The Laplace transform pairs a function of a real coordinate, call it
, with , with a different function of a complex
coordinate :
Use tables to find the equations satisfied by from these satisfied by . Solve for and look up the corresponding in the tables.
Table 7.1 lists important properties of the Laplace
transform and table 7.2 gives example Laplace transform
pairs. In the tables, , , , and are constants, normally
positive, is a natural number, and
In many cases, is physically time, since time is most likely to satisfy the constraints and coefficients independent of . Also, the Laplace transform likes initial conditions at , not boundary conditions at both and .