Subsections

### 7.1 Overview of the Procedure

The Laplace transform pairs a function of a real coordinate, call it , with , with a different function of a complex coordinate :

The pairing is designed to get rid of derivatives with respect to in equations for the function . This works as long as the coefficients do not depend on (or at the very most are low degree powers of .) The transformation is convenient since pairings can be looked up in tables.

#### 7.1.1 Typical procedure

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

Table 7.1 assumes that and are positive.

Table 7.1: Properties of the Laplace transform.
0.
1.
2.
3.
4.
5.
6.
7.

Table 7.2: Selected Laplace transform pairs.
 1. 2. 3. 4. 5. 6. 7. 8. 9.

#### 7.1.2 About the coordinate to be transformed

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 .