Page 450 - Handbook Of Integral Equations

The Laplace transform exists for any continuous or piecewise-continuous function satisfying the
condition |f(x)| < Me σ 0 x with some M > 0 and σ 0 ≥ 0. In the following, σ 0 often means the greatest
lower bound of the possible values of σ 0 in this estimate; this value is called the growth exponent of
the function f(x).
˜
For any f(x), the transform f(p)isdefined in the half-plane Re p > σ 0 and is analytic there.
For brevity, we shall write formula (1) as follows:

˜
˜
f(p)= L f(x) , or f(p)= L f(x), p .
˜
Given the transform f(p), the function can be found by means of the inverse Laplace transform

1 c+i∞ px 2
˜
f(x)= f(p)e dp, i = –1, (2)
2πi
c–i∞
where the integration path is parallel to the imaginary axis and lies to the right of all singularities
˜
of f(p), which corresponds to c > σ 0 .
The integral in (2) is understood in the sense of the Cauchy principal value:
c+iω
c+i∞
˜
˜
f(p)e px dp = lim f(p)e px dp.
ω→∞ c–iω
c–i∞
In the domain x < 0, formula (2) gives f(x) ≡ 0.
Formula (2) holds for continuous functions. If f(x)hasa(finite) jump discontinuity at a point
1
x = x 0 > 0, then the left-hand side of (2) is equal to [f(x 0 – 0) + f(x 0 + 0)] at this point (for x 0 =0,
2
the first term in the square brackets must be omitted).
For brevity, we write the Laplace inversion formula (2) as follows:

–1 ˜ –1 ˜
f(x)= L f(p) , or f(x)= L f(p), x .
7.2-2. The Inverse Transforms of Rational Functions

Consider the important case in which the transform is a rational function of the form

R(p)
˜
f(p)= , (3)
Q(p)
where Q(p) and R(p) are polynomials in the variable p and the degree of Q(p) exceeds that of R(p).
Assume that the zeros of the denominator are simple, i.e.,

Q(p) ≡ const (p – λ 1 )(p – λ 2 ) ... (p – λ n ).

Then the inverse transform can be determined by the formula

n
R(λ k )
f(x)= exp(λ k x), (4)
Q (λ k )

k=1
where the primes denote the derivatives.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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