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CHAPTER 6

Integral Transforms: The Laplace

Transform

Integral transforms are a powerful method of obtaining solutions to both ordinary and partial
differential equations. They are used to change ordinary differential equations into algebraic
equations and partial differential into ordinary differential equations. The general idea is to
multiply a function f (t) of some independent variable t (not necessarily time) by a Kernel
function K(t, s ) and integrate over some t space to obtain a function F(s )of s which one hopes
is easier to solve. Of course one must then inverse the process to find the desired function f (t).
In general,

b

F(s ) = K(t, s ) f (t)dt (6.1)
t=a

6.1 THE LAPLACE TRANSFORM
A useful and widely used integral transform is the Laplace transform, defined as

∞

L[ f (t)] = F(s ) = f (t)e −st dt (6.2)
t=0

Obviously, the integral must exist. The function f (t) must be sectionally continuous and of

exponential order, which is to say f (t) ≤ Me kt when t > 0 for some constants M and k.For

2
example neither the Laplace transform of t −1 nor exp(t ) exists.
The inversion formula is
γ +iL
1 lim
−1 ts
L [F(s )] = f (t) = F(s )e ds (6.3)
2πi L →∞
γ −iL