Definitions of the Laplace Transform (1A) Young Won Lim 1/31/15

Transcription

1 Definitions of the Laplace Transform (A)

2 Copyright (c) 24 Young W. Lim. Permission is granted to copy, distriute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version pulished y the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced y using OpenOffice and Octave.

3 Improper Integral Hiding the limiting process lim + a I = a f ( x) dx lim I + L converge diverge + a lim a a I = a f ( x) dx lim I a L converge diverge lim c c a c I = a lim I c L converge diverge a lim + c a c I = c lim c a + I L converge diverge a + Definitions (A) 3

4 Improper Integral Examples lim + a + a lim c a c a + x 2 dx = lim [ dx = lim 2 x x ] x dx = lim + a a x dx = lim [2 x ] a a + = lim ( + ) = converge Definitions (A) 4 = lim a + (2 2 ) = 2 converge lim f (x) = lim x x x = f ()

5 An Improper Integration F (s) = f (t)e s t dt Complex Numer Real Numer Real Numer s = σ + i ω t Integration Variale R{s} real part I {s} imag part The improper integral converges if the limit defining it exists. Definitions (A) 5

6 F(s) : a function of s For a given function f(t) F (s) = f (t)e s t dt g(s, t) = f (t )e st t G (s, t ) = g(s, t ) G: an antiderivative of g with respect to t g(s, t)dt = lim = lim g(s, t)dt = F(s) [G(s, t)] [G(s, ) G(s, )] During integration, complex variale s is treated as a constant In the result, the literal t vanishes a function of s Definitions (A) 6

7 An Integration Function For a given function f(t) F (s) = f (t)e st dt Complex Numer Real Numer s = σ + i ω t Real Numer Integration Variale s s 2 s 3 F (s ) = F (s 2 ) = F (s 3 ) = f (t )e s t f (t )e s 2 t f (t)e s 3 t dt dt dt F (s ) F (s 2 ) F (s 3 ) Complex Numers Complex Numers Definitions (A) 7

8 F(s) : a Complex Function For a given function f(t) Complex Numer Real Numer s F (s ) = f (t )e s t dt F (s ) s = σ + i ω Complex Numers Complex Function Complex Numers I I s = σ + iω ω R{F (s )} σ R R I{F (s )} F (s ) = R{F (s )} + i I{F (s )} Definitions (A) 8

9 f(t) : a real-valued or complex-valued function t f f (t ) t f (t) f (t ) Real Numers Real Numers t t Real-valued Function f (t ) t f f (t ) t I{s} Real Numers Complex Numers R{f (t )} t t R{s} I{f (t )} Complex-valued Function f (t ) = R{f (t )} + i I{f (t )} Definitions (A) 9

10 Complex Function Plot I F I R{F (s )} F (s ) R R I s = σ + iω F (s ) = R{F (s )} + i I{F (s )} R s I F I I{F (s )} arg {F (s )} R R I s s = σ + iω F (s ) = R{F (s )} + i I{F (s )} R Definitions (A)

11 Two Functions: f(t) & F(s) For a given function f(t) there exits a unique F(s) s-domain function F(s) R{F (s )} F (s ) f (t) F (s) s I{s} R{s} I{F (s )} arg {F (s )} t-domain function f(t) f (t) I{s} t R{s} s Real numer domain function f(t) Complex numer domain function F(s) Definitions (A)

12 Laplace Transform f a (x) L F A (s) f (t)e st dt = F (s) s f (t ) L F B (s) e a t f 2 (t)e st dt = F 2 (s) s+a f c (t) L F C (s) cos(k t) f 3 (t)e st dt = F 3 (s) s s 2 +k 2 f a (x) Real-valued Function F A (s) Complex Function f (t) I I t s = σ + iω f (t ) ω R{F (s )} t t σ R R I{F (s )} f (t ) F (s ) = R{F (s )} + i I{F (s )} Definitions (A) 2

13 Laplace Transform L s F(s) = e s t dt [ = lim s e st] [ = lim s e s + ] s e s s < lim e s = s > F(s) = s e at L s+a F(s) = e a t e s t dt [ = lim e (s+ = lim [ (s+a) a)t] (s+a) e (s+ a) + (s+a) e (s+ a)] (s+a) < lim e (s+a ) = s > a F(s) = (s+a) Definitions (A) 3

