The Laplace Transform (Intro)

Transcription

1 4 The Laplace Tranform (Intro) The Laplace tranform i a mathematical tool baed on integration that ha a number of application It particular, it can implify the olving of many differential equation We will find it particularly ueful when dealing with nonhomogeneou equation in which the forcing function are not continuou Thi make it a valuable tool for engineer and cientit dealing with real-world application By the way, the Laplace tranform i jut one of many integral tranform in general ue Conceptually and computationally, it i probably the implet If you undertand the Laplace tranform, then you will find it much eaier to pick up the other tranform a needed 4 Baic Definition and Example Definition, Notation and Other Baic Let f be a uitable function (more on that later) The Laplace tranform of f, denoted by either F or L[ f ], i the function given by F() L[ f ] f (t)e t dt (4)! Example 4: For our firt example, let u ue f (t) { if t if < t Thi i the relatively imple dicontinuou function graphed in figure 4a To compute the Laplace tranform of thi function, we need to break the integral into two part: F() L[ f ] f (t)e t dt f (t) e }{{} t dt + e t dt + f (t) e }{{} t dt dt e t dt 475

2 476 The Laplace Tranform T (a) (b) S Figure 4: The graph of (a) the dicontinuou function f (t) from example 4 and (b) it Laplace tranform F() So, if, F() And if, e t dt e t F() F() t e t dt [ e e ] dt [ e ] Thi i the function ketched in figure 4b (Uing L Hôpital rule, you can eaily how that F() F() a So, depite our need to compute F() eparately when, F i a continuou function) A the example jut illutrated, we really are tranforming the function f (t) into another function F() Thi proce of tranforming f (t) to F() i alo called the Laplace tranform and, unurpriingly, i denoted by L Thu, when we ay the Laplace tranform, we can be referring to either the tranformed function F() or to the proce of computing F() from f (t) Some other quick note: There are tandard notational convention that implify bookkeeping The function to be tranformed are (almot) alway denoted by lower cae Roman letter f, g, h, etc and t i (almot) alway ued a the variable in the formula for thee function (becaue, in application, thee are typically function of time) The correponding tranformed function are (almot) alway denoted by the correponding upper cae Roman letter F, G, H, ETC and i (almot) alway ued a the variable in the formula for thee function Thu, if we happen to refer to function f (t) and F(), it i a good bet that F L[ f ] Oberve that, in the integral for the Laplace tranform, we are integrating the inputted function f (t) multiplied by the exponential e t over the poitive T axi Becaue of the ign in the exponential, thi exponential i a rapidly decreaing function of t when > and i a rapidly increaing function of t when < Thi will help determine both the ort of function that are uitable for the Laplace tranform, and the domain of the tranformed function

3 Baic Definition and Example It i alo worth noting that, becaue the lower it in the integral for the Laplace tranform i t, the formula for f (t) when t < i completely irrelevant In fact, f (t) need not even be defined for t < For thi reaon, ome author explicitly it the value for t to being nonnegative We won t do thi explicitly, but do keep in mind that the Laplace tranform of a function f (t) i only baed on the value/formula for f (t) with t Thi will become a little more relevant when we dicu inverting the Laplace tranform (in chapter 6) 4 A indicated by our dicuion o far, we are treating the in F() f (t)e t dt a a real variable, that i, we are auming denote a relatively arbitrary real value Be aware, however, that in more advanced development, i often treated a a complex variable, σ + iξ Thi allow the ue of reult from the theory of analytic complex function But we won t need that theory (a theory which few reader of thi text are likely to have yet een) So, in thi text (with one very brief exception in chapter 6), will alway be aumed to be real Tranform of Some Common Function Before we can make much ue of the Laplace tranform, we need to build a repertoire of common function whoe tranform we know It would alo be a good idea to compute a number of tranform imply to get a better grap of thi whole Laplace tranform idea So let get tarted! Example 4 (tranform of favorite contant): Let f be the zero function, that i, f (t) for all t Then it Laplace tranform i F() L[] e t dt Now let h be the unit contant function, that i, dt (4) Then H() L[] h(t) for all t e t dt e t dt What come out of thi integral depend trongly on whether i poitive or not If <, then < and e t dt e t dt e t t t e t e

4 478 The Laplace Tranform If, then Finally, if >, then So, e t dt e t dt e t L[] t e t dt dt t t t e t e + e t dt if < if (43) A illutrated in the lat example, a Laplace tranform F() i often a well-defined (finite) function of only when i greater that ome fixed number ( in the example) Thi i a reult of the fact that the larger i, the fater e t goe to zero a t (provided > ) In practice, we will only give the formula for the tranform over the interval where thee formula are well-defined and finite Thu, in place of equation (43), we will write L[] for > (44) A we compute Laplace tranform, we will note uch retriction on the value of To be honet, however, thee retriction will uually not be that important in practice What will be important i that there i ome finite value uch that our formula are valid whenever > Keeping thi in mind, let go back to computing tranform! Example 43 (tranform of ome power of t ): L [ t n] t n e t dt We want to find t n e t dt for n,, 3, With a little thought, you will realize thi integral will not be finite if So we will aume > in thee computation Thi, of coure, mean that t e t It alo mean that, uing L Hôpital rule, you can eaily verify that t n e t for n t Keeping the above in mind, conider the cae where n, L[t] Thi integral jut crie out to be integrated by part : L[t] }{{} t e} t {{ dt } u dv uv t v du te t dt

