0195136047.pdf

APPENDIX

H

Laplace Transforms

Iff(t)is a piecewise continuous real-valued function of the real variablet( 0 ≤t<∞), and

|f(t)|<Meσt(t > T;M,σ,Tpositive constants), then theLaplace transformoff(t), given by

La[f(t)]=F(s)=

∫∞

0

e−stf(t) dt, (1)

exists in the half-plane of the complex variables=σ+jω, for which the real part ofsis greater

than some fixed values 0 , i.e.,Re(s)s 0.

Theinverse transformis defined as

f(t=)La−^1 [F(s)]=

1

2 πj

σ∫+j∞

σ−j∞

F(s)estds (2)

whereσ>s 0 is chosen to the right of any singularity ofF(s).

The property of Laplace transform given by

La

[

f(r)(t)

]

=

∫∞

0

e−st(

drf

dtr

)dt=srF(s)−

∑r−^1

n= 0

sr−^1 −nf(n)( 0 +)

or

La

[

drf(t)

dtr

]

=srF(s)−sr−^1 ( 0 +)−sr−^2

df

dt

( 0 +)−...−

dr−^1 f

dtr−^1

( 0 +) (3)

makes the Laplace transform very useful for solving linear differential equations with constant

coefficients, and many boundary value problems.

A summary of properties of Laplace transformation and a table of Laplace transform pairs

are given below.

Summary of Properties of Laplace Transformation

Property Time Function Laplace Transform

Linearity a 1 f 1 (t)±a 2 f 2 (t) a 1 F 1 (s)±a 2 F 2 (s)

Differentiation f′(t) sF (s)−f( 0 +)

fn(t) snF(s)−sn−^1 f( 0 +)−sn−^2 f′( 0 +)−...−fn−^1 ( 0 +)

Integration f−^1 (t)=

∫t

0

f(τ) dτ F(s)

s

+f

− (^1) ( 0 +)
s
f−n(t) F(s)
sn
+f
− (^1) ( 0 +)
sn
+f
− (^2) ( 0 +)
sn−^1
+...+f
n( 0 +)
s
851