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The Development of the Laplace Transform, 1737-1937 L Euler to Spitzer, 1737-1880 MICHAEL A. B. DEAKIN

Communicated by C.

TRUESDELL

Abstract This paper, the first of two, follows the development of the LAPLACE T r a n s f o r m from its earliest beginnings with EULER, usually dated at 1737, to the year 1 8 8 0 , when SPITZER was its major, if himself relatively m i n o r , protagonist. The coverage aims at completeness, a n d shows the state which the technique reached in the h a n d s of its greatest e x p o n e n t to that time, PETZVAL. A sequel will trace the d e v e l o p m e n t of the m o d e r n theory from its beginnings with POINCARI~ to its present form, due to DOETSCH.

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Definitions and Notation . . . . . . . . . . . . . . . . . . . . EULER'S Contributions . . . . . . . . . . . . . . . . . . . . . The Work of LA~RANGE . . . . . . . . . . . . . . . . . . . . LAPLACE'SStudies . . . . . . . . . . . . . . . . . . . . . . . Systematization by LACROIX . . . . . . . . . . . . . . . . . . FOURXErtand POISSON . . . . . . . . . . . . . . . . . . . . . Theoretical Developments by CAUCHY . . . . . . . . . . . . . . A Posthumous Paper by ABEL . . . . . . . . . . . . . . . . . L,OtrVILLE'SResearches . . . . . . . . . . . . . . . . . . . . GRtrNERT'SIntegrals. , . . . . . . . . . . . . . . . . . . . . MURPHY'S Inversion Formula . . . . . . . . . . . . . . . . . . Two papers by LOBATTO . . . . . . . . . . . . . . . . . . . . A Systematic Theory-PEYzVAL'S Work . . . . . . . . . . . . . Die Priorit/its-Ansprtiche . . . . . . . . . . . . . . . . . . . The Work of BOOLE . . . . . . . . . . . . . . . . . . . . . . A Paper by RIEMANN . . . . . . . . . . . . . . . . . . . . . . . The Years 1860-1880 • • . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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344 345 346 350 352 360 361 364 368 369 372 372 373 376 381 382 383 384 384 385

344

M . A . B . DEAKIN 1. Introduction

The LAPLACE Transform is now widely used in mathematics itself and in its applications, particularly in electrical engineering. It is employed in the solution of differential equations, of difference equations and of functional equations; it allows ready evaluation of certain integrals and claims connection with number theory via the DIRICHLET Series and the RIEMANN ~-function - all this in addition to the interest that the transform itself holds within functional analysis. The modern version of the LAPLACE Transform is most correctly attributed to DOETSCH, as will be shown in a subsequent paper, and is a relatively recent development. We may date it most conveniently from the publication of his Theorie und Anwendung der Laplace-Transformation [18] in 1937. Its ready acceptance into undergraduate mathematics courses is a remarkable story, to be taken up elsewhere. However, the theory has a long history, dating back to EULER, whose first paper on the subject [20] appeared in 1737. This paper and its sequel concern themselves with the two hundred years between this paper and DOETSCH' book. The year 1880 is a natural place at which to interrupt the account as it is here that POINCARt~ introduces to the theory the full power of the calculus o f residues. Previous histories of the LAPLACETransform have been published. The earliest is by SPITZER in the introduction to his Vorlesungen iiber lineare DifferentialGleichungen (1878) [89]. This is relatively complete for its time, but is written tendentiously, and with a view to establishing LAPLACE'S priority. It neglects (one is tempted - see Section 3 below - to say "deliberately neglects") the prior work of EULER. PINCHERLEgives two accounts [71, 72], both predating DOETSCH, and neither of them at all complete. A recent paper by PETROVA [64] discusses the work of LAPLACE and of ABEL, but is incorrect in many of its details. The best account extant is that of GRATTAN-GUINNESS[31 ], although this, largely on account of its brevity, is also incomplete. DOETSCH' Kapitel 2 and Historische Anmerkungen [18, pp. 6-12, 404-424] and the associated bibliography provide much useful material, particularly for the second of the two periods (1880-1937), and there is also valuable material to be found in works by SCHLESINGER [78, 79]. Of the many potted histories to be found in modern texts, mention need only be made of the best - that of CARSLAW • JAEGER [7]. It is typical of most such accounts in that it concentrates almost exclusively on a line of development beginning with HEAVISIDE,and ending with VAN DER POE, and thus neglects entirely the mainstream of the theory's development. As a result of this eccentric approach, it sees the development of the LAPLACE Transform as an attempt to rigorize the Calculus of Operators, although, in fact, the two branches of theory are quite distinct for much of their history. This paper and its sequel attempt to provide a full account of the development of the LAPLACE Transform, not only to complete and correct earlier histories, but also to provide, in convenient form, a coherent narrative of both mathematical and human interest, Attention will focus primarily on the use of the LAPLACE Transform in the solution of linear differential equations, as that aspect of the theory has proved most fruitful in its development. It is hoped that a full account

