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The initial- and final-value theorems in Laplace transform theory
Journal of
The Pranklin lnslilule V o l . 274
THE
SEPTEMBER,
1962
No. 3
INITIAL- AND FINAL-VALUE THEOREMS IN L A P L A C E T R A N S F O R M T H E O R Y * BY
BERNARD RASOF 1 ABSTRACT
The initial- and final-value theorems, generally neglected in Laplace transform theory, for some purposes are among the most powerful results in that subject. Here are advanced some useful applications of these theorems: ~,,e show how they may be employed as necefsar~ checks on the accuracy of response transtbrms, to evaluate (directly, in a uniform manner, and with remarkable ease) certain definite integrals, to derive an important general result, and to evaluate (again uniformly and easily) certain limits and indeterminate forms (this is believed to be a new application). T h e n the theorems are generalized, and the first of these is applied to generate, directly from a Laplace transform itself, the Maclaurin expansion of the inverse Laplace transform. INTRODUCTION
When a function f(t) is piecewise continuous and of exponential order it possesses (1)z a Laplace transform if(s) related to f(t) by way of the definition
F(s) = 2 U(t)} =
e-./(t)dt,
(1)
where Re(s) is large enough to insure convergence of the integral. The Laplace transform has an extensive literature on theory and application. Its numerous utilizations are summarized in (2), where it is shown to be useful in treating an impressively large and varied class of important pure and applied problems. * This work was sponsored by the Educational Development Program, Department of Engineering, University of California, with financial support from the Ford Foundation. 1 Department of Engineering, University of California, Los Angeles, Calif. 2 T h e boldface numbers in parentheses refer to fhe references appended to this paper. (Note
T h e Franklin Institute is not responsible for the statements and opinions advanced by contributors in the
,IouuNAt.)
I(J 5
I66
BERNARD
RASOF
[ ].I:.1.
THE INITIAL- AND FINAL-VALUE THEOREMS
Most of the theorems establishing basicproperties of the Laplace transformation present the Laplace transform F(s) of specific functions f(t) or of operations on f(t). The exceptions are two unique theorems which assert the equality of l i m f ( t ) and lim sF(s) for two pairs of particular values of t and s. These are the so-called initial- and final-value theorems, stated below without proof. 3
Initial-value_ theorem. If both f(t) and f'(t) possess Laplace transforms and lim sF(s) exists, then lim f(t) = lim sF(s).
(2)
Final-value theorem. If both f ( t ) and f'(t) possess Laplace transforms and sF(s) has no poles on or to the right of the imaginary axis in the s-plane, then lim f(t) = lira sF(s). (3) The last hypothesis requires existence of l i m f ( t ) .
For, if any pole sp of
sF(s) has R e ( s p ) > 0, then f ( t ) ---, ~ as t ---~ ~; if sF(s) has poles on Re(s) = 0, t h e n f ( t ) has a stable oscillation as t ----' ~ ; in neither instance does l i m f ( t ) exist. T h e purpose of this paper is to advance some applications of (2) and (3). To this end, we show how the theorems m a y be employed as part of the procedure of solving ordinary and partial differential equations, and parenthetically suggest that this is an essential aspect of the solution. Then we apply (3) alone to evaluate (with a single stroke) some formidable definite integrals. After this we present the application of these theorems to the evaluation of certain limits and indeterminate forms, in what is believed to be a new application of these theorems. Lastly, (2) and (3) are generalized, and the first of these is used to obtain a new method of generating directly from a Laplace transform F(s) itself the Maclaurin expansion of the inverse Laplace transform f(t) = £-1 {-F(,f)}. USE OF THESE THEOREMS IN PARTIALLY SUBSTANTIATING RESPONSE TRANSFORMS
W h e n an initial-value problem inf(t) is solved by use of Laplace transform methods, an intermediate result is a response transform F(s) = ~ { f ( t ) l which must be inverted to yield the s o l u t i o n f ( t ) of the original problem. The inversion of F(s) to givef(t) is the most difficult part of the solution, and if an error is made in obtaining F(s), the effort and time expended in obtaining its inverse will be wasted. For this reason, before proceeding with the inversion of F(s) it is useful to have a way of at least partially establishing its accuracy. W h e n they can be employed, (2) and (3) supply just such a control which can prevent a serious error. The first 3 These theorems are derived in a number of textbooks; see especially (1) and (3).
Sept., ,96':,.]
LAPLACE TRANSFORM THEORY
i6 7
theorem provides a rapid w a y of determining directly from F(s) itself the initial condition on ,E-I{F(s)}, for comparison with the given initial condition o n f ( t ) . The conditions on (3) are more stringent than those on (2), so the former cannot be employed as frequently as the latter; however, when (3) can be utilized it offers a means of easily computing, also directly from F(s), the limiting value ( t h a t is, as t --' ~) of the function which would be obtained f r o m ~ - l { F ( s ) ] , and comparing this with a possibly explicitly stated asymptotic value o f f ( t ) , or one inferred from physical considerations. If application of (2) and (3) to F(s) recovers f ( 0 ) and gives a result in accordance with l i m f ( t ) , then F(s) may be correct and its inversion to yield f(t) can and should proceed. O n the other hand, should the application of either or both of these theorems fail to confirm known results, it would be concluded that F(s) is in error and that its calculation should be reviewed and corrected before attempting the inversion. This requirement that these theorems applied to F(s) produce given, known or inferred results is in the nature of a necessary but not sufficient condition on the correctness ofF(s). Testing the accuracy of a response transform by use of (2) and (3) usually is straightforward, direct and rapid; the habitual use of these theorems for this purpose is invaluable in detecting errors before the often laborious process of inversion is attempted.
USE OF THESE THEOREMS AS NECESSARY CHECKS ON THE ACCURACY OF RESPONSE TRANSFORMS
Example / A circuit has a constant voltage source in series with lumped, constant resistance and inductance R and L. A switch in the circuit is closed at ! = 0; what is the current i(t) for t > 0? This initial-value problem requires solving
L[di(t)/dt] + Ri(t) = U(t),
i(0) = 0;
(4)
U(t) is a unit step voltage originating at t = 0. Inductance is proportional to di(t)/dt, and this decreases to zero; hence, as t --' ~ the effect of the inductance will become vanishingly small and it is physically evident that
i(t) --~ 1/R as t --' ~. Taking the Laplace transform of (4) and solving for I(s) = ~ {i(t)l gives
I(s) = llsL (s + R / L ) .
(5)
Before obtaining i(t) by inverting its Laplace transform, I(s) is tested to determine if i(0) = 0 and i(t)--~ 1/R as t --~ :¢ agree with lim sT(s) and lira if(s), respectively, representing application of (2) and (3) to this s
"0
problem. The current ,(t) possesses a Laplace transform (4); £ li'(t)] follows
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BERNARD RASOF
[J.V.I.
from (4) and (5); from (5), lim sF(s) exists: thus, the hypotheses of (2) are met and lim s[(s) = lim s[1/sL (s + R/L)] = 0 = lim i(t), recovering i(0) in (4). Since both R and L are positive, s-[(s) has no poles in the right half of the complex s-plane: that is, the hypotheses of (3) are met and thus lim si(s) = lim s[1/sL (s + R/L)] = 1/R = lim i(t), s~O
s~O
in accord with physical intuition.
l
*~
Application of (2) and (3) directly to
I--(s) leads to the anticipated results 0 and 1/R, respectively, so ](s) given by (5) is not obviously incorrect.
