Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.2 LAPLACE TRANSFORMS

f(t) f ̄(s) s 0

cc/s 0

ctn cn!/sn+1 0

sinbt b/(s^2 +b^2 )0

cosbt s/(s^2 +b^2 )0

eat 1 /(s−a) a

tneat n!/(s−a)n+1 a

sinhat a/(s^2 −a^2 ) |a|

coshat s/(s^2 −a^2 ) |a|

eatsinbt b/[(s−a)^2 +b^2 ] a

eatcosbt (s−a)/[(s−a)^2 +b^2 ] a

t^1 /^212 (π/s^3 )^1 /^20

t−^1 /^2 (π/s)^1 /^20

δ(t−t 0 ) e−st^00

H(t−t 0 )=

{

1fort≥t 0

0fort<t 0

e−st^0 /s 0

Table 13.1 Standard Laplace transforms. The transforms are valid fors>s 0.

Comparing this with the standard Laplace transforms in table 13.1, we find that the inverse
transform of 3/sis 3 fors>0 and the inverse transform of 2/(s+1) is 2e−tfors>−1,
and so

f(t)=3− 2 e−t, ifs> 0 .

13.2.1 Laplace transforms of derivatives and integrals

One of the main uses of Laplace transforms is in solving differential equations.

Differential equations are the subject of the next six chapters and we will return

to the application of Laplace transforms to their solution in chapter 15. In

the meantime we will derive the required results, i.e. the Laplace transforms of

derivatives.

The Laplace transform of the first derivative off(t) is given by

L

[

df

dt

]

=

∫∞

0

df

dt

e−stdt

=

[

f(t)e−st

]∞

0 +s

∫∞

0

f(t)e−stdt

=−f(0) +sf ̄(s), fors> 0. (13.57)

The evaluation relies on integration by parts and higher-order derivatives may

be found in a similar manner.