The Laplace Transform, 1737-1880

345

will be useful pedagogically, for example to demonstrate the power of some of the earlier approaches. (These indeed work, as I have pointed out elsewhere [17], in cases where our modern technique fails.)

2. Definitions and Notation

Let F(t) be a function of the independent variable t. Its Laplace Transform is then defined as co

.g,e{F(t)} = f(p) = f e-ptF(t) dt,

(2.1)

0

provided the integral converges. The LAPLACETransform (2.1) is closely related to a number of other integral transforms, notably the p-multiplied Laplace Transform, or Laplace-Carson Trans-

form f(p) = p f e-Ptr(t) dt,

(2.2)

0

the Fourier Transform

f(p) ---- ~ e-iptF(t) dt u

(2.3)

O0

and the Mellin Transform oo

f(p) =

f

tP-XF(t) dt.

(2.4)

0

A number of closely related forms will not be given explicitly at this stage. In the above formulae, t is often taken to be real and F(t) is assumed to be a realvalued function, p however is best viewed as being complex. The most general integral transform to be considered will be one in which all variables are complex, so that we have

f(p) = f K(p, z) r(z) dz c

(2.5)

for some contour C in the complex z-plane. However, only the specializations of Equation (2.5) relevant to the study of the LAX'LACETransform proper will here be considered. There are close relations between Equations (2.1)-(2.4). If H(t) is the unit step function or HEAVlSlDE function

H(t)=

[10 if t ~ 0 , if t < 0 ,

then Equations (2.1) and (2.3) may be related by taking the FOURIER Transform of F(t) H(t) and replacing ip by p. Similarly the MELUN Transform may be reduced to the LAPLACE Transform by the change of variable t = eL It follows that some of the history of these transforms is relevant to the discussion of the LAI'LACETransform. It is unnecessary here to list the properties

346

M . A . B . DEAKIN

of the various transforms. These are now widely available. See, for example, texts by DOETSCH [18], WIDDER [94], CARSLAW t~ JAEGER [7], MCLACHLAN [59] or ERDt~LYI [19]. As applied in particular to a linear ordinary differential equation in F(t), a LAPLACE Transform solution allows a new equation to be set up involving instead f(p). In many cases, this is of lower order than the original and so may the more readily be solved. Other cases allow reduction of partial differential equations to ordinary ones to the same effect. The original equation may be solved if the integral (2.1) may be solved for F(t) when f(p) is known. The key to this is the FOURIER Integral Theorem

F(t) ---- ~ -~o -

F(u) e ip(t-u) du dp,

(2.6)

which allows us to solve Equation (2.2) by

F(t) = ~ - o o

e'ptf(P) dp.

(2.7)

A relatively elementary argument thus allows the solution, or inversion, o f Equation (2.1) by 1

v+ioo fi ePtf(p) dp

(2.8)

where the real part of V exceeds the real parts of all singularities off(p). Similarly the LAPLACE-CARSONand MELLIN Transforms may be inverted, giving in the latter case 1

F(t) ----

7+i~

f

2eri r-ioo

t-Ptf(p) dp.

(2.9)

The history of these formulae will also occupy us later. It will be noted that the integral (2.8) in particular is a contour integral in the complex plane. Its contour is thus not rigidly determined to be the line Re (p) = 7, but may take on many forms - even, after transformation, in certain cases the form of a real definite integral. Thus, the solution of the original differential equation is given as a contour or definite integral. Historically the development begins at this point - the search for solutions to differential equations in the form of definite integrals of certain types. It is this line of research that SPITZER [89] regarded as giving priority to LAPLACE in the development of the method. There are, however, important prior contributions by EULER and LAGRANGE, as we shall see.

