Introduction

Description:

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

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

Typical procedure:

Use tables to find the equations satisfied by from these satisfied by u. Solve for and look up the corresponding u in the tables.

About coordinate t:

In many cases, t is physically time, since time is most likely to satisfy the constraints and coefficients independent of t. Also, the Laplace transform likes initial conditions at t=0, not boundary conditions at both t=0 and .

Table 6.3: Properties of the Laplace transform:

Here a>0, b>0, c are constants, and n is a natural number.

Table 6.4: Laplace transform pairs:

Here k>0, a and b are constants, n is a natural number, and

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