Example 2 A semi-infinite homogeneous block with thermal diffusivity k 2 > 0 and coefficient of surface heat transfer H > 0 is at uniform temperature v0 > 0 w h e n t > 0. A t t = 0 the s u r f a c e x = 0 is placed in contact w i t h a fluid at temperature zero and possessing infinite heat capacity. Heat flows from the surface x = 0 in accordance with Newton's law of cooling. W h a t is the temperature v(x,t) for all x > 0, t > 0? This initial-value problem requires the solution of
ov(x,t)/ot = k2o2v(x,t)/Ox 2 v(x,O) = Vo,
ov(O,t)/ox = Hv(O,t).~
(6)
As t ----' ~, it is physically evident that at every point in the block v(x, t) ---" O. Taking the Laplace transform of (6) and solving for V(x,s) = ,,g{v(x,t)} gives
V(x,s) = (vo/s) 11 - [ H / ( H + (s/k)'/2)l] exp [ - x(s/k)'/2l .
(7)
Before inverting V(x,s) to give v(x,t), V(x,s) is subjected to the_tests of determining if v (x,O) = v o and v (x, t) ---* 0 as t ---* ~ agree with lim s V(x, s) and lim s-V(x,s), respectively. From (4), ~2[v(x,t)}
exists;
~2{ov(3c, t)/ot}
s~0
m a y be obtained from (7); further, from (7), lim s V(x,s) exists: the hypotheses of (2) are fulfilled and lim s-V(x,s) = lim S(Vo/S)11 - [ H / ( H + s-~
(s/k),/2)l}
exp [ - x(s/k)l/2]
s~
=
v0 =
limv(x,t), t~O+
recovering v(x,O). met and
Since sV(x,s) has no poles, the hypotheses of (3) are
Sept., 1962 ]
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LAPLACE TRANSFORM THEORY
m
lim sV(x,s) = lim S(Vo/S ) 11 - [ H / ( H + (s/k)'/2)]l exp [ - x(,/k)'/2] '0
~ ~0
= 0 = limv(t,t), in accord with physical intuition. Again, application of (2) and (3) directly to V(x,s) has produced anticipated results, so V(x,s) is not obviously wrong; and, as before, now it would be appropriate to invert V~(x,s) to obtain v (x, t). USE OF THE INITIAL-VALUE THEOREM TO DETERMINE ARBITRARY CONSTANTS OF INTEGRATION
Sometimes (2) m a y be employed to evaluate arbitrary constants arising from the solution of certain ordinary differential equations, as in the example below. Laplace transform methods used to solve Bessel's equation of index 0,
tJ,;'(t) + Jo(t) + t j o ( t ) = o, subject to the initial condition J0(O) = 1, result in the differential equation
(,2+ in_ which Z(s) = ~ g(s) is
[J0(t)}.
+ ,g-(s) = 0, T h e solution of this differential equation in
Z ( S ) = C(S 2 -{- 1) I / 2
(8)
where C is an arbitrary constant of integration. Using the initial-value theorem and the given J0(0) = 1 to determine C, application of (2) to (8) gives C = 1, since lim sZ(s) = C l i m s(s z + 1)-'/2 = C = lim Jo(t) = 1. APPLICATION OF THE FINAL-VALUE THEOREM TO THE EVALUATION OF CERTAIN DEFINITE INTEGRALS
Certain definite integrals whose lower and upper limits are 0 and t, respectively, possess tabulated Laplace transforms (5). We restrict ourselves here to such integrals, and use the final-value theorem to evaluate several of these. Although the same results can be obtained by conventional methods, these procedures generally are lengthy and unrelated; when (3) m a y be applied to such problems the unity, directness and simplicity of the necessary operations demonstrate the power of the method. As illustrations of this use of (3), almost in a single step and in a uniform manner such well-known integrals as those below may be evaluated: lim, -.
f
(sin xlx)dx = lim,. s £
{f
(sin x/x)dx
}
- lim s[s -I arc cos s] = rr/2
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BERNARD RASOF
[J.P.l.
e x, dx = lim Erf(tv2) = lim s
lim (21ze llz) f " "
t
'~
s
*0
= lim s[s-'(s + 1) -'/2 ] = 1 s
lim t~
f
+0
,
J,(ax)dx = lim s
Jo
= lim sla"/s(s 2 + a2)V2[s + 02 + a2)'/21. } = a-' s~0
lim f '
[J,(ax)/x]dx
= lim s £
ax)/x
s~O
= lim s[a"/ns[s + 02 + a2),/2].} = n - , . s~0
C e r t a i n other integrals also m a y be evaluated by utilizing
Theorem I. If
f0'
f ( x ) d x a n d f ( t ) possess L a p l a c e transforms a n d F(s) =
.Elf(t)} has no poles on or to the right of the i m a g i n a r y axis in the s-plane, then
f0'
f ( x ) d x --- lim F(s).
lim
(9)
Proof. A s t a n d a r d o p e r a t i o n - L a p l a c e t r a n s f o r m pair (1) is
<,t2
i/0,
f(x)
= F(s)/s ;
-
(10)
w h e n the hypotheses of T h e o r e m I hold, setting (10) into (3) leads to (9). T o illustrate the use of T h e o r e m I, consider the p r o b l e m of evaluating
I(a,b) =
fo
te -°' cos bt dt,
a > O.
T h e p r o b l e m requires a n u m b e r of integrations by parts; but, from a table in (3), £ {re -°' cos bt} = [(s + a) 2 - b2]/[(s + a) 2 + b2]2, which satisfies the hypotheses of T h e o r e m I, leading i m m e d i a t e l y to lim ,.
fo .
xe - = c o s b x d x .
.
.
= lim (s + a) 2 - b2 _ a 2 - b2 0 [(s + a) 2 + b2]2 (a 2 + b2) 2
T h e most extensive table of function-Laplace t r a n s f o r m pairs (5) contains u p w a r d s of 60 such pairs which, u p o n p r e l i m i n a r y e x a m i n a t i o n ,
Sept., ~962.]
LAPLACE TRANSFORM THEORY
I7I
satisfy the hypotheses of T h e o r e m I. Application of this T h e o r e m to these pairs should permit evaluation of a n u m b e r of definite integrals which probably have not been evaluated previously. Thus, definite integrals which may be evaluated by the use of T h e o r e m I include lim t ~
lim f '
t
f
[0rx) -'/2 - a e x p ( a 2 x ) Erfc(axl/2)]dx = lim (a + s'/2) -' = a-',
-,tO
~
'0
e x p [ - (a + b)x/2]Io[(a - b ) x / 2 ] d x = lim (s + a) -1/2(s + b) -~n = (ab)-~/2
s~0
lim t
*~
£
x"-~J~_~(ax)dx = (2a)"-~Tr-'nr(n) lim 0 2 + a2) -~ s 40
= (2/a)"+~Tr ' / z I ' ( n ) / 2 . APPUCATION OF THE THEOREMS TO DERIVE A GENERAL RESULT IN LAPLACE TRANSFORM THEORY
Previously (2) and (3) have been applied to particular functions to yield specific results. Here we employ these theorems to derive an operationtransform pair. Apply the initial-value theorem to the function-Laplace transform pair (5) x -1
x
= s-'
/07
e -s'
t)dt,
and the final-value theorem to the function Laplace transform pair (5) 0~
x-if(x)
= s-1
e-,tf(t)dt,
in both cases obtaining immediately 2
{f~x-lf(x)dx} : f ~ f ~e - " f ( t ) d t ,
which is considerably more difficult to prove by standard methods (2). APPLICATION OF THE THEOREMS TO THE EVALUATION OF CERTAIN LIMITS
T h e theorems (2) and (3) may be used to determine, directly and simply, a number of limits which are not easy to establish by conventional means: for example, these other methods m a y require knowledge of the series expansions of the function involved. With appropriate notation, when a limit to be evaluated may be written as l i m f ( t ) , l i m f ( t ) , lim sF(s) or lira sF(s), and the hypotheses of the apt~0+
t ~
s ~
~ ~0
plicable theorem are satisfied, instead of evaluating the limit as originally
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BERNARD RASOF
[J.F.I.