3. Euler's Contributions A number of investigations by EULER have been, from time to time, cited as bearing on the subject. Thus, ENESTRt3M,in a footnote to an account by P1NCHERLE [72], mentions two [20, 25]. PETROVA [64] lists the first of these and two others

The Laplace Transform, 1737-1880

347

([22, 27]. GRATTAN-GUINNESS cites one of ENESTR6M'S tWO references and two others [21, 24] and one might add to these another [26] of marginal relevance. SPITZER [89] mentions (as irrelevant!) a further paper [23] whose content is most germane to the account. The paper de Constructione Aequationum [20] aims to construct differential equations which shall be soluble by known techniques. Here EULER considers the transformation z = f ea~ X(x) dx.

(3.1)

The integral (3.1) is, however, more explicitly written in modern notation as x

z(x; a) = f eat X(t) dr,

(3.2)

0

in other words rather as an indefinite than as a definite integral. This is clear from the succeeding manoeuvers which involve various suppositions as to the behaviour of X and differentiation with respect to x. In other cases, differentiation with respect to a is performed, so that a genuine transformation is involved, but the viewpoint is one considerably removed from that of subsequent work on integral transforms of the type given by Equation (2.5). The paper of 1753 [21] is of marginal interest. It gives in a systematic fashion the solutions of second order linear differential equations with constant coefficients. These take on forms reminiscent of the LAPLACETransform, but involve indefinite, rather than definite, integrals. They also relate closely to BOOLE'S [5] interpretation of certain formulae involving differential operators, but, even in this area, EULER'S work is perhaps best viewed as an anticipation of the later clarity achieved by CAUCHV [9, 12, 13, •4]. Of much more importance is the paper [23] dismissed by SPITZER [89] as not involving the LAPLACE Transform. True it does not - but SVITZER, in the same discussion, regards the key issue as the use of definite integrals in the solution of linear differential equations. He himself, in mentioning the work at all, makes eminently clear the case for EULER'S priority. However, it was not to his taste to recognize this point, as he was by that time wholly committed to his championship of LAPLACE. Here EULER seeks to solve the differential equation (in his notation) A y du 2 + (B + Cu) du dy + (D + Eu + Fuu) ddy = O,

(3.4)

which he sets about doing by putting y = f P dx (u + x) n,

(3.5)

where P is a function of x. The ensuing discussion has much in common with his later work in Institutiones Calculi Integralis, where the transformation (3.5) is also discussed. However, the question of the limits of integration is not entirely clarified in this paper. PETROVA [64] also mentions as relevant a further paper by EULER [22] on the propagation of sound. It is quite unclear to me why she regards this paper as pertinent. Possibly she confused it with the approximately contemporaneous

348

M.A.B. DEAKIN

discussion by LAGRANGE [42] on the same topic. The significance of this paper is discussed in Section 4 below. The first volume of Institutiones Calculi Integralis [24] appeared in 1768 and contains in Chapter IV Some work on the integration of differential equations using exponential forms, which are, however, indefinite. Further on, in Chapter V, a number of integrals are evaluated, including forms such as f ea~ sin 2 4~d~b (in modern notation), which are dearly relevant to the study of the LAPLACE Transform, but which do not for EULER'S purposes form part of it. By far the most important of EULER'S papers on the LAPLACETransform is Section 1053 in Volume II of his Institutiones Calculi Integralis [25, esp. pp. 242-3]. Here he considers the transformation (here given in a notation differing only trivially from his): y(u) = f er(u)o(x) P(x) dx. b He then forms the expression L(u) d2y ~u 2 + M(u) ~dy -}- N(u) y,

(3.6)

(3.7)

which is readily found to be f erQP(N + (MK' -+- L K " ) Q -}- L(K') z Q2) dx.

(3.8)

b

This expression he seeks to evaluate in the form x~a

f

d(e K("~oO:)R(x)),

(3.9)

x=b

where R(b) e r(u)°(b) : O.

(3.10)

Equating the integrands of Expressions (3.8), (3.10), he then has dR-~KRdQ:Pdx(N~-(MK'+LK")Qq-L(K')2QE}.

(3.11)

Under the conditions L(K') 2 : A + o~K, M K ' -}- L K " :- B -}- ilK,

(3.12)

N = C -}- ~,K, where A, B, C, ~, fl, ~ are constants, he then replaces Equation (3.11) by the pair of equations dR = P dx(C q- BQ + AQ2),

R dO. = P dx(~, + ~Q + c,Q 2). It follows immediately that dR C -+- BQ + AQ 2 --~ = ~' + flQ + ocQZ ,

(3.13)

(3.14)

The Laplace Transform, 1737-1880

349

which may be integrated to give R. Then, by Equations (3.13),

RdQ

Pdx =

V -k flQ -k ~xQz

(3.t5)

and hence x=a

y(u) =

f

RdQ (3.16)

e ~°

Equation (3.16) thus gives the solution of the equation

L~uz -k M

-? Ny = U(u)

(3,17)

if

U(u) = R(a) ejr(II)°(").

