Mathematical Methods for Physics and Engineering - Matematica.NET

13.3 CONCLUDING REMARKSThe properties of the Laplace transform derived in this section can sometimesbe useful in finding the Laplace transforms of particular functions.◮Find the Laplace transform of f(t) =t sin bt.Although we could calculate the Laplace transform directly, we can use (13.62) to give¯f(s) =(−1) d ds L [sin bt] = − d ( ) b 2bs= , for s>0. ◭ds s 2 + b 2 (s 2 + b 2 )213.3 Concluding remarksIn this chapter we have discussed Fourier and Laplace transforms in some detail.Both are examples of integral transforms, which can be considered in a moregeneral context.A general integral transform of a function f(t) takes the formF(α) =∫ baK(α, t)f(t) dt, (13.65)where F(α) is the transform of f(t) with respect to the kernel K(α, t), and α isthe transform variable. For example, in the Laplace transform case K(s, t) =e −st ,a =0,b = ∞.Very often the inverse transform can also be written straightforwardly andwe obtain a transform pair similar to that encountered in Fourier transforms.Examples of such pairs are(i) the Hankel transformF(k) =f(x) =∫ ∞0∫ ∞0f(x)J n (kx)xdx,F(k)J n (kx)kdk,where the J n are Bessel functions of order n, and(ii) the Mellin transformF(z) =∫ ∞0f(t) = 12πit z−1 f(t) dt,∫ i∞−i∞t −z F(z) dz.Although we do not have the space to discuss their general properties, thereader should at least be aware of this wider class of integral transforms.459