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Chapter 2 The Laplace Transformation 2 16 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (2.42) We will now derive the basic properties of the gamma function, and its relation to the well known factorial function The integral of (2.42) can be evaluated by performing integration by parts. Thus, in (2.42) we let Then, and (2.42) is written as (2.43) With the condition that , the first term on the right side of (2.43) vanishes at the lower limit . It also vanishes at the upper limit as . This can be proved with L’ Hôpital’s rule by differentiating both numerator and denominator times, where . Then, Therefore, (2.43) reduces to and with (2.42), we have (2.44) By comparing the integrals in (2.44), we observe that Γ n () x n1 e x x d 0 = n ! nn 1 n2 ⋅⋅ 321 = ue x and dv x = = du e x dx v x n n ---- = = Γ n x n e x n ------------ x0 = 1 n -- x n e x x d 0 + = n0 > = x mm n x n e x n x lim x n ne x ------- x lim x m m d d x n x m m d d x ------------------- x lim x m1 d d nx x d d x ------------------------------------ x lim == = = nm 1 + x x ------------------------------------------------------------------------------------- x lim 1 + x mn e x -------------------------------------------------------------------- x lim 0 = Γ n 1 n x n e x x d 0 = Γ n x e x x d 0 1 n x n e x x d 0

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 2 17 Copyright © Orchard Publications The Laplace Transform of Common Functions of Time (2.45) or (2.46) It is convenient to use (2.45) for , and (2.46) for . From (2.45), we see that becomes infinite as . For , (2.42) yields (2.47) and thus we have obtained the important relation, (2.48) From the recurring relation of (2.46), we obtain (2.49) and in general (2.50) for The formula of (2.50) is a noteworthy relation; it establishes the relationship between the function and the factorial We now return to the problem of finding the Laplace transform pair for , that is, (2.51) To make this integral resemble the integral of the gamma function, we let , or , and thus . Now, we rewrite (2.51) as Therefore, we have obtained the transform pair Γ n () Γ n1 + n -------------------- = n Γ n Γ + = n0 < n = Γ 1 e x x d 0 e x 0 1 == = Γ 1 1 = Γ 2 1 Γ 1 1 Γ 3 2 Γ 2 21 2 ! = Γ 4 3 Γ 3 32 3 ! = Γ + n ! = 2 3
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