Definition of the aplace Tranform (A) 2/9/5
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Improper Integral Hiding the limiting proce lim + a I = a f ( x) dx lim I + converge diverge + a lim a a I = a f ( x) dx lim I a converge diverge lim c c a c I = a lim I c converge diverge a lim + c a c I = c lim c a + I converge diverge a + Definition (A) 3 2/9/5
Improper Integral Example http://en.wikipedia.org/wiki/ http://en.wikipedia.org/wiki/ 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 Definition (A) 4 = lim a + (2 2 ) = 2 converge lim f (x) = lim x x x = f () 2/9/5
An Improper Integration F () = f (t)e t dt Complex Numer Real Numer Real Numer = σ + i ω t Integration Variale R{} real part I {} imag part The improper integral converge if the limit defining it exit. Definition (A) 5 2/9/5
F() : a function of For a given function f(t) F () = f (t)e t dt g(, t) = f (t )e t t G (, t ) = g(, t ) G: an antiderivative of g with repect to t g(, t)dt = lim = lim g(, t)dt = F() [G(, t)] [G(, ) G(, )] During integration, complex variale i treated a a contant In the reult, the literal t vanihe a function of Definition (A) 6 2/9/5
An Integration Function For a given function f(t) F () = f (t)e t dt Complex Numer Real Numer = σ + i ω t Real Numer Integration Variale 2 3 F ( ) = F ( 2 ) = F ( 3 ) = f (t )e t f (t )e 2 t f (t)e 3 t dt dt dt F ( ) F ( 2 ) F ( 3 ) Complex Numer Complex Numer Definition (A) 7 2/9/5
F() : a Complex Function For a given function f(t) Complex Numer Real Numer F ( ) = f (t )e t dt F ( ) = σ + i ω Complex Numer Complex Function Complex Numer I I = σ + iω ω R{F ( )} σ R R I{F ( )} F ( ) = R{F ( )} + i I{F ( )} Definition (A) 8 2/9/5
f(t) : a real-valued or complex-valued function t f f (t ) t f (t) f (t ) Real Numer Real Numer t t Real-valued Function f (t ) t f f (t ) t I{} Real Numer Complex Numer R{f (t )} t t R{} I{f (t )} Complex-valued Function f (t ) = R{f (t )} + i I{f (t )} Definition (A) 9 2/9/5
Complex Function Plot I F I R{F ( )} F ( ) R R I = σ + iω F ( ) = R{F ( )} + i I{F ( )} R I F I I{F ( )} arg {F ( )} R R I = σ + iω F ( ) = R{F ( )} + i I{F ( )} R Definition (A) 2/9/5
Two Function: f(t) & F() For a given function f(t) there exit a unique F() -domain function F() R{F ( )} F ( ) f (t) F () I{} R{} I{F ( )} arg {F ( )} t-domain function f(t) f (t) I{} t R{} Real numer domain function f(t) Complex numer domain function F() Definition (A) 2/9/5
aplace Tranform f a (x) F A () f (t)e t dt = F () f (t ) F B () e a t f 2 (t)e t dt = F 2 () +a f c (t) F C () co(k t) f 3 (t)e t dt = F 3 () 2 +k 2 f a (x) Real-valued Function F A () Complex Function f (t) I I t = σ + iω f (t ) ω R{F ( )} t t σ R R I{F ( )} f (t ) F ( ) = R{F ( )} + i I{F ( )} Definition (A) 2 2/9/5
aplace tranform of and exp( at) F() = e t dt [ = lim e t] [ = lim e + ] e < lim e = > F() = e at +a F() = e a t e t dt [ = lim e (+ = lim [ (+a) a)t] (+a) e (+ a) + (+a) e (+ a)] (+a) < lim e (+a ) = > a F() = (+a) Definition (A) 3 2/9/5
aplace tranform of exp(+at) and exp( at) e a t +a F() = e a t e t dt [ = lim e (+ = lim [ (+a) a)t] (+a) e (+ a) + (+a) e (+ a)] (+a) < lim e (+a ) = > a F() = (+a) e +at a F () = e +a t e t dt [ = lim ( a) e ( a)t] = lim [ ( a) e ( a ) + ( a) e ( a)] ( a) < lim e ( a) = > +a F () = ( a) Definition (A) 4 2/9/5
aplace tranform of coh(kt) and inh(kt) coh(k t ) coh(k t ) = (e+k t + e k t ) 2 2 k 2 F () = (e +k t +e k t ) e t dt 2 > +k ( k) F () = ( 2 ( k) + )) (+k = = 2 ( 2 k 2 ) e +k t e t dt + 2 > k e k t e t dt (+k) inh(k t) inh(k t) = (e+ k t e k t ) 2 k 2 k 2 F () = (e +k t e k t ) e t dt 2 > +k ( k) = 2 F () = ( 2 ( k) )) (+k = k ( 2 k 2 ) e +k t e t dt 2 > k e k t e t dt (+k) Definition (A) 5 2/9/5
