0195136047.pdf

852 APPENDIX H

∫t

0

f(τ)g(t−τ ) dτ F (s)G(s)

Multiplication byt

(frequency differentiation)

tf (t) −

d

ds[F(s)]

Division byt

(frequency integration)

1

t

f(t)

∫∞

s

F(s) ds

Time delay or shift f(t−T)·u(t−T) e−sTF(s)

Periodic function

f(t)=f(t+nT )

f(t), 0 ≤t≤T F (s)/[1−eTs],whereF(s)=

∫T

0

f(t)e−stdt

Exponential translation

(frequency shifting)

e−atf(t) F(s+a)

Change of scale

(time scaling)

f(at), a > 0

1

a

F(

s

a

)

Initial value f( 0 +)=lim

t→ 0

f(t) slim→∞sF (s)

Final value f(∞)=tlim→∞f(t), lim

s→ 0

sF (s)

where limit exists [sF (s)has poles only inside the left half of thes-plane.]

Table of Laplace Transform Pairs

f(t)=La−^1 [F(s)] F(s)=La[f(t)]

δ(t)(unit impulse or delta function) 1

δ(t−T) e−sT

a a

s

u(t)or 1 (unit step function)

1

s

u(t−T)

e−sT

s

t

1

s^2

(t−T ) u(t−T)

e−sT

s^2

tu(t−T)

( 1 +sT )e−sT

s^2

tn,(n−integer)

n!

sn+^1

tn−^1 /(n− 1 )!,nan integer^1

sn

e−at

1

s+a

te−at

1

(s+a)^2

e−attn n!

(s+a)n+^1

e−at−e−bt

b−a

1

(s+a)(s+b)

sinωt ω

s^2 +ω^2