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Poularikas, A. D., Seely, S. “Laplace Transforms.” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000

5
Laplace Transforms
Alexander D. Poularikas
University of Alabama in Huntsville

Samuel Seely
(Deceased)

5.1 5.2 5.3 5.4 5.5

Introduction Laplace Transform of Some Typical Functions Properties of the LaplaceTransform The Inverse Laplace Transform Solution of Ordinary Linear Equations with Constant Coefficients 5.6 The Inversion Integral 5.7 Applications to Partial Differential Equations 5.8 The Bilateral or Two-Sided Laplace Transform Appendix

5.1

Introduction*

The Laplace transform has been introduced into the mathematical literature by a variety of procedures. Among these are: (a) in itsrelation to the Heaviside operational calculus, (b) as an extension of the Fourier integral, (c) by the selection of a particular form for the kernel in the general Integral transform, (d) by a direct definition of the Laplace transform, and (e) as a mathematical procedure that involves multiplying the function f (t ) by e – s t dt and integrating over the limits 0 to ∞. We will adopt this latterprocedure. Not all functions f (t ), where t is any variable, are Laplace transformable. For a function f (t ) to be Laplace transformable, it must satisfy the Dirichlet conditions — a set of sufficient but not necessary conditions. These are 1. f (t ) must be piecewise continuous; that is, it must be single valued but can have a finite number of finite isolated discontinuities for t > 0. 2. f (t ) must beof exponential order; that is, f(t) must remain less than Me – a o t as t approaches ∞, where M is a positive constant and a o is a real positive number. For example, such functions as: tan β t, cot β t, et 2 are not Laplace transformable. Given a function f (t) that satisfies the Dirichlet conditions, then

F s =

( ) ∫ f (t )e
0

∞

−st

dt written

{ f (t )}

(1.1)

is called theLaplace transformation of f (t ). Here s can be either a real variable or a complex quantity. Observe the shorthand notation {f (t )} to denote the Laplace transformation of f (t ). Observe also that only ordinary integration is involved in this integral.
*

All the contour integrations in the complex plane are counterclockwise.

© 2000 by CRC Press LLC

To amplify the meaning ofcondition (2), we consider piecewise continuous functions, defined for all positive values of the variable t, for which

lim f t e − c t = 0, c = real constant .
t→∞

()

Functions of this type are known as functions of exponential order. Functions occurring in the solution for the time response of stable linear systems are of exponential order zero. Now we can recall that the integral

∫ f (t ) e∞ 0

– st

dt converges if

∫ f (t )e
0 ∞

∞

−st

dt < ∞ , s = σ + jω

If our function is of exponential order, we can write this integral as

∫ f (t ) e
0

−ct

e

− σ −c t

(

) dt .

This shows that for σ in the range σ > 0 (σ is the abscissa of convergence) the integral converges; that is

∫ f (t )e
0

∞

−st

dt < ∞ , Re s > c .

()

Therestriction in this equation, namely, Re(s) = c , indicates that we must choose the path of integration in the complex plane as shown in Figure 5.1.

FIGURE 5.1

Path of integration for exponential order function.

© 2000 by CRC Press LLC

5.2

Laplace Transform of Some Typical Functions

We illustrate the procedure in finding the Laplace transform of a given function f(t). In all cases it isassumed that the function f(t) satisfies the conditions of Laplace transformability. Example 5.2.1 Find the Laplace transform of the unit step function f (t ) = u (t ), where u (t ) = 1, t > 0, u (t ) = 0, t < 0. Solution By (1.1) we write

{u(t )} = ∫ u(t )e
∞ 0

−st

dt =

∫e
0

∞

−st

e−st dt = s

∞

0

1 = . s
∞

(2.1)

The region of convergence is found from the...