5 Baic Definition and Example 479 ( ) t e t [ t te t }{{} t e }{{} ( ) e t dt ] e t dt e t dt Admittedly, thi lat integral i eay to compute, but why bother ince we computed it in the previou example! In fact, it i worth noting that combining the lat computation with the computation for L[] yield L[t] e t dt e t dt L[] [ ] So, L[t] for > (45) Now conider the cae where n Again, we tart with an integration by part: L [ t ] t e t dt t }{{} u e} t {{ dt } dv uv t v du ( t [ t e t t }{{} ) e t t e }{{} ( ) e t t dt ] te t dt te t dt But remember, te t dt L[t] Combining the above computation with thi (and referring back to equation (45)), we end up with L [ t ] te t dt L[t] [ ] (46) 3 Clearly, a pattern i emerging I ll leave the computation of L [ t 3] to you? Exercie 4: Auming >, verify (uing integration by part) that L [ t 3] 3 L[ t ], and from that and the formula for L [ t ] computed above, conclude that L [ t 3] 3 4 3! 4

6 48 The Laplace Tranform? Exercie 4: More generally, ue integration by part to how that, whenever > and n i a poitive integer, L [ t n] n L[ t n ] Uing the reult from the lat two exercie, we have, for >, L [ t 4] 4 L[ t 3] ! 5, L [ t 5] 5 L[ t 4] ! 6, In general, for > and n,, 3,, L [ t n] n! n+ (47) (If you check, you ll ee that it even hold for n ) It turn out that a formula very imilar to (47) alo hold when n i not an integer Of coure, there i then the iue of jut what n! mean if n i not an integer! Since the dicuion of that iue may ditract our attention away from one the main iue at hand that of getting a baic undertanding of what the Laplace tranform i by computing tranform of imple function let u hold off on that dicuion for a few page Intead, let compute the tranform of ome exponential:! Example 44 (tranform of a real exponential): tranform of e 3t, L [ e 3t] e 3t e t dt e 3t t dt Conider computing the Laplace e ( 3)t dt If 3 i not poitive, then e ( 3)t i not a decreaing function of t, and, hence, the above integral will not be finite So we mut require 3 to be poitive (that i, > 3 ) Auming thi, we can continue our computation So L [ e 3t] e ( 3)t dt 3 e ( 3)t t [ 3 t e ( 3)t e ( 3)] [ ] 3 L [ e 3t] 3 Replacing 3 with any other real number i trivial for 3 <? Exercie 43 (tranform of real exponential): Let α be any real number and how that L [ e αt] for α < (48) α

7 Linearity and Some More Baic Tranform 48 Complex exponential are alo eaily done:! Example 45 (tranform of a complex exponential): of e i3t lead to Computing the Laplace tranform L [ e i3t] e i3t e t dt e ( i3)t dt i3 e ( i3)t t i3 [ t e ( i3)t e ( i3)] Now, and t e ( i3)t t e t+i3t e ( i3) e t e t[ co(3t) + i in(3t) ] Since ine and coine ocillate between and a t, the lat it doe not exit unle t e t, and thi occur if and only if > In thi cae, t e ( i3)t t e t[ co(3t) + i in(3t) ] Thu, when >, L [ e i3t] [ i3 t e ( i3)t e ( i3)] i3 [ ] i3 Again, replacing 3 with any real number i trivial? Exercie 44 (tranform of complex exponential): Let α be any real number and how that L [ e iαt] for < (49) iα 4 Linearity and Some More Baic Tranform Suppoe we have already computed the Laplace tranform of two function f (t) and g(t), and, thu, already know the formula for F() L[ f ] and G() L[g]

8 48 The Laplace Tranform Now look at what happen if we compute the tranform of any linear combination of f and g : Letting α and β be any two contant, we have L[α f (t) + βg(t)] α [α f (t) + βg(t)]e t dt [ α f (t)e t f (t)e t dt + β + βg(t)e t] dt g(t)e t dt αl[ f (t)] + βl[g(t)] αf() + βg() Thu, the Laplace tranform i a linear tranform; that i, for any two contant α and β, and any two Laplace tranformable function f and g, L[α f (t) + βg(t)] αl[ f ] + βl[g] Thi fact will implify many of our computation, and i important enough to enhrine a a theorem While we are at it, let note that the above computation can be done with more function than two, and that we, perhap, hould have noted the value of for which the integral are finite Taking all that into account, we can prove: Theorem 4 (linearity of the Laplace tranform) The Laplace tranform tranform i linear That i, L[c f (t) + c f (t) + + c n f n (t)] c L[ f (t)] + c L[ f (t)] + + c n L[ f n (t)] where each c k i a contant and each f k i a Laplace tranformable function Moreover, if, for each f k we have a value k uch that then, letting max be the larget of thee k, F k () L[ f k (t)] for k <, L[c f (t) + c f (t) + + c n f n (t)] c F () + c F () + + c n F n () for max <! Example 46 (tranform of the ine function): Let u conider finding the Laplace tranform of in(ωt) for any real value ω There are everal way to compute thi, but the eaiet tart with uing Euler formula for the ine function along with the linearity of the Laplace tranform: [ ] e L[in(ωt)] L iωt e iωt i i L[ e iωt e iωt] [L [ e iωt] i L [ e iωt] ] From example 45 and exercie 44, we know L [ e iωt] iω for >

