition

Example 1.1

Find the Laplace transform ft=1 for t=0.

Solution

We know that Lft=?08e-stftdt

Here, ft=1:

?L1=1-s0-1=1s=Fs Thus, L1=1s, s>0.

Example 1.2

Find the Laplace transform of f(t)=eat for t=0, where "a" is a constant.

Solution

We know that Lft=?08e-stftdt

Here f(t)=eat

Thus, Leat=1s-a, s>a.

Note: Similarly, Le-at=1s+a, s>-a.

Example 1.3

Find the Laplace transform of ft=sinat, where "a" is a real constant.

Solution

We know that

Now, ft=sinat and Lft= ?08e-stft dt

Thus, Lsinat=as2+a2.

Example 1.4

Find the Laplace transform of ft=cosat, where "a" is a real constant.

Solution

We know that

Now, ft=cosat and Lft=?08e-stft dt ,

Thus, Lcosat=ss2+a2.

1.3.1 Linearity of Laplace transforms

Functions f1t and f2t have Laplace transforms F1s and F2s, respectively.

Also, if c1 and c2 are any constants, then,

Proof. We know that

Now,

Example 1.5

Find the Laplace transform of ft=sinhat, where "a" is a real constant.

Solution

We know that

Now,

(by linearity property)

Thus, Lsinhat=as2-a2

Example 1.6

Find the Laplace transform of ft=coshat, where "a" is a real constant.

Solution

We know that

Now,

(by linearity property)

Thus, Lcoshat=ss2-a2.

Note

1. The gamma function is defined by the improper integral ?08e-xxn-1 dx=|n? for n>0

2. If n is a positive integer, then |n+1?=n!

3. |n+1?=n|n?

4. ...