Page 228 - Schaum's Outline of Differential Equations

CHAPTER 21

The Laplace

Transform

DEFINITION

Letf(x) be defined for 0 <.r <•* and let i denote an arbitrary real variable. The Laplace (rans/brni off(x).
designated b\ either %{f(x)} or F(s). is

l
for all values of s for which the improper integral converges. Convergence occurs when she imit

exists. If this limit does not exist, the improper integral diverges and/(.v) has nol_aplace transform. When evaluating
w
the integral in Eq. (21.1), the variable s is treated as a constant because the integration is ith respect to _v.
The Laplace transforms for a number of elementary functions are calculated in Problems 21.4 through 21.8:
additional transforms are gi\en in Appendix A.

PROPERTIES OF LAPLACE TRANSFORMS

c
Properly 21.1. (Linearity). If f{f(x)} = F(s) and %{g(x)} = G'(.v). then for any two constants c, and c 2

Property 21.2. If !£{f(x)} = F(s\ then for an\ constant a

Property 21.3. If ?£{f(x)} = F(.s), then for an\ posili\e integer n

Properly21.4.

2 1
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