The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 136 Friday, March 12, 2004 12:39 PM

136 The Mathematics of Financial Modeling and Investment Management

∞

Lafx[ ()+ bg x()]= ∫ e –sx(af x()+ bg x())dx

• ∞
  ∞ ∞

∫

= ae –sx fx()dx+ be–sxgx

∫ ()dx

• ∞ –∞
  = aL[fx()]+ bL[gx()]

Laplace transforms turn differentiation, integration, and convolu-

tion (defined below) into algebraic operations. For derivatives the fol-

lowing property holds for the two-sided transform:

L

df x()

-------------- = sL[fx()]

dx

and

L -------------df x() - = sL[fx()]– f() 0

dx

for the one-sided transform. For higher derivatives the following for-

mula holds for the two-sided transform

()

x

n

0 f

(n– 1 )

()]– s ()

n– 1

L[f f 0

n

()] = sL[fx ()– s

n– 2

f' ()– ...– 0

An analogous property holds for integration for one-sided trans-

forms

t

L (^) ∫fx
1
() = ---L[fx()] for the one-sided transform
0 s^
t
L (^) ∫fx
1
() = ---L[fx()] for the two-sided transform
0 s^
Consider now the convolution. Given two functions fand g, their
convolution h(x) = f(x) ∗g(x) is defined as the integral