9 Table and a Few More Tranform 483 Thu, alo, L [ e iωt] L [ e i( ω)t] i( ω) + iω for > Plugging thee into the computation for L[in(ωt)] (and doing a little algebra) yield, for >, L[in(ωt)] [L [ e iωt] i L [ e iωt] ] [ ] i iω + iω [ ] + iω i ( iω)( + iω) iω ( + iω)( iω) [ ] ( + iω) ( iω) i i i ω [ ] iω i ω, which immediately implifie to L[in(ωt)] ω + ω for > (4)? Exercie 45 (tranform of the coine function): Show that, for any real value ω, L[co(ωt)] + ω for > (4) 43 Table and a Few More Tranform In practice, thoe uing the Laplace tranform in application do not contantly recompute baic tranform Intead, they refer to table of tranform (or ue oftware) to look up commonly ued tranform, jut a o many people ue table of integral (or oftware) when computing integral We, too, can ue table (or oftware) after and you have computed enough tranform on your own to undertand the baic principle, we have computed the tranform appearing in the table o we know our table i correct The table we will ue i table 4, Laplace Tranform of Common Function (Verion ), on page 484 Checking that table, we ee that we have already verified all but two or three of the entrie, with thoe being the tranform of fairly arbitrary power of t, t α, and the hifted tep function, tep(t α) So let compute them now

10 484 The Laplace Tranform Table 4: Laplace Tranform of Common Function (Verion ) In the following, α and ω are real-valued contant, and, unle otherwie noted, > f (t) F() L[ f (t)] Retriction t t n n! n+ n,, 3, t π t α Ŵ(α + ) α+ e αt α < α α < e iαt iα co(ωt) in(ωt) + ω ω + ω tep α (t), tep(t α) e α α Arbitrary Power (and the Gamma Function) Earlier, we aw that L [ t n] t n e t dt n! n+ for > (4) when n i any nonnegative integer Let u now conider computing L [ t α] t α e t dt for > when α i any real number greater than (When α, you can how that t α blow up too quickly near t for the integral to be finite) The method we ued to find L[t n ] become awkward when we try to apply it to find L[t α ] when α i not an integer Intead, we will cleverly implify the above integral for L[t α ] by uing the ubtitution u t Since t i the variable in the integral, thi mean t u and dt du

11 Table and a Few More Tranform 485 So, auming > and α >, L [ t α] t α e t dt ( u ) α e u du u α α+ e u du α+ u α e u du Notice that the lat integral depend only on the contant α we ve factored out any dependence on the variable Thu, we can treat thi integral a a contant (for each value of α ) and write L [ t α] C α where C α+ α u α e u du It jut o happen that the above formula for C α i very imilar to the formula for omething called the Gamma function Thi i a function that crop up in variou application (uch a thi) and, for x >, i given by Ŵ(x) Comparing thi with the formula for C α, we ee that C α u α e u du u x e u du (43) u (α+) e u du Ŵ(α + ) So our formula for the Laplace tranform of t α (with α > ) can be written a L [ t α] Ŵ(α + ) α+ for > (44) Thi i normally conidered the preferred way to expre L[t α ] becaue the Gamma function i conidered to be a well-known function Perhap you don t yet conider it well known, but you can find table for evaluating Ŵ(x), and it i probably one of the function already defined in your favorite computer math package That make graphing Ŵ(x), a done in figure 4, relatively eay A it i, we can readily determine the value of Ŵ(x) when x i a poitive integer by comparing our two formula for L[t n ] when n i a nonnegative integer the one mentioned at the tart of our dicuion (formula (4)), and the more general formula (formula (44)) jut derived for L[t α ] with α n : Thu, n! n+ L [ t n] Letting x n +, thi become Ŵ(n + ) n+ when n,,, 3, 4, Ŵ(n + ) n! when n,,, 3, 4, Ŵ(x) (x )! when x,, 3, 4, (45) In particular: Ŵ() ( )!!, Ŵ() ( )!!,

12 486 The Laplace Tranform X Figure 4: The graph of graph of the Gamma function over the interval (, 4) A x + or x +, Ŵ(x) + very rapidly and Ŵ(3) (3 )!!, Ŵ(4) (4 )! 3! 6, Ŵ() ( )!! 39,96,8 Thi how that the Gamma function can be viewed a a generalization of the factorial Indeed, you will find text where the factorial i redefined for all poitive number (not jut integer) by x! Ŵ(x + ) We won t do that Computing Ŵ(x) when x i not an integer i not o imple It can be hown that ( ) Ŵ π (46) Alo, uing integration by part (jut a you did in exercie 4 on page 48), you can how that Ŵ(x + ) xŵ(x) for x >, (47) which i analogou to the factorial identity (n + )! (n + )n! We will leave the verification of thee to the more adventurou (ee exercie 43 on page 54), and go on to the computation of a few more tranform! Example 47: Conider finding the Laplace tranform of, t and 3 t t For the firt, we ue formula (44) with α /, along with equation (46): ] [ ] Ŵ ( L[ ) ( ) + Ŵ L t / π t / + / For the econd, formula (44) with α / give t ] [ ] L[ L t / Ŵ ( + ) / + ( ) 3 Ŵ 3 /

