Page 451 - Handbook Of Integral Equations

If Q(p) has multiple zeros, i.e.,

s 2
s m
Q(p) ≡ const (p – λ 1 ) (p – λ 2 ) ... (p – λ m ) ,
s 1
then
m s k –1
1 d
s k ˜
f(x)= lim (p – λ k ) f(p)e px .
(s k – 1)! p→s k dp k –1
s
k=1
7.2-3. The Convolution Theorem for the Laplace Transform
x
The convolution of two functions f(x) and g(x)isdefined as the integral f(t)g(x – t) dt, and is
0
usually denoted by f(x) ∗ g(x). The convolution theorem states that

L f(x) ∗ g(x) = L f(x) L g(x) ,

and is frequently applied to solve Volterra equations with kernels depending on the difference of the
arguments.

7.2-4. Limit Theorems

˜
Let 0 ≤ x < ∞ and f(p)= L f(x) be the Laplace transform of f(x). If a limit of f(x)as x → 0
exists, then

˜
lim f(x) = lim pf(p) .
x→0 p→∞
If a limit of f(x)as x →∞ exists, then

˜
lim f(x) = lim pf(p) .
x→∞ p→0
7.2-5. Main Properties of the Laplace Transform
The main properties of the correspondence between functions and their Laplace transforms are
gathered in Table 1.
There are tables of direct and inverse Laplace transforms (see Supplements 4 and 5), which are
handy in solving linear integral and differential equations.

7.2-6. The Post–Widder Formula

˜
In applications, one can find f(x) if the Laplace transform f(t) on the real semiaxis is known for
t = p ≥ 0. To this end, one uses the Post–Widder formula

n
(–1) n
n+1 (n)
n
f(x) = lim f ˜ t . (5)
n→∞ n! x x
Approximate inversion formulas are obtained by taking sufficiently large positive integer n in (5)
instead of passing to the limit.
•
References for Section 7.2: G. Doetsch (1950, 1956, 1958), H. Bateman and A. Erd´ elyi (1954), I. I. Hirschman and
D. V. Widder (1955), V. A. Ditkin and A. P. Prudnikov (1965), J. W. Miles (1971), B. Davis (1978), Yu. A. Brychkov and
A. P. Prudnikov (1989), W. H. Beyer (1991).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 432