Understanding Engineering Mathematics

is called theLaplace transform off.t/. Other notations used areF(s)(lower case goes to
upper case when we transform) orL[f(t)], the square brackets denoting that the Laplace
transform acts on afunction. The appearance of the exponential function here should
perhaps come as no surprise when one considers the predominant role that this function
plays in, for example, differential equations.
Note that improper infinite integrals of the type occurring in the definition above are
defined by:

∫∞

0

g(t)dt=lim

a→∞

∫a

0

g(t)dt

and this shows how they are evaluated in practice – i.e. do the integral first and then take
the limit, as we did in Problem 17.1.

Exercises on 17.2

1. Write down the values of the following limits:

(i) lim

t→∞

e−st

s

s> 0 (ii) lim

t→∞

e−st

s^2

s> 0

(iii) lim

t→∞

te−st

s

s> 0 (iv) lim

t→∞

tne−st s> 0 ,npositive

2. Find the Laplace transform off(t)=3.

Answers

1. (i) 0 (ii) 0 (iii) 0 (iv) 0 2.

3

s

17.3 Laplace transforms of the elementary functions

Assembling the Laplace transforms of the elementary functions is simply a matter of inte-
gration. We flagged up the main points in Problem 17.1, and now you can try developing
a table of Laplace transforms for yourself.

Problem 17.2

Find the Laplace transform of the constant functionf.t/=1.

From the definition we have

L[1]=

∫∞

0

1 e−stdt

= lim

a→∞

[

e−st

−s

]a

0

=

1

s

provideds> 0

Problem 17.3

EvaluateL[t],L[t^2 ],L[t^3 ] and look for a pattern forL[tn].