(3.18)

It is not a difficult matter to invert this analysis in many important cases. Two examples are given in my earlier study [17]. Particularly in the case U(u) = 0, we have some latitude in the specification of the function K(u), the simplest nontrivial choice being X(u) = u.

Notice, in particular, that if L, M, N are linear in u, then Conditions (3.12) are satisfied under this choice of K. Almost certainly, this is the basis of WEILER'S [93] claim that EULER integrated the equation

(a2 -~- b2x) y" -~ (a, -~ blX) y' -~ (ao + box)y : 0

(3.19)

in the form U2

y = f eilxV(u)du.

(3.20)

IIl

(The notation has here been altered to accord with WE~LER'S.) SPITZER [89] does not discuss this point, contenting himself with his attempts to show the irrelevance of EULER'S previous paper [23]. WEILER, however, has clearly demonstrated EULER'S priority over LAPLACE in SPITZER'S own terms. Indeed EULER considers a number of other integral transforms in the same discussion (see Sections 1017-1058 of Institutiones Calculi Integralis). SPITZER is correct in stating that WE1LER gives no reference in EULER for his claim and it is indeed possible that SPITZER did not know of the above passages. His otherwise extremely thorough scholarship and his vigorous espousal of LAPLACE'S priority, however, mean that we cannot overlook the possibility that the omission was deliberate. Furthermore, SPITZER must also have overlooked, ignored, or dismissed as unimportant, LAPLACE'S own mention of EULER [49, p. 88], although here it must be said that LAPLACE'S vague form of reference ("... j'ai insist6 particuli6rement sur ces passages qu'Euler consid6rait en m~me temps que moi, et dont il a fait plusieurs applications curieuses, mais qui n'ont paru que depuis la publication des M6moires cit6s") leaves the reader in considerable doubt as to what he had in mind.

350

M . A . B . DEAKIN

Of EULER'S subsequent publications little need be said. The investigation of 1781 [26] considers integrals of the form oo

f f(x, a) dx. 0

Particularly, he evaluates

• e--kyyn-1 dy,

(3.21)

0

for complex k. He thus evaluates a number of LAPLACEand FOURIER Transforms, and, in particular, produces a theory of the /'-function co

r(u) = f e-Xx u-' dx.

(3.22)

0

This is, in the light of modern knowledge, a necessary, but hardly a sufficient condition for a theory of the LAPLACE Transform as now taught. Exactly similar comments apply to an investigation in 1812 by BESSEL [2], who, but for the fact that he was briefly credited with the LAPLACE Transform, would not otherwise figure in this discussion. Finally, a paper written in 1779 and published in 1813 [27] considers the solution of partial differential equations. The equations involved are linear and solutions involving exponentials are discovered. Sums and integrals of such forms are clearly allowable as solutions also. However, there is little in common between this theory and the use of definite integrals involving parameters. Certainly LACROIX in his Traitd des diffdrences et des sdries [40, 41] followed modern practice in regarding the two as quite distinct. (See Section 6 below.)

4. The Work of Lagrange Three investigations by LAGRANGEadvance the theory. The first [42], of which a translation by CROSS [16] is now available, concerns the propagation of sound. Here LAGRANGE considers the wave equation

d2z dt 2

d2z =

c

dx 2

(4.1)

(LAGRANGE'S notation). This equation is multiplied on both sides by M dx, where M is an arbitrary function of x and the result is then integrated between x = 0 and x = a. The assumption is made that z vanishes when x = 0 and since

fd2z dz dM r d2M dx z M dx =-~x M -- Z-d-S+ j Z~xz dX

(4.2)

we set M(0) = 0 to achieve consistency at the lower limit. The upper limit is also chosen so that z vanishes when x = a. The term

dz

dxx M is also made to vanish there by suitable choice of M. Hence, in the equa-

The Laplace Transform, 1737-1880

351

tion

/ + Jf"- ~d2z s M dx = e (dz._~xM -- z dMt-if-x-x

r. d2M dx, cjz-9-U

(4.3)

which we would write with the limits x = 0, x = a, the first term on the right has been made to vanish and we are left with f d2z

r

d2M

--~ M dx = c J z-hTx~ dX,

(4.4)

where the integrals are definite integrals on the range (0, a). As M is still largely unspecified, he assumes

dZM dx z

--

kM

(4.5)

and so finds

f d2z dtz M dx = k c f z M d x .

