25.5 INVERSE LAPLACE TRANSFORM
Γ R ΓΓR R
L L L
(a) (b) (c)Figure 25.7 Some contour completions for the integration pathLof the
inverse Laplace transform. For details of when each is appropriate see the
main text.plausible:
f(x)=1
2 πi∫λ+i∞λ−i∞ds esx∫∞0e−suf(u)du, Re(s)> 0 ,i.e.λ> 0 ,=1
2 πi∫∞0du f(u)∫λ+i∞λ−i∞es(x−u)ds=1
2 πi∫∞0du f(u)∫∞−∞eλ(x−u)eip(x−u)i dp, puttings=λ+ip,=1
2 π∫∞0f(u)eλ(x−u) 2 πδ(x−u)du={
f(x) x≥ 0 ,0 x< 0.(25.26)Our main purpose here is to demonstrate the use of contour integration. To
employ it in the evaluation of the line integral (25.25), the pathLmust be made
part of a closed contour in such a way that the contribution from the completion
either vanishes or is simply calculable.
A typical completion is shown in figure 25.7(a) and would be appropriate if
̄f(s) had a finite number of poles. For more complicated cases, in whichf ̄(s)has
an infinite sequence of poles but all to the left ofLas in figure 25.7(b), a sequence
of circular-arc completions that pass between the poles must be used andf(x)is
obtained as a series. If ̄f(s) is a multivalued function then a cut plane is needed
and a contour such as that shown in figure 25.7(c) might be appropriate.
We consider here only the simple case in which the contour in figure 25.7(a)is used; we refer the reader to the exercises at the end of the chapter for others.