Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 408

a result that, by repeated application, can be used to construct similar formulas for
higher order derivatives, for example,

f 00 ð sf 0 ð s 2 L L f t Þg ¼ f f ð t Þg 0 Þ f ð 0 Þ: (A.209)

The Laplace transform of an integral is given by

8
<

9
=
; ¼

ð

t

1

s f

L f ð x Þ d x ð s Þ; (A.210)

:

0

for s > a . It follows that, for a function whose value at t ¼ 0 is zero, and whose

higher order derivatives are zero at t 0, differentiation corresponds to multipli-

cation by s and integration corresponds to division by s . If the function f ( t )isof
exponential order, then

¼

Þg ¼ f f e at f

L ð t ð s þ a Þ; (A.211)

a , and if f

for s > a ð s Þ has derivatives of all orders for s > a ,

or f ð n Þ

f 0 ð

n f

s Þ¼ L f tf ð t Þg ð s Þ¼ L fð t Þ ð t Þg: (A.212)

In the solution of differential equations using the Laplace transform it is often
necessary to expand the transform into partial fractions before finding the inverse
transform. The rational function P ( s )/ Q ( s ), where Q ( s ) is a polynomial with n
distinct zeros,

a 1 , a 2 , ... , a n , and P ( s ) is a polynomial of degree less than n , can be

written in the form

P ð s Þ A 1 A 2 A n

Þ ¼ a 1 þ a 2 þþ

(A.213)

Q ð s s s s a n

where the A i are constants. The constants A i are determined by multiplying both
sides of the equation above by s

a i and letting s approach a i , thus

ð s a i Þ P ð s Þ Þ

Q ð s Þ Q ða i Þ
sa i

P ð s ða i Þ

Q 0 ða i Þ :

P

A i ¼ lim it

s>a i

¼ lim it

s>a i

¼ (A.214)

Q ð s Þ

Combining the two previous equations one may write

P ð s Þ ða 1 Þ

Q 0 ða 1 Þ

P ða n Þ

Q 0 ða n Þ

P

1 1

Þ ¼ a 1 þþ a n ;

(A.215)

Q ð s s s

whose inverse Laplace transform is given by