(4.6)

s =fzMdx

(4.7)

He now sets

and so reaches d2s

dt---i=kcs.

(4.8)

The result of this investigation has thus been to transform a partial differential equation to an ordinary one by means of what we would now term a FOURIER Transform. The integration by parts (4.2) and the subsequent use of limits chosen to cause the vanishing of the integrated terms is a direct anticipation of the technique adopted by LAPLACE [47] and PETZVAL [65, 67] and widely used before DOETSCH introduced the modern approach to the LAPLACE Transform. A later discussion by LAGRANGE [43], also translated by CROSS [16], applies similar ideas to the three-dimensional case. Here three simultaneous partial differential equations are involved and three multipliers L, M, N (each functions of the three spatial variables X, I1, Z) are used. The technique involves reduction of volumetric integrals over a box to surface integrals over the ends and the imposition of three conditions analogous to Equation (4.5). The work is not entirely successful as the three conditions, to quote CROSS' translation ,"are not susceptible to integration or they require other methods which we do not know". LAGRANGE does, however, consider certain special cases, in particular the exponential substitution

L = Ae (px+qv+rz)'('#, M = Be (px+qr+rz)(#, N

=

Cc (px+qY+rz)l/-~,

(4.9)

352

M. A, B. DEAKIN

where A, B, C, p, q, r are constants. This is a clear case of the use of a LAPLACE Transform, although LAGRANGEnotes that the solutions so produced are inapplicable to the theory of sound propagation. The third of LAGRANGE'Sworks on the subject [44] has a quite different source. GRATTAN-GUINNESS [31] draws attention to an investigation of integrals of the forms

f yaXdx

r X(x) ax

and

J ~

dx.

The former can be expanded as

f ya x dx

-- a x ]~g a

dx (log a) 2 + dx z (log a) s

"'" + const.

The series terminates in the case y = x m, and in some other cases is useful for series expansions. Similar comments apply to the latter form, which is in fact identical. This work was known to LAPLACE, some of whose investigations have a similar flavour.

5. Laplace's Studies O f LAPLACE'S works on the subject, the earliest is a paper on series [46]. Here he introduces the notion of a generating function for the sequence whose typical member is Yx. The generating function is then u(t) =

2~ yx tx.

(5.1)

x--O

He is concerned with a number of questions - initially the problem of interpolation. A Yx defined over all positive x would allow a continuous analogue of Equation (5.1)

u(t) = ~f-y~t x dx,

(5.2)

0

an integral transform related to the LAPLACE Transform. Although LAPLACE does not take this route, we find later in the paper, in Section XVIII, a discussion of the partial differential equation cqZu ~u ~u 9s ~s-----(+ m ~s -~ n-~s~ + lu = 0.

(5.3)

He now sets, for arbitrary functions 4~(s), ~0(s), 4 h ( s ) = f 4~(s)ds, 4 2 ( s ) = f dpl(s)ds, etc. and similarly defines ~pl(sx), ~p2(Sa) etc. and seeks a solution of the form 12 =

A q ~ ( S ) -~-

A(1)qbl(S)

it- A(Z)q~2(s ) -~- ... -}- B~0(sl) + B(1)'~I(S1) -~- B(2)~2(SI) ~-i .--,

(5.4)

The Laplace Transform, 1737-1880

3 53

where the coefficients are functions of s, sl. This produces a set of equations for the coefficient functions, whose solution is found to be A(~) B ~)

=

e -msI-ns

(5.5)

(ran - - 1) ~ s~,

e m*,-ns (ran - - l) ~

S F~ ,

in the case where m, n, l are constants. He thus reaches . = e-' .....

If

+(z) 3 [sl(s - z)l dz + f ~ ( z ) J [ s ( s l

}

- - z)l dz ,

(5.6)

(mn - - l) 2 s~(s - - z) 2 ÷ . . . .

(5.7)

0

where d [ s l ( s - - z)] = 1 + (ran - - 1) s l ( s - - z) 47

2!