14 Laplace Transform s cos(k t) L F(s) = s 2 +k 2 s k cos(k t ) e s t dt sin(k t) e s t dt {[ = lim k sin(k t )e s t] = lim s {[ k cos(k t)e s k t] s k sin(k t) e s t dt } s k cos(k } t) e s t dt {[ lim k sin(k } t)e s t] { = lim k sin (k )e s } k sin(k )e s = lim {[ k cos(k t } { )e s = lim t] k cos(k )e s + k cos(k )e s } = k F(s) = cos(k t ) e s t dt = lim s { k k s k cos(k t) e st dt } F(s) = s k 2 s2 k 2 F(s) s2 (+ k )F (s) = s ( 2 k 2 s2 +k 2 ) k F (s) = s 2 k 2 F(s) = s (s 2 +k 2 ) Definitions (A) 4

15 Integration y parts f (x)g(x) d d x d dx (f g) = f ' (x)g(x) + f ( x) g' (x) d f dx g + f d g dx f (x)g(x) d x f ' (x)g(x) + f ( x) g' (x) f g = f ' g dx + f g ' dx f (x)g ' (x) dx = f (x)g(x) f ' (x)g(x) dx Definitions (A) 5

16 Region of Convergence L s e st dt [ = lim s e st] [ = lim s e s + ] s e s [ = lim s e s + ] s s < (σ +iω) < R{s} < σ < t > Right-sided function lim e s = lim e (σ+i ω) = lim e σ e +i ω = e +i ω = σ > Right-sided ROC (s+a) < lim e (s+a ) = s > a F(s) = (s+a) Definitions (A) 6

17 Laplace Transform e a t L s+a F(s) = e a t e s t dt [ = lim e (s+ = lim [ (s+a) a)t] (s+a) e (s+ a) + (s+a) e (s+ a)] (s+a) < lim e (s+a ) = s > a F(s) = (s+a) Definitions (A) 7

18 Laplace Transform a > > exponential order exponential order Right-sided function e at u(t) Laplace transform exists e +t u(t) Laplace transform exists exponential order exponential order Left-sided function e a t u( t ) Laplace transform exists e +t u( t ) Laplace transform exists Definitions (A) 8

19 Exponential Order Laplace Transform Exponential Order α F (s) = f (t)e st d t a function f has exponential order α for s > R(s) > the integral converges if f(t) does not grow too rapidly the growth rate of a function f(t) there exist constants M > and such that for some t > t f (t) M e α t, t > t α f (t) e σt d t < for some σ f (t)e st d t = f (t)e x t e i y t d t = f (t)e xt d t f (t ) exponential order σ F (s) = < f (t) e σ t f (t)e s t d t d t < for s > σ asolutely converges for R(s) > σ s > σ Definitions (A) 9

20 Convergence of the Laplace Transform Laplace Transform F (s) = = f (t)e st d t {f (t )e xt } e iy t d t f st (t )e d t = f (t) xt e d t < ( e st = e xt e iyt = e xt ) f(t) continuous on [, ) f(t) = for t < f(t) has exponential order α f'(t) piecewise continuous on [, ) {f (t) e xt } = g(t) asolutely integrale for x > α Use Fourier Inversion F(s) converges asolutely for Re(s) > α f s t (t )e d t < Definitions (A) 2

21 Fourier Transform g (t ) = f (t)e xt asolutely integrale for x > α F (x, y) = F (x, y) = Fourier Transform { f (t)e xt } e iyt d t g (t) e iyt d t g (t ) = f (t)e xt + g (t ) = F (x, y)e iyt d y 2 π f (t) = + F ( x, y)e xt e iyt d y 2π Inverse Fourier Transform y = + α s = x + i fixed x s = x + i y d s = i d y f (t ) = + 2π F (x, y)e ( x+ iy)t d y = x+ i F (s)e s t d s 2 πi x i y = (x > α) s = x i = lim y x+ i y 2 πi x i y F (s)e s t d s x i y Definitions (A) 2

22 Forward and Inverse Laplace Transform Forward Laplace Transform f (t) F (s) F(s) = f (t)e s t dt Inverse Laplace Transform f (t) F (s) f (t) = σ+ j 2π j σ j F(s)e +s t ds Definitions (A) 22

23 References [] [2] [3] M.L. Boas, Mathematical Methods in the Physical Sciences [4] E. Kreyszig, Advanced Engineering Mathematics [5] D. G. Zill, W. S. Wright, Advanced Engineering Mathematics [6] T. J. Cavicchi, Digital Signal Processing [7] F. Waleffe, Math 32 Notes, UW 22/2/ [8] J. Nearing, University of Miami [9]