given it may be easier to evaluate its mate as given by the relevant theorem. To illustrate, (2) may be used to evaluate certain limits of the type l i m f ( t ) by evaluating instead the corresponding l i m s £ { f ( t ) } . The t~O+
~ ~*
method is only useful when s £ { f ( t ) } is a less complicated function than f ( t ) itself. From (5), one learns that in general the functions f whose Laplace transforms F are simpler t h a n f itself are the higher transcendental functions; among these are the complementary error function Erfc(t) = 1 - Erf(t), where Erf(t) = (2/a-v2)
f,
e -~ dx; the Laguerre polynomial
L°(t) = (e'/n !) d " ( e - ' t " ) / d t " ; the Bessel functionJ,(t) = ~
( - 1)k(t/2)"+2k/
k-O
k! (n + k)[, of the first kind and non-negative integral index n; the modified Bessel function Io(t ) = Jo(it), i = ( - 1)v2, of the first kind and index 0; and the Legendre polynomial Pm(t) = (2mm!)-~d m(t 2 - 1)m/dlm. The method is illustrated by a few examples selected to present the idea dramatically: for example, a complicated limit is determined by evaluation of its simpler mate given by either (2) or (3). Because the hypotheses of (2) are met in the examples below, (2) m a y be used to show directly, uniformly, and with only appeal to elementary operations on limits, that lim J~,(at) = lim s • l J o ( a t ) }
= lim [s(s 2 + a2) -'/21 = 1
lim J , ( a t ) = lim s £ l J , , ( a t ) } t~O+
s +*
= lim s , ~*
( 3 `2
lim P2,(cost) = lim s t..0 +
....
+ a2)'/2[s -}- (s 2 q- a2)'/2]"
= 0,
i s i + 12)(s2 -}- 3z) "'' [s2 + (2n z 1)2].~ (S2
+ 22)(s2 7 4~i
+
(in) a] J
n # 0
= 1
lim L,,(t) = lim s[(s - 1)"s -"-~] = 1 I ~0+
s **
lim e x p [ - ( a + b)t/2]Io[(a - b)t/2] = lim s[(s + a) '/2(s + b) ,/2] = 1 t~O+
s~
lim[ae .' Erfc(a'/Zt~/z) + (ab)V2e~' Erfc(b~/2t v2) - be bt] t~O+
= lim s[(a - b)s'/2(s '/z + a '/2) '(s - b) ~] = a - b. In the examples below the hypotheses of (3) are fulfilled, so immediately lira J,, (at) = lim s = 0 . . . . . o (s ~ + ai)'/2[s + (s ~ + aa)v~l,,J
Sept., ,962.1
LAPLACE TRANSFORM THEORY
173
lim Erfc(a'/2/2t '/2) = lim s[s 'exp (-a'/is'/2)] = 1 . t
'0
, ~
The more interesting limits which m a y be evaluated by application of the initial- and final-value theorems are indeterminate forms (see the following Section). APPLICATION OF THE THEOREMS TO THE EVALUATION OF INDETERMINATE FORMS
Indeterminate forms of various types usually are evaluated by procedures which are varied, apparently unrelated, often roundabout and frequently tedious. In problems requiring the evaluation of certain indeterminate forms, (2) or (3) may be employed. This application refers back to the Introduction, where it was observed that (2) and (3) assert that under certain conditions l i m f ( t ) and lim s£{f(t)} are equal for two pairs of values of t and s. In this use of (2) and (3), when the limiting value of the variable is 0 or ,:, an indeterminate form to be evaluated is written as lim f(t), lim f(t), - -
__
l
lim sF(s) or lim sF(s), whichever is appropriate.
q}+
t
',
In many instances, es-
pecially those involving higher transcendental functions, the indeterminate form when written as l i m f ( t ) will possess a tabulated Laplace transform or the F(s) in the indeterminate form written as lim sF(s) will possess a tabulated inverse Laplace transform f ( t ) ; in some instances the indeterminate form written either as lim f(t) or lim sF(s) will be one-half of a tabulated function-Laplace transform pair. W h e n an indeterminate form is written as lim f(t) but £ {f(/)} is tabut
~0+
lated and lim s£{f(t)} is more readily evaluated; or conversely, when an indeterminate form is written as lim sF(s) but f(t) = £ - ' { F ( s ) }
is tabu-
lated and lira f ( t ) is easier to evaluate; and the hypotheses of (2) are met, then (2) can be employed to evaluate the given indeterminate form. An analogous statement applies to (3). Although indeterminate forms at 0 and :~ may be evaluated by L'Hgpital's rule, when the method described can be used it produces desired results in a uniform manner, directly and easily. The method changes the evaluation of a complicated indeterminate form to the determination of a comparatively simple one, by utilizing tabulated results originally derived for a completely different purpose. We illustrate the applicability of the method with some examples. There is no advantage to replacing an indeterminate form by one which is equally, or even more, complicated, so the examples chosen have this desired simplification. We first use (2) directly to evaluate an indeterminate form of type lim f(l) by evaluating instead the simpler lim sF(s). Then we employ (2)
x 74
BERNARD RASOF
[,J.V.l,
in its converse form to evaluate some indeterminate forms of type lim sF(s) by evaluating instead l i m f ( t ) . Finally, we employ both the direct and cont ~0+
verse forms of (3) to evaluate other indeterminate forms. If Jl' (0+) is not known, the indeterminate form l im [J1 (at)/t] cannot ,-0+ be evaluated by L'HSpital's rule. However, with the method presented here, l i m [ J ~ ( a t ) / t ] = lim s{a[s + (s 2 + a2)V2]-~} = a / 2 which m a y be verified using more elaborate techniques. In the first illustration of the converse application of (2), we use the well-known fact that lim (sin a t ) I t = a. From (5), with a m i n i m u m of ,40+ labor we have lim[s arc tan (a/s)] = lim ( s i n a t ) / t = a s--~
t--0+
lim s[ln (1 + 4ag/s2)] = 4 lim ( s i n 2 a t ) / t = O. s--~
t +0+
Some indeterminate forms evaluated in this way require one or more uses of L'Hbpital's rule on indeterminate forms much simpler than the originals; thus, lira s[ln (s + b) - In (s + a)] = lim (e-"~ - e-b~)/t = b - a lira s ,~.
{s
In
P2 + a2~ ks2 +
b a} + 2b arc t a n - - - 2a arc tan
b~j
s = 2 lim
t~0+
cos at - cos bt
= b2 -
a 2.
12
T h e direct application of (3) to the evaluation of indeterminate forms is illustrated by determining lira {[exp(a2t)] E r f c ( a W 2 + b/2t~/2)}, which ordinarily would require four applications of L'H6pital's rule. From (5) we obtain £{[exp(aZt)] Erfc(at 1/2 + b/2t'/2)} = e - ° b e x p ( - b s l / 2 ) / [ s ' / 2 ( s '/2 + a)].
