Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.2 LAPLACE TRANSFORMS

A similar result may be obtained for two-dimensional Fourier transforms in

whichf(r)=f(ρ), i.e.f(r) is independent of azimuthal angleφ. In this case, using

the integral representation of the Bessel functionJ 0 (x) given at the very end of

subsection 18.5.3, we find

̃f(k)=^1

2 π

∫∞

0

2 πρf(ρ)J 0 (kρ)dρ. (13.52)

13.2 Laplace transforms

Often we are interested in functionsf(t) for which the Fourier transform does not

exist becausef→0ast→∞, and so the integral defining ̃fdoes not converge.

This would be the case for the functionf(t)=t, which does not possess a Fourier

transform. Furthermore, we might be interested in a given function only fort>0,

for example when we are given the value att= 0 in an initial-value problem.

This leads us to consider the Laplace transform,f ̄(s)orL[f(t)],off(t), which

is defined by

f ̄(s)≡

∫∞

0

f(t)e−stdt, (13.53)

provided that the integral exists. We assume here thatsis real, but complex values

would have to be considered in a more detailed study. In practice, for a given

functionf(t) there will be some real numbers 0 such that the integral in (13.53)

exists fors>s 0 but diverges fors≤s 0.

Through (13.53) we define alineartransformationLthat converts functions

of the variabletto functions of a new variables:

L[af 1 (t)+bf 2 (t)]=aL[f 1 (t)]+bL[f 2 (t)]=af ̄ 1 (s)+bf ̄ 2 (s). (13.54)

Find the Laplace transforms of the functions(i)f(t)=1,(ii)f(t)=eat,(iii)f(t)=tn,

forn=0, 1 , 2 ,....

(i) By direct application of the definition of a Laplace transform (13.53), we find

L[ 1 ]=

∫∞

0

e−stdt=

[

− 1

s

e−st

]∞

0

=

1

s

, ifs> 0 ,

where the restrictions>0 is required for the integral to exist.

(ii) Again using (13.53) directly, we find

f ̄(s)=

∫∞

0

eate−stdt=

∫∞

0

e(a−s)tdt

=

[

e(a−s)t

a−s

]∞

0

=

1

s−a

ifs>a.