Page 229 - Schaum's Outline of Differential Equations

212 THE LAPLACE TRANSFORM [CHAP. 21

Property 21.5. If, = F(s), then

Property 21.6. If f(x) is periodic with period co, that is,f(x + co) =f(x), then

FUNCTIONS OF OTHER INDEPENDENT VARIABLES

For consistency only, the definition of the Laplace transform and its properties, Eqs. (21.1) through (21.8),
are presented for functions of x. They are equally applicable for functions of any independent variable and
are generated by replacing the variable x in the above equations by any variable of interest. In particular, the
counterpart of Eq. (21.1) for the Laplace transform of a function of t is

Solved Problems

21.1. Determine whether the improper integral converges.

Since

the improper integral converges to the value •!•.

21.2. Determine whether the improper integral converges.

Since

the improper integral diverges.

21.3. Determine those values of s for which the improper integral converges.
For 5 = 0,