Mathematical Methods for Physics and Engineering - Matematica.NET

INTEGRAL TRANSFORMS◮Find the Laplace transform of d 2 f/dt 2 .Using the definition of the Laplace transform and integrating by parts we obtain[ ] dL2 ∫f ∞d 2 f=dt 2 dt 2 e−st dt0[ ] ∞ ∫ df∞=dt e−st + s0 0= − dfdt (0) + s[s ¯f(s) − f(0)],dfdt e−st dtfor s>0,where (13.57) has been substituted for the integral. This can be written more neatly as[ ] dL2 f= s 2 df¯f(s) − sf(0) −dt 2 dt (0), for s>0. ◭In general the Laplace transform of the nth derivative is given by[ d n ]fLdt n = s n ¯f − s n−1 n−2 dff(0) − sdt (0) − ···− dn−1 f(0), for s>0.dtn−1 (13.58)We now turn to integration, which is much more straightforward. From thedefinition (13.53),[∫ t] ∫ ∞ ∫ tL f(u) du = dt e −st f(u) du000=[− 1 ∫ t] ∞ ∫ ∞1s e−st f(u) du +00 0 s e−st f(t) dt.The first term on the RHS vanishes at both limits, and so[∫ t]L f(u) du = 1 L [f] . (13.59)s013.2.2 Other properties of Laplace transformsFrom table 13.1 it will be apparent that multiplying a function f(t) bye at has theeffect on its transform that s is replaced by s − a. This is easily proved generally:L [ e at f(t) ] ==∫ ∞0∫ ∞0f(t)e at e −st dtf(t)e −(s−a)t dt= ¯f(s − a). (13.60)As it were, multiplying f(t) bye at moves the origin of s by an amount a.456