(11)
The left-hand m e m b e r of (11) exists and the right-hand m e m b e r has no poles, so (3) m a y be applied to (11) and leads directly to the result lim {[exp(a2t)] E r f c ( a W z + b/2t~/2)} = lira s{e -"~ e x p ( - b s l / Z ) / [ s ' / 2 ( s '/2 + a)]} -- 0. s~0
The converse application of (3) to the evaluation of indeterminate forms is illustrated by determining lira [s-V2e"/" Erfc(a/s)V2], which is difficult to s~0
Sept., ,962.1
LAPLACE TRANSFORM THEORY
I75
determine by s t a n d a r d methods. Again, in (5) we find
s-3/Ze°/'Erfc(a/s) '/2 =1~ {[1 - exp(-2[at]~/2)]/(a~r)'/2}.
(12)
T h e right-hand m e m b e r of (12) exists and the left-hand m e m b e r has no poles; so (3) m a y be applied, with the result that lira s[ s- 3/:~e~/" Erfc (a/s) ~/2] = lira {[1 - exp ( - Z[at]~/2 ) ]/ (azr) '/2} = (a~r) ,/2. GENERALIZATION OF THE INITIAL- AND FINAL-VALUE THEOREMS
W e generalize the initial-value t h e o r e m so that initial values of derivatives of the solution to an initial-value problem, as well as that of the solution, m a y be recovered and confirmed before inverting the response transform.
Theorem II. Generalized Initial- Value Theorem. I f f ( t ) , f ' ( t ) , f " ( t ) , . . . , have Laplace transforms, F(s) = ~ {f(t)l, and the lim exists, then lim s
"F-(s) -
s ° k-'f(~)(O+
f("+~)(t)
= limf(~)(t).
(13)
Proof: Assume_ T h e o r e m II_to be true for some n > 0. L e t f ( t ) = g'(t), so that F(s) = sG(s) - g ( 0 + ) , G(s) = • ]g(t)}; then (13) b e c o m e s lim~ .~ s f "+' G-(s) - s"g(O+) - y " S"-k-'g(k+l)(0+
= limgC"+1)(t).
(t4)
t ~0~ k=O
After a change in notation fromg, G to f, F, respectively, (14) becomes lim s
"+iF(s) -
~
s"-'f (k)(O+ k=0
= lim f("+')(t). /~0+
Thus, if (13) holds for n it also holds for n + 1; and the initial-value t h e o r e m (2) asserts that (13) is true for n = 0: hence (13) is true for n = 1,2,3,.... T h e final-value t h e o r e m also can be generalized; the essential result is
Theorem Ill." Generalized Final- Value Theorem. If f (t), f'(t), f " ( t ) , . . . , f("+'~ (t) have L a p l a c e transforms, F(s) = • {f(t)}, and s"+lff(s) has no poles on or to the right of the i m a g i n a r y axis in the s-plane, then lim s "+1F(s) = l i m f f "1 (t). s~0
I
(15)
~
Proof: A s s u m e that T h e o r e m III is true for some n > 0. Let f(t) = g'(t); then as before F ( s ) = sG(s) - g ( 0 + ) , G-(s) = £{g(t)}" and (15) becomes
1 76
BERNARD RASOF
[J.F.I.
lim . < + 2 G ( s ) = lim g("+'> (t). T h u s , after changing from g, G to f, F, if (15) holds for n, it also holds for n + 1. But (3) shows (15) to be true when n = 0; therefore (15) holds for n = 1,2,3,.... APPLICATION OF THE GENERALIZED INITIAL-VALUE THEOREM
Heaviside (6) and C a r s o n (7) o b t a i n e d M a c l a u r i n expansions of ~-~ {F(s)} directly from F ( s ) itself by expanding F ( s ) in negative powers of s and then e m p l o y i n g the transform-function pair £-~{s -/"+~/} = t " / F(n + 1), n > - 1 , on each term of the resulting series. W h e n F ( s ) is a rational function, this p r o c e d u r e requires e x p a n d i n g a polynomial in s 1 for several positive integral exponents a n d collecting the coefficients of e a c h s ~,m > 0. Thus, the H e a v i s i d e - C a r s o n p r o c e d u r e for obtaining the first few terms of the inverse L a p l a c e t r a n s f o r m of F(s) = 1 / ( s 5 - s 4 - s ~ - 1) requires rewriting F ( s ) as F(s)
= 7 7 ( 1 - s-' - s - 2 -
s -5) = s -5
(s - 1 + s - 2 + s - s ) k,
(16)
k=0
e x p a n d i n g (s -1 + s -2 + s-S) k for k = 0, 1, 2 , . . . , and collecting coefficients of corresponding powers o f s -1. W i t h the factor s -5 in (16), the coefficient ofs -~° necessitates e x p a n d i n g (s -1 + s -2 + s-5) k fork = 1, 2, 3, 4, 5. T h e o r e m II provides a n o t h e r w a y of generating these coefficients, also directly from F ( s ) itself. W h e n the hypotheses of that theorem are fulfilled, the coefficients f(k> ( O ) / k ! of the M a c l a u r i n expansion of f ( t ) are given explicitly as f(t)
=~... f ( k[> ( l0 ) /!k = m ..,k -~,1, k =0
,
s
kF(s)-
k =0
Y'~s*-~-lffJl(
tk
j =0
m
W i t h F ( s ) = 1 / ( s 5 - s 4 - s 3 - 1 ) , f ( 0 ) -- 0 from (2). Applying Theo r e m II to this F ( s ) _ t h e next several coefficients in the M a c l a u r i n expansion o f f ( t ) = £ - l { F ( s ) } are lim f'(t)(=f'(O+))
= lim f"(t)(=f"(O+))
t ~0+
= lim f'"(t)(=f'"(O+))
t~O+
t--~O+
lira f~v (t) = fir ( 0 + ) t ~0+
= lim s [ s 4 F ( s ) = lim [ s S / ( s 5 -
-
s3f(O+) s4 -
s3 -
-
s2f'(0+) - sf"(O+)
1)] = 1
-f"'(0+)]
= 0
Sept., f96:z.J
LAPLACE TRANSFORM T t t E O R Y limfv(t)
= fv(0+)
-- l i r a s [ s S F ( s )
= lim s{[sS/(s ~limf~'(t) t
Thus,
f(t)
the =
= fv~(O+)
= lims{[s~/(s
'0+
s
first three
£~ ' { 1 / @
5-
s3 -
5-
-/v(0+)]
s3 -
s4 -
1)] s 3-
1} =
1)] -
1
s -
1} = 2.
~ •
non-vanishing s" -
s4 -
I77
terms
of the
Maclaurin
expansion
of
1)} are
f ( t ) = t4/4! + ?/5! +
2t6/6!.
REFERENCES
(1) RULE V. CHURCHILL,"Operational Mathematics," New York, McGraw-Hill Book Co., Inc., second edition, 1958. (~) Louis A. PIPES, "Applied Mathematics for Engineers and Physicists," New' York, McGrawHill Book Co., Inc., second edition, 1958, Chapter 21. (3) MURRAYS. GARDNERANDJOHN L. BARNES,"'Transients in l.inear Systems," New York..Iulm Wiley & Sons, Inc., 1942. (4) E. C. TITCHMARSH,"Introduction to the Theory of Fourier Integrals," Oxford, Oxford University Press, 1937. (5) "Tables of Integral Transforms," Vol. I, compiled by the Staff" of the Bateman MantJs(ript Project, New York, McGraw-Hill Book Co.. Inc., 1954. (6) OLIVERHEAVlSmE,"Electromagnetic Theory." New York, Dover Publications, 1950. (7) JOHN R. CARSON,"Electric Circuit Theory and the Operational Calculus," New York, McGrawHill Book Co., Inc., 1926.