Although the LAPLACE T r a n s f o r m is not directly involved here the occurrence of the exponential factor m a y have suggested the study of forms such as the expression ; e -*x +(x) dx in place of the more obvious t r a n s f o r m of Equation (5.2). 0

LAPLACE'S later M d m o i r e sur les approximations des f o r m u l e s qui sont f o n c t i o n s de trks grands hombres [47] is a remarkable p a p e r in which the g r o u n d w o r k of the subject is laid. The relevant sections begin with the start of Article II (Section V I I I ) where solutions to difference and differential equations are considered. LAPLACE begins with the equation involving finite differences S = Ays + B A y ,

(5.8)

@ CAZys @ ...,

where S, A , B, C, etc. are functions of s and y is an u n k n o w n function of that variable. There are finitely m a n y terms on the right, but the n u m b e r of these need not be specified at this stage, as the method is quite general. A , B, C, etc. are assumed developable as power series in s, so that, for example, A = a + a(1)s + a(Z)s 2 -[- a(3)s 3 -~- . . . .

(5.9)

y, is assumed to be of the form y~ = f e -*~ 4(x) d x ,

(5.10)

where the integral is a definite integral but the limits are not yet specified. Then, in effect, he reaches Any, = f e-*X(e - x - - 1)" ~b(x) d x , (5.11) and substitutes into Equation (5.8). The result is, in effect, s = f e p d x { [ a + b(e - ~ - - 1) + e(e - x - - 1) 2 + ...] de sx d x [a(l) -+- b ° ) ( e - X - - 1) q- c°)(e - x - - 1)2 -~- ...]

(5.12)

d2e-SX

}

+ --h-SY-..2[a{2) + b~2)(e - x - - 1) + cC2)(e-~ -- 1)2 + ...] q- . . . . ax

354

M . A . B . DEAKIN

An analogous form is also produced for the kernel x s, this applying to slightly different forms for A, B, C, etc. In both cases, the kernel is represented by dy, so that in each there results

s=f+&

(

My+

N dr}y

~d2@

-d3dy

)

dx @ r-d~x2 + Q--d~x3 + . . . .

(5.13)

M, N, P, Q, etc. being functions of x. To each of the second and successive terms an integration by parts is carried out sufficiently often to produce an integral involving r}y, the kernel, without differentiation, so that there results S = f dy M~b -- ~xx (N~b) -k ~

(P4~) -- ~

(Q4) dx

+ C + r}y N(h -- ~x(P4)) -k ~Z2 (Q4))z +

l 2,y

Pqb--~x(Q~b)+...

...

(5.14)

q-~[Q4~-...]+

....

Here C is an arbitrary constant, which we would nowadays absorb by using a different notation for the limits of the integration. LAPLACE now attacks Equation (5.14) by setting the integrand equal to zero to reach

d

dz

0 = Mq5 - - ~xx (Nqb) -5 ~

d3 (P~b) -- ~

(Q4)) + ....

(5.15)

with consequent simplification of Equation (5.14). F r o m Equation (5.15), q5 is to be determined, while Equation (5.14) now gives the limits of integration. LAPLACE also notes here that (in essence) Equation (5.15) is the adjoint of the Equation

Mr}y+

N d@

d2@

~ d3r}y

~x +P--~xZ +(2d---~+

....

0,

(5.16)

an observation now generally traced back to a later derivation by MELLIN [60]. In Section IX, LAPLACE directs attention to the matter of the limits of integration, first in the case where S = 0. Put C = 0. I f now @ and all its derivatives vanish for some value h of x (h being independent of s), we make progress in determining the limits. For the case @ = x s, h = 0 (at least for sufficiently large s), and for the case @ = e -~x, h = ~ (supposing s to be positive). He now seeks values of x which will satisfy the algebraic equations d d2 0 = N4, -- ff-£x(P40 + ~ (Q4,) - ..., d 0 = P4~ -- ~x (Q4~) -k ..., 0 = Qqb -- ...,

(5.17)

The Laplace Transform, 1737-1880

355

there being i such equations if i is the order of the differential equation (5.15), whose solution for 4) will contain i arbitrary constants. Since we have assumed S = 0, 4) is a linear combination of these constants each multiplying a corresponding particular integral. The first particular integral let us call, not following LAPLACE'S notation, 4)1 and substitute this into the first of Equations (5.17). Suppose this to be satisfied for x = q. The same process is repeated for the others, and a solution, which we would now write

n=lh

f° y4).(x)dx

(5.18)

emerges. The next sections consider the non-homogeneous case. The attempt is to find limits h, p on the integration in Equations (5.10) and its analogue for which dy = x'. Most of the ensuing discussion concerns that case, and will not be given here. LAPLACE also considers the use of the same method for differential equations, and remarks (at the end of Section XIII) that the kernel e -~x leads to similar, but often more convenient forms. Section XV begins the discussion of what is now known as the LAPLACE Equation