13 Table and a Few More Tranform 487 Uing formula (47) and (46), we ee that ( ) ( ) 3 Ŵ Ŵ + ( ) Ŵ π Thu For the tranform of t ] [ ] L[ L t / [ ] L 3 t 3 t, we imply have [ ] L t / 3 ( ) 3 Ŵ 3 / ( ) Ŵ 3 + / 3 + π 3 / ( ) 4 Ŵ 3 4 / 3 Unfortunately, there i not a formula analogou to (46) for Ŵ ( ) ( 4 /3 or Ŵ /3 ) There i the approximation ( ) 4 Ŵ , 3 which can be found uing either table or a computer math package, but, ince thi i jut an approximation, we might a well leave our anwer a ( ) 4 [ ] [ ] Ŵ L 3 t L t / / 3 The Shifted Unit Step Function Step function are the implet dicontinuou function we can have function, which we will denote by tep(t), i defined by The (baic) unit tep tep(t) { if t < if < t It graph ha been ketched in figure 43a For any real value α, the correponding hifted unit tep function, which we will denote by tep α, i given by { } { if t α < if t < α tep α (t) tep(t α) if < t α if α < t It graph, with α >, ha been ketched in figure 43b Do oberve that the baic tep function and the the tep function at zero are the ame, tep(t) tep (t) You may have noted that we ve not defined the tep function at their point of dicontinuity ( t for tep(t), and t α for tep α (t) ) That i becaue the value of a tep function right at The unit tep function i alo called the Heaviide tep function, and, in other text, i often denoted by u and, occaionally, by h or H

14 488 The Laplace Tranform T α T (a) (b) Figure 43: The graph of (a) the baic tep function tep(t) and (b) a hifted tep function tep α (t) with α > it ingle dicontinuity will be completely irrelevant in any of our computation or application Oberve thi fact a we compute the Laplace tranform of tep α (t) when α : L [ tep α (t) ] α α tep α (t)e t dt tep α (t)e t dt + e t dt + α α tep α (t)e t dt e t dt α e t dt You can eaily how that the above integral i infinite if < or But if >, then the above become L [ tep α (t) ] α e t dt e t tα t e t e α + eα Thu, L [ tep α (t) ] e α for > and α (48) 44 The Firt Tranlation Identity (And More Tranform) The linearity of the Laplace tranform allow u to contruct tranform from linear combination of known tranform Other identitie allow u to contruct new tranform from other formula involving known tranform One particularly ueful identity i the firt tranlation identity (alo called the tranlation along the S axi identity for reaon that will oon be obviou) The derivation of thi identity tart with the obervation that, in the expreion F() L[ f (t)] f (t)e t dt for >,

15 The Firt Tranlation Identity (And More Tranform) 489 the i imply a place holder It can be replaced with any ymbol, ay, X, that doe not involve the contant of integration t, F(X) L[ f (t)] X f (t)e Xt dt for X > In particular, let X α where α i any real contant Uing thi for X in the above give u F( α) L[ f (t)] α But α > > + α and f (t)e ( α)t dt f (t)e ( α)t dt for α > f (t)e t e αt dt e αt f (t)e t dt L [ e αt f (t) ] So the expreion above for F( α) can be written a F( α) L [ e αt f (t) ] for > + α Thi give u the following identity: Theorem 4 (Firt tranlation identity) If L[ f (t)] F() for >, then, for any real contant α, L [ e αt f (t) ] F( α) for > + α (49) Thi i called a tranlation identity becaue the graph of the right ide of the identity, equation (49), i the graph of F() tranlated to the right by α,3 (The econd tranlation identity, in which f i the function hifted, will be developed later)! Example 48: Let u ue thi tranlation identity to find the tranform of t e 6t Firt we mut identify the f (t) and the α and then apply the identity: L [ t e 6t] L [ ] e 6t t L [ e 6t f (t) ] F( 6) (4) }{{} f (t) Here, f (t) t and α 6 From either formula (47) or table 4, we know So, for any X, F() L[ f (t)] L [ t ]! + 3 F(X) X 3 More preciely, it hifted to the right by α if α >, and i hifted to the left by a if α < 3 Some author prefer to ue the word hifting intead of tranlation

16 49 The Laplace Tranform Uing thi with X 6, the above computation of L [ t e 6t] (equation et (4)) become L [ t e 6t] F( }{{ 6 } ) F(X) X 3 X ( 6) 3 Notice that, in the lat example, we carefully rewrote the formula for F() a a formula of another variable, X, and ued that to get the formula for F( 3), F() 3 F(X) X 3 F( }{{ 6 } ) X ( 6) 3 Thi help to prevent dumb mitake It replace the with a generic placeholder X, which, in turn, i replaced with ome formula of So long a you remember that the in the firt equation i, itelf, imply a placeholder and can be replaced throughout the equation with another formula of, you can go traight from the formula for F() to the formula for F( 6) Unfortunately, thi i often forgotten in the heat of computation, epecially by thoe who are new to thee ort of computation So I trongly recommend including thi intermediate tep of replacing F() with F(X), and uing the formula for F(X) with X 6 (or X whatever formula of i appropriate ) Let try another:! Example 49: Find L [ e 3t in(t) ] Here, L [ e 3t in(t) ] L [ ] e 3t in(t) L [ e }{{} f (t) ] F( 3) f (t) In thi cae, f (t) in(t) Recalling the formula for the tranform of uch a function (or peeking back at formula (4) or table 4), we have So, for any X, F() L[ f (t)] L[in(t)] F(X) X Uing thi with X 3, the above computation of L [ e 3t in(t) ] become L [ e 3t in(t) ] F( }{{ 3 } ) F(X) X In the homework, you ll derive the general formula for X + 4 ( 3) 4 L [ t n e αt], L [ e αt in(ωt) ] and L [ e αt co(ωt) ] Thee formula are found in mot table of common tranform (but not our)