The Pranklin lnslilule V o l . 274
THE
SEPTEMBER,
1962
No. 3
INITIAL- AND FINAL-VALUE THEOREMS IN L A P L A C E T R A N S F O R M T H E O R Y * BY
BERNARD RASOF 1 ABSTRACT
The initial- and final-value theorems, generally neglected in Laplace transform theory, for some purposes are among the most powerful results in that subject. Here are advanced some useful applications of these theorems: ~,,e show how they may be employed as necefsar~ checks on the accuracy of response transtbrms, to evaluate (directly, in a uniform manner, and with remarkable ease) certain definite integrals, to derive an important general result, and to evaluate (again uniformly and easily) certain limits and indeterminate forms (this is believed to be a new application). T h e n the theorems are generalized, and the first of these is applied to generate, directly from a Laplace transform itself, the Maclaurin expansion of the inverse Laplace transform. INTRODUCTION
When a function f(t) is piecewise continuous and of exponential order it possesses (1)z a Laplace transform if(s) related to f(t) by way of the definition
F(s) = 2 U(t)} =
e-./(t)dt,
(1)
where Re(s) is large enough to insure convergence of the integral. The Laplace transform has an extensive literature on theory and application. Its numerous utilizations are summarized in (2), where it is shown to be useful in treating an impressively large and varied class of important pure and applied problems. * This work was sponsored by the Educational Development Program, Department of Engineering, University of California, with financial support from the Ford Foundation. 1 Department of Engineering, University of California, Los Angeles, Calif. 2 T h e boldface numbers in parentheses refer to fhe references appended to this paper. (Note
T h e Franklin Institute is not responsible for the statements and opinions advanced by contributors in the
,IouuNAt.)
I(J 5
I66
BERNARD
RASOF
[ ].I:.1.
THE INITIAL- AND FINAL-VALUE THEOREMS
Most of the theorems establishing basicproperties of the Laplace transformation present the Laplace transform F(s) of specific functions f(t) or of operations on f(t). The exceptions are two unique theorems which assert the equality of l i m f ( t ) and lim sF(s) for two pairs of particular values of t and s. These are the so-called initial- and final-value theorems, stated below without proof. 3
Initial-value_ theorem. If both f(t) and f'(t) possess Laplace transforms and lim sF(s) exists, then lim f(t) = lim sF(s).
(2)
Final-value theorem. If both f ( t ) and f'(t) possess Laplace transforms and sF(s) has no poles on or to the right of the imaginary axis in the s-plane, then lim f(t) = lira sF(s). (3) The last hypothesis requires existence of l i m f ( t ) .
For, if any pole sp of
sF(s) has R e ( s p ) > 0, then f ( t ) ---, ~ as t ---~ ~; if sF(s) has poles on Re(s) = 0, t h e n f ( t ) has a stable oscillation as t ----' ~ ; in neither instance does l i m f ( t ) exist. T h e purpose of this paper is to advance some applications of (2) and (3). To this end, we show how the theorems m a y be employed as part of the procedure of solving ordinary and partial differential equations, and parenthetically suggest that this is an essential aspect of the solution. Then we apply (3) alone to evaluate (with a single stroke) some formidable definite integrals. After this we present the application of these theorems to the evaluation of certain limits and indeterminate forms, in what is believed to be a new application of these theorems. Lastly, (2) and (3) are generalized, and the first of these is used to obtain a new method of generating directly from a Laplace transform F(s) itself the Maclaurin expansion of the inverse Laplace transform f(t) = £-1 {-F(,f)}. USE OF THESE THEOREMS IN PARTIALLY SUBSTANTIATING RESPONSE TRANSFORMS
W h e n an initial-value problem inf(t) is solved by use of Laplace transform methods, an intermediate result is a response transform F(s) = ~ { f ( t ) l which must be inverted to yield the s o l u t i o n f ( t ) of the original problem. The inversion of F(s) to givef(t) is the most difficult part of the solution, and if an error is made in obtaining F(s), the effort and time expended in obtaining its inverse will be wasted. For this reason, before proceeding with the inversion of F(s) it is useful to have a way of at least partially establishing its accuracy. W h e n they can be employed, (2) and (3) supply just such a control which can prevent a serious error. The first 3 These theorems are derived in a number of textbooks; see especially (1) and (3).
Sept., ,96':,.]
LAPLACE TRANSFORM THEORY
i6 7
theorem provides a rapid w a y of determining directly from F(s) itself the initial condition on ,E-I{F(s)}, for comparison with the given initial condition o n f ( t ) . The conditions on (3) are more stringent than those on (2), so the former cannot be employed as frequently as the latter; however, when (3) can be utilized it offers a means of easily computing, also directly from F(s), the limiting value ( t h a t is, as t --' ~) of the function which would be obtained f r o m ~ - l { F ( s ) ] , and comparing this with a possibly explicitly stated asymptotic value o f f ( t ) , or one inferred from physical considerations. If application of (2) and (3) to F(s) recovers f ( 0 ) and gives a result in accordance with l i m f ( t ) , then F(s) may be correct and its inversion to yield f(t) can and should proceed. O n the other hand, should the application of either or both of these theorems fail to confirm known results, it would be concluded that F(s) is in error and that its calculation should be reviewed and corrected before attempting the inversion. This requirement that these theorems applied to F(s) produce given, known or inferred results is in the nature of a necessary but not sufficient condition on the correctness ofF(s). Testing the accuracy of a response transform by use of (2) and (3) usually is straightforward, direct and rapid; the habitual use of these theorems for this purpose is invaluable in detecting errors before the often laborious process of inversion is attempted.
USE OF THESE THEOREMS AS NECESSARY CHECKS ON THE ACCURACY OF RESPONSE TRANSFORMS
Example / A circuit has a constant voltage source in series with lumped, constant resistance and inductance R and L. A switch in the circuit is closed at ! = 0; what is the current i(t) for t > 0? This initial-value problem requires solving
L[di(t)/dt] + Ri(t) = U(t),
i(0) = 0;
(4)
U(t) is a unit step voltage originating at t = 0. Inductance is proportional to di(t)/dt, and this decreases to zero; hence, as t --' ~ the effect of the inductance will become vanishingly small and it is physically evident that
i(t) --~ 1/R as t --' ~. Taking the Laplace transform of (4) and solving for I(s) = ~ {i(t)l gives
I(s) = llsL (s + R / L ) .
(5)
Before obtaining i(t) by inverting its Laplace transform, I(s) is tested to determine if i(0) = 0 and i(t)--~ 1/R as t --~ :¢ agree with lim sT(s) and lira if(s), respectively, representing application of (2) and (3) to this s
"0
problem. The current ,(t) possesses a Laplace transform (4); £ li'(t)] follows
168
BERNARD RASOF
[J.V.I.
from (4) and (5); from (5), lim sF(s) exists: thus, the hypotheses of (2) are met and lim s[(s) = lim s[1/sL (s + R/L)] = 0 = lim i(t), recovering i(0) in (4). Since both R and L are positive, s-[(s) has no poles in the right half of the complex s-plane: that is, the hypotheses of (3) are met and thus lim si(s) = lim s[1/sL (s + R/L)] = 1/R = lim i(t), s~O
s~O
in accord with physical intuition.
l
*~
Application of (2) and (3) directly to
I--(s) leads to the anticipated results 0 and 1/R, respectively, so ](s) given by (5) is not obviously incorrect.