0 = V q- sT,

(5.19)

V, T being linear functions of y~ and its differences (or derivatives). The above analysis now yields

dby\ 0 = f 4) dx ( M + Iv " -~-x)

(5.20)

as before in the more general Equation (5.13). The differential case is considered explicitly in Section XVI and here the kerneI by = e -~x is adopted explicitly. This section is the one on which SPITZER [89] bases his case for LAPLACE'S priority and questions WEILER'S [92] attribution o f Equation (3.19), the second order differential case of Equation (5.19), to EULER. The method, however, is clearly implicit in EULER, as we have seen, although LAPLACE'Streatment is obviously more general and is considerably more developed. Later sections of the LAPLACE paper are also relevant. Section XXV considers explicitly both the transforms

y, : A f x%(x) dx

(5.21)

y, = A f e-~X4)(x) dx,

(5.22)

and

where A is a constant and the limits of integration are allowed to depend on s. Considerably more emphasis is placed on the form (5.21) than on the form (5.22). Formulae for differences and derivatives are then derived. For example, from Equation (5.22), he has

d"y~ ds" = (--1)" A f x'*e-SX4) dx @ ....

(5.23)

356

M . A . B . D~AKIN

the dots indicating the effect of dependence of the limits on s. However, the section immediately following assumes the limits 0, e~ of modern theory. (In this connection, note that LAPLACEin the original version uses ... for terms deducible algorithmically, and &c for terms produced by persistence of form. The editors of Oeuvres Complktes do not preserve this distinction.) This part of the study deals with the differential equation

dys S'~s -k iys ~- O,

(5.24)

where, as before, i is not 1/--1, but is in fact real. To investigate Equation (5.24), he sets Ys = f e-SXcb(x) d x ,

(5.25)

where the integral is definite but its limits are as yet undertermined. Substitution of this form into Equation (5.24) and an integration by parts produces [x~b(x) e-SX] bb, ~-

e -sx ich(x)

~xx (x~b(x)) dx z 0,

(5.26)

a

where I have used a, b for the unknown limits. Each term in Equation (5.26) is required to vanish independently, so the limits are given as 0, e~ to ensure the disappearance of the first term. The second term vanishes also, giving d

icb(x) = -~x ( x~b(x)) .

(5.27)

Equation (5.27) is both the adjoint and the LAPLACE Transform of Equation (5.24). The solutions of both equations are readily found by other methods, and we have dp(x) = A x i-1 (5.28) and (5.29)

Ys ~-- s - i .

(The constant of integration in the second equation is unimportant.) Hence oo

s -i = A

f

x i - l e -sx d x ,

(5.30)

o

and if, with LAPLACE, we set s = 1, we obtain, on substituting back, oo

s_ix

f xi-le o co

sx dx

(5.31)

f x i - l e - x dx o

an equation which we may now view as giving the LAPLACE Transform of x i-1 in terms of the /'-function.

The Laplace Transform, 1737-1880

357

It now follows f r o m Equations (5.23) (truncated because o f the choice of limits) and (5.31) that

f

xi-le-SX(e - x -- 1)~ dx

A~s_i = 0

(5.32)

f xi--lC x dx 0

If we n o w suppose that the function

xi-le-~(e -x-

1)n

(5.33)

is maximized for x = a, we m a y write

xi-le-SX(e - x -- 1)n = ai-le-Sa(e -a -- 1)" e -t2.

(5.34)

LAI'LACE n o w sets x = a + 0 and takes logarithms o f both sides. This gives a series for t 2 in powers o f 0:

t 2 = hOz + h'O 3 + h"O 4 Jr-

.

.

.

(5.35)

.

Values o f h, h', h", etc. m a y be f o u n d and expressions for the first three are given in terms o f a, which in its turn satisfies a transcendental equation f o u n d by setting the derivative o f Expression (5.33) equal to zero. Equation (5.35) m a y be recast by expressing 0 as a power series in t, the result being 0 =

~h (

h't

1

+

5 h ' 2 - - 4 h ' h ''

2h

)

t2 +

8h 3

.

.

.

(5.36)

.

