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Chapter 1 - I LAPLACE TRANSFORM Contents 1 Definition of LT...

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I LAPLACE TRANSFORM Contents 1 Definition of LT 2 1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Laplace Transform of Periodic Functions . . . . . . . . . . . . . . . . . . . . 4 1.4 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 DIFFERENTIATION, INTEGRATION AND LAPLACE TRANSFORM 6 2.1 LT of the derivative of a function . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 LT of the integral of a function . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The Derivative of LT of a Function . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 The Integral of LT of a Function . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 SHIFTING THEOREMS 14 3.1 The First Shifting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 The Second Shifting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 CONVOLUTION AND THE DIRAC DELTA DISTRIBUTION 17 4.1 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1
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1 Definition of LT 1.1 Definition The Laplace Transform is widely used in engineering applications, such as solving linear ordinary differential equations. It transforms the equation in ”t-space” to one in ”s-space”. This makes the problem much easier to solve. Let f ( t ) be a function defined on [0 , ). We may assume f ( t ) = 0 when t < 0. The Laplace transform (LT) of f ( t ) is the function F ( s ), defined by: F ( s ) = L { f ( t ) } = 0 e st f ( t ) dt. This is named for Pierre-Simon Laplace, one of the best French mathematicians in the mid-to-late 18th century.The LT transforms functions of t to functions of another variable s . If F ( s ) is the Laplace transform of f ( t ), then f ( t ) is the inverse Laplace transform of F ( s ): f ( t ) = L 1 { F ( s ) } . Property 1 The LT and L 1 are linear : L { af ( t ) + bg ( t ) } = aL { f ( t ) } ( s ) + bL { g ( t ) } L 1 { aF ( s ) + bG ( s ) } = aL 1 { F ( s ) } + bL 1 { G ( s ) } 2
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Examples of Laplace Transforms f ( t ) for t 0 L ( f ) 1 1 s e at 1 s - a t n n ! s n +1 ( n = 0 , 1 , . . . ) sin at a s 2 + a 2 cos at s s 2 + a 2 For example, L { 1 } = 0 e st dt = { - 1 s e st } 0 = 1 s L { e at } = 0 e ( a s ) t dt = { - 1 s - a e ( a s ) t } 0 = 1 s - a 1.2 Existence For which functions f is the LT actually defined on? We want the indefinite integral to converge, of course. A function f ( t ) is piecewise continuous on a finite interval [ a, b ] if [ a, b ] can be subdivided into a finite number of subintervals such that f is continuous in each of the subintervals, and it approaches a finite limit when t approaches an end of any of the subintervals. Function f is piecewise continuous on an infinite interval if it is piecewise continuous on any finite subinterval of its domain. A function f ( t ) is of exponential order α if there exist constants t 0 and M such that | f ( t ) | < Me αt , for all t > t 0 . For example, t n and e t are of exponential order, e t 2 is not of exponential order. Example 2 L { t p } exists if and only if p > - 1 . 3
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To prove this we just need to a variable substitution x = st . Theorem 3 Suppose that f ( t ) is piecewise continuous and of exponential order with | f ( t ) | < Me αt , for all t > t 0 . Then the Laplace transform of f ( t ) exists for all s > α . This condition is sufficient but not necessary. For example, the function f ( t ) = t n , - 1 n < 0 is not piecewise continuous. But the Laplace transform of f ( t ) exists.
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