aplace tranform of co(kt) and in(kt) co(k t ) co(k t ) = (e+ jk t + e j k t ) 2 2 +k 2 F () = (e + j k t +e j k t ) e t dt 2 > = 2 ( j ω) F () = ( 2 ( jω) + (+ j ω)) = e + j k t e t dt + 2 ( 2 +k 2 ) > e jk t e t dt (+ jω) in(k t ) in(k t ) = (e+ jk t e j k t ) 2 j k 2 +k 2 F () = (e + j k t e j k t ) 2 j > e t dt ( j ω) F () = ( 2 ( jω) (+ j ω)) = = 2 j k ( 2 +k 2 ) e +j k t e t dt 2 j > (+ jω) e j k t e t dt Definition (A) 6 2/9/5
aplace tranform of co(kt) co(k t) F() = 2 +k 2 k co(k t ) e t dt in(k t) e t dt {[ = lim k in(k t )e t] = lim {[ k co(k t)e k t] k in(k t) e t dt } k co(k } t) e t dt {[ lim k in(k } t)e t] { = lim k in (k )e } k in(k )e = lim {[ k co(k t } { )e = lim t] k co(k )e + k co(k )e } = k F() = co(k t ) e t dt = lim { k k k co(k t) e t dt } F() = k 2 2 k 2 F() 2 (+ k )F () = ( 2 k 2 2 +k 2 ) k F () = 2 k 2 F() = ( 2 +k 2 ) Definition (A) 7 2/9/5
aplace tranform of in(kt) in(k t ) k F () = 2 +k 2 k in(k t) e t dt co(k t) e t dt {[ = lim k co(k t)e t] = lim {[ k + in (k t)e k t] + k co(k t) e t dt } k in(k } t) e t dt {[ t] lim } co(k t )e k { = lim k co(k )e + } k co(k )e = k {[ lim + k in(k t)e t ] } = lim { + k in(k )e } k in(k )e = F () = in(k t) e t dt { = lim k 2 k 2 in(k t) e t dt } F () = k 2 k 2 F () 2 (+ k 2) F () = ( 2 +k 2 ) k k F () = 2 k F () = k ( 2 +k 2 ) Definition (A) 8 2/9/5
Integration y part 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 Definition (A) 9 2/9/5
Region of Convergence e t dt [ = lim e t] [ = lim e + ] e [ = lim e + ] < R{} < f (t) (σ +iω) < σ < ω domain t lim e = lim e (σ+i ω) = lim e σ e +i ω = e +i ω = domain σ right-ided function t > right-ided ROC σ > (+a) < lim e (+a ) = > a F() = (+a) Definition (A) 2 2/9/5
Converging Improper Integral t 2 e +k t [ dt = t 2 t k ek = t]t k (ek t 2 k t e ) e +k t dt = [ k ek t] = k (e k e k ) k = σ + j ω k + k (e j ω ) R{k} < e k R{k} = e k e j ω Definition (A) 2 2/9/5
Exponential Order aplace Tranform Exponential Order α F () = f (t)e t d t a function f ha exponential order α for > R() > the integral converge if f(t) doe not grow too rapidly the growth rate of a function f(t) there exit contant M > and α uch that for ome f (t) M e α t, t > t t > t right-ided f (t) e σt d t < for ome σ f (t)e t 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 () = < f (t) e σ t f (t)e t d t d t < for > σ aolutely converge for R() > σ > σ Definition (A) 22 2/9/5
Convergence of the aplace Tranform aplace Tranform F () = = f (t)e t d t {f (t )e xt } e iy t d t f t (t )e d t = f (t) xt e d t < ( e t = e xt e iyt = e xt ) f(t) continuou on [, ) f(t) = for t < f(t) ha exponential order α f'(t) piecewie continuou on [, ) F() converge aolutely for Re() > α f t (t )e d t < right-ided function t > right-ided ROC σ > Definition (A) 23 2/9/5
Exponential Order and ROC Right-ided function Right-ided function a > α > β < exponential order e αt α > ω +a exponential order α > e αt ω a a α e at u(t) e +at u(t ) σ +a α σ Right-ided function exponential order β < ω +a e a t u(t) a β σ Definition (A) 24 2/9/5
Forward and Invere aplace Tranform Forward aplace Tranform f (t) F () F() = f (t)e t dt Invere aplace Tranform f (t) F () f (t) = σ+ j 2π j σ j F()e + t d Definition (A) 25 2/9/5
Reference [] http://en.wikipedia.org/ [2] http://planetmath.org/ [3] M.. Boa, Mathematical Method in the Phyical Science [4] E. Kreyzig, Advanced Engineering Mathematic [5] D. G. Zill, W. S. Wright, Advanced Engineering Mathematic [6] T. J. Cavicchi, Digital Signal Proceing [7] F. Waleffe, Math 32 Note, UW 22/2/ [8] J. Nearing, Univerity of Miami [9] http://cipp.ucc.edu/~haer/ph6a/complexfunbranchtheory.pdf 2/9/5