17 What I Laplace Tranformable? 49 t t + f (t) f (t) t t Y jump midpoint of jump Y t T t t t T (a) (b) Figure 44: The graph of (a) a function with a jump dicontinuity at t and (b) a function with everal jump dicontinuitie 45 What I Laplace Tranformable? (and Some Standard Terminology) When we ay a function f i Laplace tranformable, we imply mean that there i a finite value uch that the integral for L[ f (t)], f (t)e t dt, exit and i finite for every value of greater than Not every function i Laplace tranformable For example, t and e t are not Unfortunately, further developing the theory of Laplace tranform auming nothing more than the Laplace tranformability of our function i a bit difficult, and would lead to ome rather ungainly wording in our theorem To implify our dicuion, we will uually init that our function are, intead, piecewie continuou and of exponential order Together, thee two condition will enure that a function i Laplace tranformable, and they will allow u to develop ome very general theory that can be applied uing the function that naturally arie in application Moreover, thee two condition will be relatively eay to viualize So let find out jut what thee term mean Jump Dicontinuitie and Piecewie Continuity The phrae piecewie continuity ugget that we only require continuity on piece of a function That i a little mileading For one thing, we want to it the dicontinuitie between the piece to jump dicontinuitie Jump Dicontinuitie A function f i aid to have a jump dicontinuity at a point t if the left- and right-hand it t t f (t) and t t + f (t) exit, but are different finite number The jump at thi dicontinuity i the difference of the two it, jump f (t) f (t), t t + t t

18 49 The Laplace Tranform Y Y Y T T (a) (b) (c) T Figure 45: Function having at leat one point with an infinite left- or right-hand it at ome point and the average of the two it i the Y-coordinate of the midpoint of the jump, y midpoint [ ] f (t) + f (t) t t + t t A generic example i ketched in figure 44a And right beide that figure (in figure 44b) i the graph of a function with multiple jump dicontinuitie The implet example of a function with a jump dicontinuity i the baic tep function, tep(t) Jut looking at it graph (figure 43a on page 487) you can ee that it ha a jump dicontinuity at t with jump, and y / a the Y -coordinate of the midpoint On the other hand, conider the function f (t) (t ) and g(t) if t < (t ) if < t ketched in figure 45a and 45b, repectively Both have dicontinuitie a t In each cae, however, the it of the function a t from the right i infinite Hence, we do not view thee dicontinuitie a jump dicontinuitie Piecewie Continuity We ay that a function f i piecewie continuou on an finite open interval (a, b) if and only if both of the following hold: f i continuou on the interval except for, at mot, a finite number of jump dicontinuitie in (a, b), The endpoint it exit and are finite f (t) and f (t) t a+ t b We extend thi concept to function on infinite open interval (uch a (, ) ) by defining a function f to be piecewie continuou on an infinite open interval if and only if f i piecewie continuou on every finite open ubinterval In particular then, a function f being piecewie continuou on (, ) mean that t + f (t)

19 What I Laplace Tranformable? 493 i a finite value, and that, for every finite, poitive value T, f (t) ha at mot a finite number of dicontinuitie on the interval (, T), with each of thoe being a jump dicontinuity For ome of our dicuion, we will only need our function f to be piecewie continuou on (, ) Strictly peaking, thi ay nothing about the poible value of f (t) when t If, however, we are dealing with initial-value problem, then we may require our function f to be piecewie continuou on [, ), which imply mean f i piecewie continuou on (, ), defined at t, and f () t + f (t) Keep in mind that a finite number of jump dicontinuitie can be zero, in which cae f ha no dicontinuitie and i, in fact, continuou on that interval What i important i that a piecewie continuou function cannot blow up at any (finite) point in or at the end of the interval At wort, it ha only a few jump dicontinuitie in each finite ubinterval The function ketched in figure 44 are piecewie continuou, at leat over the interval in the figure And any tep function i piecewie continuou on (, ) On the other hand, the function ketched in figure 45a and 45b, are not piecewie continuou on (, ) becaue they both blow up at t Conider even the function f (t) t, ketched in figure 45c Even though thi function i continuou on the interval (, ), we do not conider it to be piecewie continuou on (, ) becaue t + t Two imple obervation will oon be important to u: If f i piecewie continuou on (, ), and T i any poitive finite value, then the integral T f (t) dt i well defined and evaluate to a finite number Remember, geometrically, thi integral i the net area between the graph of f and the T axi over the interval (, T) The piecewie continuity of f aure u that f doe not blow up at any point in (, T), and that we can divide the graph of f over (, T) into a finite number of fairly nicely behaved piece (ee figure 44b) with each piece encloing finite area The product of any two piecewie continuou function f and g on (, ) will, itelf, be piecewie continuou on (, ) You can eaily verify thi yourelf uing the fact that t t ± f (t)g(t) t t ± f (t) t t ± g(t) Combining the above two obervation with the obviou fact that, for any real value of, g(t) e t i a piecewie continuou function of t on (, ) give u:

20 494 The Laplace Tranform Lemma 43 Let f be a piecewie continuou function on (, ), and let T be any finite poitive number Then the integral T f (t)e t dt i a well-defined finite number for each real value Becaue of our interet in the Laplace tranform, we will want to enure that the above integral converge to a finite number a T That i the next iue we will addre Exponential Order Let f be a function on (, ), and let be ome real number We ay that f i of exponential order if and only if there are finite contant M and T uch that f (t) Me t whenever T t (4) Often, the precie value of i not particularly important In thee cae we may jut ay that f i of exponential order to indicate that it i of exponential order for ome value Saying that f i of exponential order i jut aying that the graph of f (t) i bounded above by the graph of ome contant multiple of ome exponential function on ome interval of the form [T, ) Note that, if thi i the cae and i any real number, then f (t)e t f (t) e t Me t e t Me ( )t whenever T t Moreover, if >, then i poitive, and f (t)e t Me ( )t a t (4) Thu, in the future, we will automatically know that f t (t)e t whenever f i of exponential order and > Tranform of Piecewie Continuou Function of Exponential Order Now, uppoe f i a piecewie continuou function of exponential order on the interval (, ) A already oberved, the piecewie continuity of f aure u that T f (t)e t dt i a well-defined finite number for each T > And if >, then inequality (4), above, tell u that f (t)e t i hrinking to a t at leat a fat a a contant multiple of ome decreaing exponential It i eay to verify that thi i fat enough to enure that T T f (t)e t dt More preciely: Exponential Order a t One can have exponential order a t and even a t 3 ) However, we are not intereted in thoe cae, and it i illy to keep repeating a t

21 What I Laplace Tranformable? 495 converge to ome finite value And that give u the following theorem on condition enuring the exitence of Laplace tranform Theorem 44 If f i both piecewie continuou on (, ) and of exponential order, then i a well-defined function for > F() L[ f (t)] f (t)e t dt In the next everal chapter, we will often aume that our function of t are both piecewie continuou on (, ) and of exponential order Mind you, not all Laplace tranformable function atify thee condition For example, we ve already een that t α with < α i Laplace tranformable But t + tα if α < So thoe function given by t α with < α < (uch a / t ) are not not piecewie continuou on (, ), even though they are certainly Laplace tranformable Still, all the other function on the left ide of table 4 on page 484 are piecewie continuou on (, ) and are of exponential order More importantly, the function that naturally arie in application in which the Laplace tranform may be ueful are uually piecewie continuou on (, ) and of exponential order By the way, ince you ve probably jut glanced at table 4 on page 484, go back at look at the function on the right ide of the table Oberve that and thee function have no dicontinuitie in the interval on which they are defined, they all hrink to a It turn out that you can extend the work ued to obtain the above theorem to how that the above obervation hold much more generally More preciely, the above theorem can be extended to: Theorem 45 If f i both piecewie continuou on (, ) and of exponential order, then F() L[ f (t)] i a continuou function on (, ) and f (t)e t dt F() We will verify thi theorem at the end of the next ection

22 496 The Laplace Tranform 46 Further Note on Piecewie Continuou and Exponentially Bounded Function Iue Regarding Piecewie Continuou Function on (, ) In the next everal chapter, we will be concerned mainly with function that are piecewie continuou on (, ) There are a few mall technical iue regarding thee function that could become ignificant later if we don t deal with them now Thee iue concern the value of uch function at jump On the Value of a Function at a Jump Take a look at figure 44b on page 49 Call the function ketched there f, and conider evaluating, ay, t f (t)e t dt The obviou approach i to break up the integral into three piece, t f (t)e t dt t f (t)e t dt + t t f (t)e t dt + t t f (t)e t dt, and ue value/formula for f over the interval (, t ), (t, t ) and (t, t ) to compute the individual integral in the above um What you would not worry about would be the actual value of f at the point of dicontinuity, t, t and t In particular, it would not matter if or f (t ) t t f (t) or f (t ) t t + f (t) f (t ) the Y-coordinate of the midpoint of the jump Thi extend an obervation made when we computed the Laplace tranform of the hifted tep function There, we found that the precie value of tep α (t) at t α wa irrelevant to the computation of L [ tep α (t) ] And the peudo-computation in the previou paragraph point out that, in general, the value of any piecewie continuou function at a point of dicontinuity will be irrelevant to the integral computation we will be doing with thee function Parallel to thee obervation are the obervation of how we ue function with jump dicontinuitie in application Typically, a function with a jump dicontinuity at t t i modeling omething that change o quickly around t t that we might a well pretend the change i intantaneou Conider, for example, the output of a one-lumen incandecent light bulb witched on at t : Until i i witched on, the bulb light output i lumen For a brief period around t the filament i warming up and the light output increae from to lumen, and remain at lumen thereafter In practice, however, the warm-up time i o brief that we don t notice it, and are content to decribe the light output by light output at time t { } lumen if t < tep (t) lumen lumen if < t