Example 2 A semi-infinite homogeneous block with thermal diffusivity k 2 > 0 and coefficient of surface heat transfer H > 0 is at uniform temperature v0 > 0 w h e n t > 0. A t t = 0 the s u r f a c e x = 0 is placed in contact w i t h a fluid at temperature zero and possessing infinite heat capacity. Heat flows from the surface x = 0 in accordance with Newton's law of cooling. W h a t is the temperature v(x,t) for all x > 0, t > 0? This initial-value problem requires the solution of
ov(x,t)/ot = k2o2v(x,t)/Ox 2 v(x,O) = Vo,
ov(O,t)/ox = Hv(O,t).~
(6)
As t ----' ~, it is physically evident that at every point in the block v(x, t) ---" O. Taking the Laplace transform of (6) and solving for V(x,s) = ,,g{v(x,t)} gives
V(x,s) = (vo/s) 11 - [ H / ( H + (s/k)'/2)l] exp [ - x(s/k)'/2l .
(7)
Before inverting V(x,s) to give v(x,t), V(x,s) is subjected to the_tests of determining if v (x,O) = v o and v (x, t) ---* 0 as t ---* ~ agree with lim s V(x, s) and lim s-V(x,s), respectively. From (4), ~2[v(x,t)}
exists;
~2{ov(3c, t)/ot}
s~0
m a y be obtained from (7); further, from (7), lim s V(x,s) exists: the hypotheses of (2) are fulfilled and lim s-V(x,s) = lim S(Vo/S)11 - [ H / ( H + s-~
(s/k),/2)l}
exp [ - x(s/k)l/2]
s~
=
v0 =
limv(x,t), t~O+
recovering v(x,O). met and
Since sV(x,s) has no poles, the hypotheses of (3) are
Sept., 1962 ]
I69
LAPLACE TRANSFORM THEORY
m
lim sV(x,s) = lim S(Vo/S ) 11 - [ H / ( H + (s/k)'/2)]l exp [ - x(,/k)'/2] '0
~ ~0
= 0 = limv(t,t), in accord with physical intuition. Again, application of (2) and (3) directly to V(x,s) has produced anticipated results, so V(x,s) is not obviously wrong; and, as before, now it would be appropriate to invert V~(x,s) to obtain v (x, t). USE OF THE INITIAL-VALUE THEOREM TO DETERMINE ARBITRARY CONSTANTS OF INTEGRATION
Sometimes (2) m a y be employed to evaluate arbitrary constants arising from the solution of certain ordinary differential equations, as in the example below. Laplace transform methods used to solve Bessel's equation of index 0,
tJ,;'(t) + Jo(t) + t j o ( t ) = o, subject to the initial condition J0(O) = 1, result in the differential equation
(,2+ in_ which Z(s) = ~ g(s) is
[J0(t)}.
+ ,g-(s) = 0, T h e solution of this differential equation in
Z ( S ) = C(S 2 -{- 1) I / 2
(8)
where C is an arbitrary constant of integration. Using the initial-value theorem and the given J0(0) = 1 to determine C, application of (2) to (8) gives C = 1, since lim sZ(s) = C l i m s(s z + 1)-'/2 = C = lim Jo(t) = 1. APPLICATION OF THE FINAL-VALUE THEOREM TO THE EVALUATION OF CERTAIN DEFINITE INTEGRALS
Certain definite integrals whose lower and upper limits are 0 and t, respectively, possess tabulated Laplace transforms (5). We restrict ourselves here to such integrals, and use the final-value theorem to evaluate several of these. Although the same results can be obtained by conventional methods, these procedures generally are lengthy and unrelated; when (3) m a y be applied to such problems the unity, directness and simplicity of the necessary operations demonstrate the power of the method. As illustrations of this use of (3), almost in a single step and in a uniform manner such well-known integrals as those below may be evaluated: lim, -.
f
(sin xlx)dx = lim,. s £
{f
(sin x/x)dx
}
- lim s[s -I arc cos s] = rr/2
170
BERNARD RASOF
[J.P.l.
e x, dx = lim Erf(tv2) = lim s
lim (21ze llz) f " "
t
'~
s
*0
= lim s[s-'(s + 1) -'/2 ] = 1 s
lim t~
f
+0
,
J,(ax)dx = lim s
Jo
= lim sla"/s(s 2 + a2)V2[s + 02 + a2)'/21. } = a-' s~0
lim f '
[J,(ax)/x]dx
= lim s £
ax)/x
s~O
= lim s[a"/ns[s + 02 + a2),/2].} = n - , . s~0
C e r t a i n other integrals also m a y be evaluated by utilizing
Theorem I. If
f0'
f ( x ) d x a n d f ( t ) possess L a p l a c e transforms a n d F(s) =
.Elf(t)} has no poles on or to the right of the i m a g i n a r y axis in the s-plane, then
f0'
f ( x ) d x --- lim F(s).
lim
(9)
Proof. A s t a n d a r d o p e r a t i o n - L a p l a c e t r a n s f o r m pair (1) is
<,t2
i/0,
f(x)
= F(s)/s ;
-
(10)
w h e n the hypotheses of T h e o r e m I hold, setting (10) into (3) leads to (9). T o illustrate the use of T h e o r e m I, consider the p r o b l e m of evaluating
I(a,b) =
fo
te -°' cos bt dt,
a > O.
T h e p r o b l e m requires a n u m b e r of integrations by parts; but, from a table in (3), £ {re -°' cos bt} = [(s + a) 2 - b2]/[(s + a) 2 + b2]2, which satisfies the hypotheses of T h e o r e m I, leading i m m e d i a t e l y to lim ,.
fo .
xe - = c o s b x d x .
.
.
= lim (s + a) 2 - b2 _ a 2 - b2 0 [(s + a) 2 + b2]2 (a 2 + b2) 2
T h e most extensive table of function-Laplace t r a n s f o r m pairs (5) contains u p w a r d s of 60 such pairs which, u p o n p r e l i m i n a r y e x a m i n a t i o n ,
Sept., ~962.]
LAPLACE TRANSFORM THEORY
I7I
satisfy the hypotheses of T h e o r e m I. Application of this T h e o r e m to these pairs should permit evaluation of a n u m b e r of definite integrals which probably have not been evaluated previously. Thus, definite integrals which may be evaluated by the use of T h e o r e m I include lim t ~
lim f '
t
f
[0rx) -'/2 - a e x p ( a 2 x ) Erfc(axl/2)]dx = lim (a + s'/2) -' = a-',
-,tO
~
'0
e x p [ - (a + b)x/2]Io[(a - b ) x / 2 ] d x = lim (s + a) -1/2(s + b) -~n = (ab)-~/2
s~0
lim t
*~
£
x"-~J~_~(ax)dx = (2a)"-~Tr-'nr(n) lim 0 2 + a2) -~ s 40
= (2/a)"+~Tr ' / z I ' ( n ) / 2 . APPUCATION OF THE THEOREMS TO DERIVE A GENERAL RESULT IN LAPLACE TRANSFORM THEORY
Previously (2) and (3) have been applied to particular functions to yield specific results. Here we employ these theorems to derive an operationtransform pair. Apply the initial-value theorem to the function-Laplace transform pair (5) x -1
x
= s-'
/07
e -s'
t)dt,
and the final-value theorem to the function Laplace transform pair (5) 0~
x-if(x)
= s-1
e-,tf(t)dt,
in both cases obtaining immediately 2
{f~x-lf(x)dx} : f ~ f ~e - " f ( t ) d t ,
which is considerably more difficult to prove by standard methods (2). APPLICATION OF THE THEOREMS TO THE EVALUATION OF CERTAIN LIMITS
T h e theorems (2) and (3) may be used to determine, directly and simply, a number of limits which are not easy to establish by conventional means: for example, these other methods m a y require knowledge of the series expansions of the function involved. With appropriate notation, when a limit to be evaluated may be written as l i m f ( t ) , l i m f ( t ) , lim sF(s) or lira sF(s), and the hypotheses of the apt~0+
t ~
s ~
~ ~0
plicable theorem are satisfied, instead of evaluating the limit as originally
I 72
BERNARD RASOF
[J.F.I.
