Extensive tables of transforms (like Table A. l) have been published that are use­ful in carrying out the inverse process. When the transform involved can be found in the tables, the function x(t) is obtained directly.

The Method of Partial Fractions

In some cases it is convenient to expand the transform x(s) in partial fractions, so that the elements are all simple ones like those in Table A. 1. The function x(t) can then be obtained simply from the table. This procedure is illustrated with an example. Let the second-order system of Sec. 7.3 be initially quiescent, that is, x(0) = 0, and i(0) = 0, and let it be acted upon by a constant unit force applied at time t = 0. Then fit) = 1, and f(s) = l/s (see Table A. l). Then (see (7.4,1))

‘To avoid ambiguity when dealing with step functions, t = 0 should always be interpreted as t = 0+.

Let us assume that the system is aperiodic; that is, that £ > 1. Then the roots of the characteristic equation are real and equal to

A, 2 = n ± со’ (A.2,6)


n = -£a)n со’ = Шяа2 ~ 1)1/2

The denominator of (A.2,5) can be written in factored form so that


s(s – A,)(s – A2) Now let (A.2,7) be expanded in partial fractions,

Heaviside Expansion Theorem

When the transform is a ratio of two polynomials in s, the method of partial frac­tions can be generalized. Let



where N(s) and D(s) are polynomials and the degree of D(s) is higher than that of N(s). Let the roots of D(s) = 0 be ar, so that

D(s) = (s – a{)(s – a2) ••• (s – an)

Then the inverse of the transform is

– ar)N(s) l

——— I ё

D(s) s=ar

The effect of the factor (5 — ar) in the numerator is to cancel out the same factor of the denominator. The substitution s = ar is then made in the reduced expression.2

In applying this theorem to (A.2,7), we have the three roots a, = 0, a2 = A,, a3 = A2, and N(s) = 1. With these roots, (A.2,9) follows immediately from (A.2,10).

The Inversion Theorem

The function x(t) can be found formally from its transform x(s’) by the application of the inversion theorem Jaeger (1949) and Carslaw and Jaeger (1947). It is given by the line integral

1 ry+ito

x{t) = —– lim es‘x(s) ds (A.2,11)

ІТГІ Jy-іш

where у is a real number greater than the real part of all values of s for which x(s) di­verges. That is, s = у is a straight line on the s plane lying parallel to the imaginary axis, and to the right of all the poles of x(.s). This theorem can be used, employing the methods of contour integrals in the complex plane, to evaluate the inverse of the transform.

Extreme Value Theorems

Equation (A.2,2) may be rewritten as



e ‘‘x(t) dt


= lim [ e~s‘x(t) dt


We now take the limit s —» 0 while T is held constant, that is, —x(0) + lim sx(s) = lim lim e~s, x(t) dt

■ s—»0 T-* 00 Jq s—*0

= lim f x(t) dt = lim [x(T) – x(0)]

T—>CO Jq T—»cc

Hence lim ix(i) = lim x(T) (A.2,12)

s—>0 T-^ 00

This result, known as the final value theorem, provides a ready means for determin­ing the asymptotic value of x{t) for large times from the value of its Laplace trans­form.

!For the case of repeated roots, see Jaeger (1949).

In a similar way, by taking the limit j —» °o at constant T, the integral vanishes for all finite x(t) and we get the initial value theorem.

lim sx(s) = x(0) (A.2,13)