23 Note on Piecewie Continuou and Exponentially Bounded Function 497 without giving any real thought a to the value of the light output the very intant we are turning on the bulb 4 What all thi i getting to i that, for our work involving piecewie continuou function on (, ), the value of a function f at any point of dicontinuity t in (, ) i irrelevant What i important i not f (t ) but the one-ided it t t f (t) and t t + f (t) Becaue of thi, we will not normally pecify the value of a function at a dicontinuity, at leat not while developing Laplace tranform If thi diturb you, go ahead and aume that, unle otherwie indicated, the value of a function at each jump dicontinuity i given by the Y -coordinate of the jump midpoint It a good a any other value Equality of Piecewie Continuou Function Becaue of the irrelevance of the value of a function at a dicontinuity, we need to lightly modify what it mean to ay f g on ome interval Henceforth, let u ay that mean f g on ome interval (a piecewie continuou function) f (t) g(t) for every t in the interval at which f and g are continuou We will not init that f and g be equal at the relatively few point of dicontinuity in the function But do note that we will till have t t ± f (t) t t ± g(t) for every t in the interval Conequently, the graph of f and g will have the ame jump in the interval By the way, the phrae a piecewie continuou function in the above definition i recommended, but i often forgotten! Example 4: The function tep (t), f (t) { if t if < t and g(t) { if t < if t all atify tep (t) f (t) g(t) for all value of t in (, ) except t, at which each ha a jump So, a piecewie continuou function, tep f g on (, ) 4 On the other hand, What i the light output of a one-lumen light bulb the very intant the light i turned on? i a nice quetion to meditate upon if you are tudying Zen

24 498 The Laplace Tranform Converely, if we know h tep on (, ) (a piecewie continuou function), then we know { if < t < h(t) if < t We do not know (nor do we care about) the value of h(t) when t (or when t < ) Teting for Exponential Order Before deriving thi tet for exponential order, it hould be noted that the order i not unique After all, if f (t) Me t whenever T t, and, then f (t) Me t proving the following little lemma: Me t whenever T t, Lemma 46 If f i of exponential order, then f i of exponential order for every Now here i the tet: Lemma 47 (tet for exponential order) Let f be a function on (, ) If there i a real value uch that f t (t)e t, If then f i of exponential order t f (t)e t doe not converge to for any real value, then f i not of exponential order PROOF: Firt, aume f t (t)e t for ome real value, and let M be any finite poitive number you wih (it could be, /, 87, whatever) By the definition of it, the above aure u that, if t i large enough, then f (t)e t i within M of Letting T be any ingle large enough value of t, we then mut have t T f (t)e t M By elementary algebra, we can rewrite thi a f (t) Me t whenever T t,

25 Note on Piecewie Continuou and Exponentially Bounded Function 499 which i exactly what we mean when we ay f i of exponential order That confirm the firt part of the lemma To verify the econd part of the lemma, aume t f (t)e t doe not converge to for any real value If f were of exponential order, then it i of exponential order for ome finite real number, and, a noted in the dicuion of expreion (4) on page 494, we would then have that f t (t)e t for > But we ve aumed thi i not poible; thu, it i not poible for f to be of exponential order Proving Theorem 45 The Theorem and a Bad Proof The baic premie of theorem 45 i that we have a piecewie continuou function f on (, ) which i alo of exponential order From the previou theorem, we know F() L[ f (t)] f (t)e t dt i a well-defined function on (, ) Theorem 45 further claim that and and F() L[ f (t)] i continuou on (, ) That i, F() F() F( ) for each > In a naive attempt to verify thee claim, you might try F() F() f (t)e t dt f (t)e t dt Unfortunately, thee computation aume α f (t)e t dt f (t)e t dt g(t, ) dt f (t)e t dt F( ), α f (t) dt g(t, ) dt which i NOT alway true Admittedly, it often i true But there are exception And becaue there are exception, we cannot rely on thi ort of witching of it with integral to prove our claim

26 5 The Laplace Tranform Preinarie There are two mall obervation that will prove helpful here and elewhere The firt concern any function f which i piecewie continuou on (, ) and atifie f (t) M T e t whenever T t, for two poitive value M T and T For convenience, let g(t) f (t)e t for t > Thi i another piecewie continuou function on(, ), but it atifie g(t) f (t)e t f (t) e t Me t e t M for T < t, On the other hand, the piecewie continuity of g on (, ) mean that g doe not blow up anywhere in or at the endpoint of (, T) So it i eay to ee (and to prove) that there i a contant B uch that g(t) B for < t < T Letting M be the larger of B and M T, we now have that g(t) M if < t < T or T t So, f (t) e t f (t)e t g(t) M for < t Multiply through by the exponential, and you ve got: Lemma 48 Aume f i a piecewie continuou function on (, ) which i alo of exponential order Then there i a contant M uch that f (t) M e t for < t The above lemma will let u ue the exponential bound M e t over all of (, ), and not jut (M T, ) The next lemma i one you hould either already be acquainted with, or can eaily confirm on your own Lemma 49 If g i an integrable function on the interval (a, b), then b a b g(t) dt g(t) dt a