given it may be easier to evaluate its mate as given by the relevant theorem. To illustrate, (2) may be used to evaluate certain limits of the type l i m f ( t ) by evaluating instead the corresponding l i m s £ { f ( t ) } . The t~O+
~ ~*
method is only useful when s £ { f ( t ) } is a less complicated function than f ( t ) itself. From (5), one learns that in general the functions f whose Laplace transforms F are simpler t h a n f itself are the higher transcendental functions; among these are the complementary error function Erfc(t) = 1 - Erf(t), where Erf(t) = (2/a-v2)
f,
e -~ dx; the Laguerre polynomial
L°(t) = (e'/n !) d " ( e - ' t " ) / d t " ; the Bessel functionJ,(t) = ~
( - 1)k(t/2)"+2k/
k-O
k! (n + k)[, of the first kind and non-negative integral index n; the modified Bessel function Io(t ) = Jo(it), i = ( - 1)v2, of the first kind and index 0; and the Legendre polynomial Pm(t) = (2mm!)-~d m(t 2 - 1)m/dlm. The method is illustrated by a few examples selected to present the idea dramatically: for example, a complicated limit is determined by evaluation of its simpler mate given by either (2) or (3). Because the hypotheses of (2) are met in the examples below, (2) m a y be used to show directly, uniformly, and with only appeal to elementary operations on limits, that lim J~,(at) = lim s • l J o ( a t ) }
= lim [s(s 2 + a2) -'/21 = 1
lim J , ( a t ) = lim s £ l J , , ( a t ) } t~O+
s +*
= lim s , ~*
( 3 `2
lim P2,(cost) = lim s t..0 +
....
+ a2)'/2[s -}- (s 2 q- a2)'/2]"
= 0,
i s i + 12)(s2 -}- 3z) "'' [s2 + (2n z 1)2].~ (S2
+ 22)(s2 7 4~i
+
(in) a] J
n # 0
= 1
lim L,,(t) = lim s[(s - 1)"s -"-~] = 1 I ~0+
s **
lim e x p [ - ( a + b)t/2]Io[(a - b)t/2] = lim s[(s + a) '/2(s + b) ,/2] = 1 t~O+
s~
lim[ae .' Erfc(a'/Zt~/z) + (ab)V2e~' Erfc(b~/2t v2) - be bt] t~O+
= lim s[(a - b)s'/2(s '/z + a '/2) '(s - b) ~] = a - b. In the examples below the hypotheses of (3) are fulfilled, so immediately lira J,, (at) = lim s = 0 . . . . . o (s ~ + ai)'/2[s + (s ~ + aa)v~l,,J
Sept., ,962.1
LAPLACE TRANSFORM THEORY
173
lim Erfc(a'/2/2t '/2) = lim s[s 'exp (-a'/is'/2)] = 1 . t
'0
, ~
The more interesting limits which m a y be evaluated by application of the initial- and final-value theorems are indeterminate forms (see the following Section). APPLICATION OF THE THEOREMS TO THE EVALUATION OF INDETERMINATE FORMS
Indeterminate forms of various types usually are evaluated by procedures which are varied, apparently unrelated, often roundabout and frequently tedious. In problems requiring the evaluation of certain indeterminate forms, (2) or (3) may be employed. This application refers back to the Introduction, where it was observed that (2) and (3) assert that under certain conditions l i m f ( t ) and lim s£{f(t)} are equal for two pairs of values of t and s. In this use of (2) and (3), when the limiting value of the variable is 0 or ,:, an indeterminate form to be evaluated is written as lim f(t), lim f(t), - -
__
l
lim sF(s) or lim sF(s), whichever is appropriate.
q}+
t
',
In many instances, es-
pecially those involving higher transcendental functions, the indeterminate form when written as l i m f ( t ) will possess a tabulated Laplace transform or the F(s) in the indeterminate form written as lim sF(s) will possess a tabulated inverse Laplace transform f ( t ) ; in some instances the indeterminate form written either as lim f(t) or lim sF(s) will be one-half of a tabulated function-Laplace transform pair. W h e n an indeterminate form is written as lim f(t) but £ {f(/)} is tabut
~0+
lated and lim s£{f(t)} is more readily evaluated; or conversely, when an indeterminate form is written as lim sF(s) but f(t) = £ - ' { F ( s ) }
is tabu-
lated and lira f ( t ) is easier to evaluate; and the hypotheses of (2) are met, then (2) can be employed to evaluate the given indeterminate form. An analogous statement applies to (3). Although indeterminate forms at 0 and :~ may be evaluated by L'Hgpital's rule, when the method described can be used it produces desired results in a uniform manner, directly and easily. The method changes the evaluation of a complicated indeterminate form to the determination of a comparatively simple one, by utilizing tabulated results originally derived for a completely different purpose. We illustrate the applicability of the method with some examples. There is no advantage to replacing an indeterminate form by one which is equally, or even more, complicated, so the examples chosen have this desired simplification. We first use (2) directly to evaluate an indeterminate form of type lim f(l) by evaluating instead the simpler lim sF(s). Then we employ (2)
x 74
BERNARD RASOF
[,J.V.l,
in its converse form to evaluate some indeterminate forms of type lim sF(s) by evaluating instead l i m f ( t ) . Finally, we employ both the direct and cont ~0+
verse forms of (3) to evaluate other indeterminate forms. If Jl' (0+) is not known, the indeterminate form l im [J1 (at)/t] cannot ,-0+ be evaluated by L'HSpital's rule. However, with the method presented here, l i m [ J ~ ( a t ) / t ] = lim s{a[s + (s 2 + a2)V2]-~} = a / 2 which m a y be verified using more elaborate techniques. In the first illustration of the converse application of (2), we use the well-known fact that lim (sin a t ) I t = a. From (5), with a m i n i m u m of ,40+ labor we have lim[s arc tan (a/s)] = lim ( s i n a t ) / t = a s--~
t--0+
lim s[ln (1 + 4ag/s2)] = 4 lim ( s i n 2 a t ) / t = O. s--~
t +0+
Some indeterminate forms evaluated in this way require one or more uses of L'Hbpital's rule on indeterminate forms much simpler than the originals; thus, lira s[ln (s + b) - In (s + a)] = lim (e-"~ - e-b~)/t = b - a lira s ,~.
{s
In
P2 + a2~ ks2 +
b a} + 2b arc t a n - - - 2a arc tan
b~j
s = 2 lim
t~0+
cos at - cos bt
= b2 -
a 2.
12
T h e direct application of (3) to the evaluation of indeterminate forms is illustrated by determining lira {[exp(a2t)] E r f c ( a W 2 + b/2t~/2)}, which ordinarily would require four applications of L'H6pital's rule. From (5) we obtain £{[exp(aZt)] Erfc(at 1/2 + b/2t'/2)} = e - ° b e x p ( - b s l / 2 ) / [ s ' / 2 ( s '/2 + a)].
