Laplace Transforms Essentials
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Laplace Transforms Essentials - Morteza Shafii-Mousavi
CHAPTER 1
THE LAPLACE TRANSFORM
1.1 INTEGRAL TRANSFORMS
A class of transformations, which are called Integral transforms, are defined by
e9780738672458_i0002.jpg(1)
Given a function K(s, t), called the kernel of the transformation, equation (1) associates with each function f(t), of the class of functions for which the above exists, a function F(s) defined by (1).
Various particular choices of the kernel function K(s, t) in (1) have led to a special transformation, each with its own properties to make it useful in specific circumstances. The transform defined choosing the kernel
e9780738672458_i0003.jpg(2)
is called Laplace transform, which is the one to which this book is devoted.
1.2 DEFINITION OF LAPLACE TRANSFORM
Let f(t) be a function for t > 0. The Laplace transform of f(t), denoted by L{f(t), is defined by
e9780738672458_i0004.jpg(3)
1.3 NOTATION
The integral in (3) is a function of the parameter s that is called F(s). It is customary to denote the functions of t by the lower case letters f, g, h, k, y, etc., and their Laplace transforms by the corresponding capital letters. Also some texts use capital letter L and show the Laplace transform of f(t) by L{f(t)}. Therefore we may write
e9780738672458_i0005.jpg(4)
If f(x) is a function of x e9780738672458_img_8807.gif 0, then its Laplace transform is denoted by L{f(x)}.
Example
Find L{sin at}.
By definition
e9780738672458_i0006.jpgBy employing integration by parts
e9780738672458_i0007.jpgTherefore
e9780738672458_i0008.jpgSince for positive s, e−st → 0 at t → ∞, and sin at and cos at are bounded functions, therefore the above yields
e9780738672458_i0009.jpg1.4 LAPLACE TRANSFORMATION OF ELEMENTARY FUNCTIONS
The following table shows Laplace transforms of some elementary functions.
Note: Factorial n. For every integer n > 0,
factorial n = n! = 1 · 2 · ... · n
and by definition 0 ! = 1.
Table 1
e9780738672458_i0010.jpg1.5 SECTIONALLY OR PIECEWISE CONTINUOUS FUNCTIONS
The function f(t) is said to be piecewise continuous or sectionally continuous over an interval a < t < b if that interval can be divided into a finite number of intervals c < t < d such that
f(t) is continuous in the open interval c < t < d,
f(t) approaches a finite limit as t approaches each end point within the interval c < t < d; that is, the limits
e9780738672458_i0011.jpgexist and are finite, where ε > 0.
An example of a sectionally continuous function is shown in the following Figure 1.1.
e9780738672458_i0012.jpgFigure 1.1
1.6 FUNCTIONS OF EXPONENTIAL ORDER
The function f(t) is said to be of exponential order as t → ∞ if there exists two constants M > 0 and b and a fixed t0 such that
e9780738672458_i0013.jpg(5)
We also say f(t) is of exponential order b as