27 Note on Piecewie Continuou and Exponentially Bounded Function 5 The Proof of Theorem 45 Now we will prove the two claim of theorem 45 Keep in mind that f i a piecewie continuou function on (, ) of exponential order, and that F() f (t)e t dt for > We will make repeated ue of the fact, tated in lemma 48 jut above, that there i a contant M uch that f (t) M e t for < t (43) Since the econd claim i a little eaier to verify, we will tart with that Proof of the Second Claim The econd claim i that which, of coure, can be proven by howing F(), F() Now let > Uing inequality (43) with the integral inequality form lemma 49, we have F() f (t)e t dt f (t)e t dt f (t) e t dt M e t e t dt M L [ e t ] M Thu, confirming the claim M F(), Proof of the Firt Claim The firt claim i that F i continuou on (, ) To prove thi, we need to how that, for each >, Note that thi it can be verified by howing F() F( ) F() F( )

28 5 The Laplace Tranform Now let and be two different point in (, ) Again uing inequality (43) with the integral inequality from lemma 49, F() F( ) f (t)e t dt f (t)e t dt f (t) [ e t e ] t dt Oberve that e t e t f (t) e t e t dt M e t e t e t dt { [ + e t e ] } t if < [ e t e ] ± [ e t e ] t t if < with the ign choen appropriately Uing thi with the preceding equence of inequalitie, we get Thu, F() F( ) F() F( ) ±M M e t e t e t dt ±M M e [ t e t e ] t dt [ ±M e t e t dt ±M [ L [ e t ] L [ e t ] ] ±M [ [ ] which i all we needed to how to confirm the firt claim ] ] e t e t dt [ ±M ], Additional Exercie 46 Sketch the graph of each of the following choice of f (t), and then find that function Laplace tranform by direct application of the definition, formula (4) on page 475 (ie, compute the integral) Alo, if there i a retriction on the value of for which the formula of the tranform i valid, tate that retriction a f (t) 4 b f (t) 3e t c f (t) { if t 3 if 3 < t d f (t) { if t 3 if 3 < t

29 Additional Exercie 53 e f (t) { e t if t 4 if 4 < t f f (t) { e t if < t 4 otherwie { t if < t { if < t g f (t) otherwie h f (t) t otherwie 47 Find the Laplace tranform of each, uing either formula (47) on page 48, formula (49) on page 48 or formula (44) on page 48, a appropriate: a t 4 b t 9 c e 7t d e 7t e e i7t 48 Find the Laplace tranform of each of the following, uing table 4 on page 484 (Tranform of Common Function) and the linearity of the Laplace tranform: a in(3t) b co(3t) c 7 d coh(3t) e inh(4t) f 3t 8t + 47 g 6e t + 8e 3t h 3 co(t) + 4 in(6t) i 3 co(t) 4 in(t) 49 Compute the following Laplace tranform: a t 3 / b t 5 / c t / 3 d 4 t e tep (t) 4 For the following, let { if t < f (t) if t a Verify that f (t) tep (t), and uing thi and linearity, b compute L[ f (t)] 4 Find the Laplace tranform of each of the following, uing table 4 on page 484 (Tranform of Common Function) and the firt tranlation identity: a te 4t b t 4 e t c e t in(3t) d e t co(3t) e e 3t t f e 3t tep (t) 4 Verify each of the following uing table 4 on page 484 (Tranform of Common Function) and the firt tranlation identity (aume α and ω are real-valued contant and n i a poitive integer): a L [ t n e αt] b L [ e αt in(ωt) ] c L [ e αt co(ωt) ] n! ( α) n+ for > α ω ( α) + ω for > α α ( α) + ω for > α d L [ e αt tep ω (t) ] α e ω( α) for > α and ω

30 54 The Laplace Tranform 43 The following problem all concern the Gamma function, Ŵ(σ) e u u σ du a Uing integration by part, how that Ŵ(σ + ) σŵ(σ) whenever σ > b i By uing an appropriate change of variable and ymmetry, verify that ( ) Ŵ e τ dτ ii Starting with the obervation that, by the above, ( ) ( ) ( )( ) Ŵ Ŵ e x dx e y dy e x y dx dy, how that Ŵ ( ) π (Hint: Ue polar coordinate to integrate the double integral) 44 Several function are given below Sketch the graph of each over an appropriate interval, and decided whether each i or i not piecewie continuou on (, ) a f (t) tep 3 (t) b g(t) tep (t) tep 3 (t) in(t c in(t) d t e tan(t) f t g i t h t t j t + k The ever increaing tair function, if t < tair(t) if < t < if < t < 3 3 if 3 < t < 4 4 if 4 < t < 5 45 Aume f and g are two piecewie continuou function on an interval (a, b) containing the point t Aume further that f ha a jump dicontinuity at t while g i continuou at t Verify that the jump in the product f g at t i given by the jump in f at t g(t )

31 Additional Exercie Uing the tet for exponential order (lemma 47 on page 498), determine which of the following are of exponential order, and, for each which i of exponential order, determine the poible value for the order a e 3t b t c te 3t d e t e in(t) 47 For the following, let α and σ be any two poitive number a Uing baic calculu, how that t α e σt ha a maximum value M α,σ on the interval [, ) Alo, find both where thi maximum occur and the value of M α,σ b Explain why thi confirm that i t α M α,σ e σt whenever t >, and that ii t α i of exponential order σ for any σ > 48 Aume f i a piecewie continuou function on (, ) of exponential order, and let α and σ be any two poitive number Uing the reult of the lat exercie, how that t α f (t) i piecewie continuou on (, ) and of exponential order + σ

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