Mathematical Methods for Physics and Engineering - Matematica.NET

INTEGRAL TRANSFORMSFigure 13.7text).Two representations of the Laplace transform convolution (seewhere the integral in the brackets on the LHS is the convolution of f and g,denoted by f ∗ g. As in the case of Fourier transforms, the convolution definedabove is commutative, i.e. f ∗ g = g ∗ f, and is associative and distributive. From(13.64) we also see thatL −1[ ∫] t¯f(s)ḡ(s) = f(u)g(t − u) du = f ∗ g.0◮Prove the convolution theorem (13.64) for Laplace transforms.From the definition (13.64),¯f(s)ḡ(s) ==∫ ∞0∫ ∞0e −su f(u) dudu∫ ∞0∫ ∞0e −sv g(v) dvdv e −s(u+v) f(u)g(v).Now letting u + v = t changes the limits on the integrals, with the result that¯f(s)ḡ(s) =∫ ∞0du f(u)∫ ∞udt g(t − u) e −st .As shown in figure 13.7(a) the shaded area of integration may be considered as the sumof vertical strips. However, we may instead integrate over this area by summing overhorizontal strips as shown in figure 13.7(b). Then the integral can be written as¯f(s)ḡ(s) ==∫ t0∫ ∞0∫ ∞du f(u) dt g(t − u) e −st0{∫ t}dt e −st f(u)g(t − u) du[∫ t]= L f(u)g(t − u) du . ◭04580