Appendix D Laplace transforms
The Laplace transform of a functionf(t) is just one of a general class of integral
transforms of the form
f ̃(s) =∫∞
0dt K(st)f(t), (D.1)whereK(x) is a kernel that is typically a smooth and rapidly decaying function ofx.
The Laplace transform corresponds to the choiceK(x) = e−x.
Interestingly, even if the integral off(t) over the intervalt∈[0,∞) does not exist,
the Laplace transformf ̃(s) nevertheless can. If there exists a positive constants 0 such
that|e−s^0 tf(t)|≤MforMfinite, thenf ̃(s) exists fors > s 0. By contrast, a function
such asf(t) = exp(t^2 ) does not have a Laplace transform.
Laplace transforms of elementary functions are generally straightforward to evalu-
ate. Some examples are given below:
f(t) =tn, f ̃(s) =n!
sn+1,
f(t) = e−at, f ̃(s) =1
s+a,
f(t) = cos(ωt), f ̃(s) =s
s^2 +ω^2,
f(t) = sin(ωt), f ̃(s) =ω
s^2 +ω^2,
f(t) = cosh(αt), f ̃(s) =s
s^2 −α^2,
f(t) = sinh(αt), f ̃(s) =
α
s^2 −α^2. (D.2)
In addition to these elementary transforms another useful result is the Laplace trans-
form of the convolutionf(t) between two functionsg(t) andh(t). Recall that this
convolution is defined as
f(t) =∫t0dτ g(τ)h(t−τ). (D.3)The Laplace transform off(t) isf ̃(s) = ̃g(s) ̃h(s), which is known as theconvolution
theorem. The proof of the convolution theorem proceeds by writing