Chapter 6 Laplace Transform 367
f(t) F(s) f(t) F(s)00 tk skk+! 1(^1) s^1 ebtcos(ωt) s (^2) − 2 bss−+bb (^2) +ω 2
eat s−^1 a ebtsin(ωt) s (^2) − 2 bsω+b (^2) +ω 2
cosh(at) s (^2) −sa 2 ebttk (s−kb!)k+ 1
sinh(at) s (^2) −aa 2 eat− (^1) s(s−aa)
cos(ωt) s (^2) +sω 2 tcos(ωt) s
(^2) −ω 2
(s^2 +ω^2 )^2
sin(ωt) s (^2) +ωω 2 tsin(ωt) (s (^2) +^2 sωω (^2) ) 2
t s^12
Table 2 Laplace transforms
transforms. The last method, which involves the least work, is the most popu-
lar. The transforms in Table 2 were all calculated from the definition or by use
of formulas in this section.
EXERCISES
- By using linearity and the transform ofeat, compute the transform of each
of the following functions.
a.sinh(at);
d.sin(ωt−φ);
b.cos(ωt);
e.e^2 (t+^1 );c. cos^2 (ωt);
f. sin^2 (ωt).- Use differentiation with respect totto find the transform of
a.teatfromL(eat),
b.sin(ωt)fromL(cos(ωt)),
c. cosh(at)fromL(sinh(at)).- Compute the transform of each of the following directly from the defini-
tion.
a.f(t)=
{ 0 , 0 <t<a,
1 , a<t;