(11)
The left-hand m e m b e r of (11) exists and the right-hand m e m b e r has no poles, so (3) m a y be applied to (11) and leads directly to the result lim {[exp(a2t)] E r f c ( a W z + b/2t~/2)} = lira s{e -"~ e x p ( - b s l / Z ) / [ s ' / 2 ( s '/2 + a)]} -- 0. s~0
The converse application of (3) to the evaluation of indeterminate forms is illustrated by determining lira [s-V2e"/" Erfc(a/s)V2], which is difficult to s~0
Sept., ,962.1
LAPLACE TRANSFORM THEORY
I75
determine by s t a n d a r d methods. Again, in (5) we find
s-3/Ze°/'Erfc(a/s) '/2 =1~ {[1 - exp(-2[at]~/2)]/(a~r)'/2}.
(12)
T h e right-hand m e m b e r of (12) exists and the left-hand m e m b e r has no poles; so (3) m a y be applied, with the result that lira s[ s- 3/:~e~/" Erfc (a/s) ~/2] = lira {[1 - exp ( - Z[at]~/2 ) ]/ (azr) '/2} = (a~r) ,/2. GENERALIZATION OF THE INITIAL- AND FINAL-VALUE THEOREMS
W e generalize the initial-value t h e o r e m so that initial values of derivatives of the solution to an initial-value problem, as well as that of the solution, m a y be recovered and confirmed before inverting the response transform.
Theorem II. Generalized Initial- Value Theorem. I f f ( t ) , f ' ( t ) , f " ( t ) , . . . , have Laplace transforms, F(s) = ~ {f(t)l, and the lim exists, then lim s
"F-(s) -
s ° k-'f(~)(O+
f("+~)(t)
= limf(~)(t).
(13)
Proof: Assume_ T h e o r e m II_to be true for some n > 0. L e t f ( t ) = g'(t), so that F(s) = sG(s) - g ( 0 + ) , G(s) = • ]g(t)}; then (13) b e c o m e s lim~ .~ s f "+' G-(s) - s"g(O+) - y " S"-k-'g(k+l)(0+
= limgC"+1)(t).
(t4)
t ~0~ k=O
After a change in notation fromg, G to f, F, respectively, (14) becomes lim s
"+iF(s) -
~
s"-'f (k)(O+ k=0
= lim f("+')(t). /~0+
Thus, if (13) holds for n it also holds for n + 1; and the initial-value t h e o r e m (2) asserts that (13) is true for n = 0: hence (13) is true for n = 1,2,3,.... T h e final-value t h e o r e m also can be generalized; the essential result is
Theorem Ill." Generalized Final- Value Theorem. If f (t), f'(t), f " ( t ) , . . . , f("+'~ (t) have L a p l a c e transforms, F(s) = • {f(t)}, and s"+lff(s) has no poles on or to the right of the i m a g i n a r y axis in the s-plane, then lim s "+1F(s) = l i m f f "1 (t). s~0
I
(15)
~
Proof: A s s u m e that T h e o r e m III is true for some n > 0. Let f(t) = g'(t); then as before F ( s ) = sG(s) - g ( 0 + ) , G-(s) = £{g(t)}" and (15) becomes
1 76
BERNARD RASOF
[J.F.I.
lim . < + 2 G ( s ) = lim g("+'> (t). T h u s , after changing from g, G to f, F, if (15) holds for n, it also holds for n + 1. But (3) shows (15) to be true when n = 0; therefore (15) holds for n = 1,2,3,.... APPLICATION OF THE GENERALIZED INITIAL-VALUE THEOREM
Heaviside (6) and C a r s o n (7) o b t a i n e d M a c l a u r i n expansions of ~-~ {F(s)} directly from F ( s ) itself by expanding F ( s ) in negative powers of s and then e m p l o y i n g the transform-function pair £-~{s -/"+~/} = t " / F(n + 1), n > - 1 , on each term of the resulting series. W h e n F ( s ) is a rational function, this p r o c e d u r e requires e x p a n d i n g a polynomial in s 1 for several positive integral exponents a n d collecting the coefficients of e a c h s ~,m > 0. Thus, the H e a v i s i d e - C a r s o n p r o c e d u r e for obtaining the first few terms of the inverse L a p l a c e t r a n s f o r m of F(s) = 1 / ( s 5 - s 4 - s ~ - 1) requires rewriting F ( s ) as F(s)
= 7 7 ( 1 - s-' - s - 2 -
s -5) = s -5
(s - 1 + s - 2 + s - s ) k,
(16)
k=0
e x p a n d i n g (s -1 + s -2 + s-S) k for k = 0, 1, 2 , . . . , and collecting coefficients of corresponding powers o f s -1. W i t h the factor s -5 in (16), the coefficient ofs -~° necessitates e x p a n d i n g (s -1 + s -2 + s-5) k fork = 1, 2, 3, 4, 5. T h e o r e m II provides a n o t h e r w a y of generating these coefficients, also directly from F ( s ) itself. W h e n the hypotheses of that theorem are fulfilled, the coefficients f(k> ( O ) / k ! of the M a c l a u r i n expansion of f ( t ) are given explicitly as f(t)
=~... f ( k[> ( l0 ) /!k = m ..,k -~,1, k =0
,
s
kF(s)-
k =0
Y'~s*-~-lffJl(
tk
j =0
m
W i t h F ( s ) = 1 / ( s 5 - s 4 - s 3 - 1 ) , f ( 0 ) -- 0 from (2). Applying Theo r e m II to this F ( s ) _ t h e next several coefficients in the M a c l a u r i n expansion o f f ( t ) = £ - l { F ( s ) } are lim f'(t)(=f'(O+))
= lim f"(t)(=f"(O+))
t ~0+
= lim f'"(t)(=f'"(O+))
t~O+
t--~O+
lira f~v (t) = fir ( 0 + ) t ~0+
= lim s [ s 4 F ( s ) = lim [ s S / ( s 5 -
-
s3f(O+) s4 -
s3 -
-
s2f'(0+) - sf"(O+)
1)] = 1
-f"'(0+)]
= 0
Sept., f96:z.J
LAPLACE TRANSFORM T t t E O R Y limfv(t)
= fv(0+)
-- l i r a s [ s S F ( s )
= lim s{[sS/(s ~limf~'(t) t
Thus,
f(t)
the =
= fv~(O+)
= lims{[s~/(s
'0+
s
first three
£~ ' { 1 / @
5-
s3 -
5-
-/v(0+)]
s3 -
s4 -
1)] s 3-
1} =
1)] -
1
s -
1} = 2.
~ •
non-vanishing s" -
s4 -
I77
terms
of the
Maclaurin
expansion
of
1)} are
f ( t ) = t4/4! + ?/5! +
2t6/6!.
REFERENCES
(1) RULE V. CHURCHILL,"Operational Mathematics," New York, McGraw-Hill Book Co., Inc., second edition, 1958. (~) Louis A. PIPES, "Applied Mathematics for Engineers and Physicists," New' York, McGrawHill Book Co., Inc., second edition, 1958, Chapter 21. (3) MURRAYS. GARDNERANDJOHN L. BARNES,"'Transients in l.inear Systems," New York..Iulm Wiley & Sons, Inc., 1942. (4) E. C. TITCHMARSH,"Introduction to the Theory of Fourier Integrals," Oxford, Oxford University Press, 1937. (5) "Tables of Integral Transforms," Vol. I, compiled by the Staff" of the Bateman MantJs(ript Project, New York, McGraw-Hill Book Co.. Inc., 1954. (6) OLIVERHEAVlSmE,"Electromagnetic Theory." New York, Dover Publications, 1950. (7) JOHN R. CARSON,"Electric Circuit Theory and the Operational Calculus," New York, McGrawHill Book Co., Inc., 1926.