This result is now substituted into Equation (5.34) and an integral taken from t = -- cx~ to t = cx~. LAPLACE has previously [47, X I X ] derived the expansion for the F-function f x ' e - x d x = i '+ ½e-'l/~-~ 0

1 + IN + '

'

(5.37)

a version of STIRLING'S Formula, f o u n d by transform methods using considerations based on the kernel x s. F r o m his new integration he determines f

xi-le-SX(e

-x

--

1) n dx

=

aa-le-Sa(e

-a

--

1)"

o

V ( +15 2

12hh + ) 16h 3 (5.38)

Expressions (5.37), (5.38) are now substituted into Equation (5.31) to yield a series expansion for Arts -i. The m e m o i r includes other calculations performed in similar vein. A l m o s t thirty years were to elapse, as he himself remarked, before LAPLACE returned to the subject [48]. Writing on definite integrals and their application to probability, he considers first a n u m b e r of integrals o f the f o r m cx~

f f ( x ) e -ax ax 0

358

M . A . B . DEAKIN

and evaluates these explicitly in cases where f ( x ) is both real and complex. The object at this point is the evaluation of the integrals themselves and to this end a number of ingenious techniques are used. Also treated are related integrals such as oo

l c_o ,_-x_ d

0

1 +x*

"

Later on, in Section V, there is an explicit use of a version of the LAPLACE Transform to solve the partial differential equation ~U 8r

~U 2U + 2#

~2U +

(5.39)

~/L2 ,

where I have written r in place of LAPLACE'S r'. The solution is obtained by setting (5.40)

U = f do(t, r) e -"t dt,

the integral being definite, but with initially unspecified limits. Then the transformed equation becomes, by the same argument that produced, for example, Equation (5.26) edo ado e-~- = t2do -- 2 t ~ -

(5.41)

with the limits so chosen that tdoe- m vanishes at both. The solution of Equation (5.41) is do =- e -g ~p

(5.42)

(~o being an arbitrary function of the specified argument), and one might now solve Equation (5.39) in the form U =

(5.43)

e - m + ~ ~ ~-~ dr. 0

However, LAPLACE takes a more interesting route and sets t = 2# + 2s t/-----i .

(5.44)

This gives _~

eZ,

ds,

(5.45)

where H is an arbitrary function. This form, but for its conversion to a real integral, is the inverse LAPLACETransform we would expect from the methodology embarked on by the substitution (5.40). Equation (5.45), closest to our modern form, is only produced from Equation (5.43) by attaching a negative sign to the value of ~/----~. Without this device, we reach the alternative form U=e_~2

_

f~ e _ S ~ l ,

s - - / z ]eZ / - r- 1

ds,

(5.46)

The Laplace Transform, 1737-1880

359

where _P is arbitrary. LAPLACE gives the value of U as the sum of the expressions (5.45), (5,40, and develops this by taking powers of e -2r. This leads to the equation 2e-~2 [

Q ( 1 ) ( 1 - 2 / , 2)

U = t/---~ 1 q-

e4r

Q(2)( i - - 4 / * 2 @ 4 / * 4)

e8r

@ L(O)#

@7

@...

L(')/* 1 ---~-/* + e6r ,

+.,

(5.47)

where the Q~i), L(o are constants depending on the initial conditions. The remainder of this interesting investigation does not involve the LAPLACE Transform explicitly and will not be pursued here. A number of orthogonality relations are derived, and a number of special cases are solved either explicitly or approximately. This interesting study was noted in connection with the LAPLACE Transform by both PETROVA [64] and GRATTAN-GIJIYNESS [31], but escaped the attention of LACROIX [41]. The subject recurs in Thdorie analytique des probabilitds [49] (1812), where LAPLACE begins the investigation in his Section 21 with Equation (5.1). He then sets (in modern notation) t = ico and multiplies the equation by e-iX% Writing U(co) for u(t) and integrating with respect to co yields:

f Ue -ix~' act =

f {yoe -'~' + yle -t(x-'> + ... q- Yx q- Yx+l d~' + . . . } do),

(5.48)

and a FOURIER Transform emerges if the integrals go from --Te to az. He finds

Yx = ~1 -=f U (cos cox -- i sin ox) do),

(5.49)

"'mais cette formule a l'inconv~nient d'introduire des imaginaires dont on peut d6barraser de la mani&e suivant". He then considers, without any further motivation, two versions of the MELLIN Transform:

Yx = f t-~-IT(t) dt,

(5.50)

y~ = f t-~T(t) dt.

(5.51)

and

Now, with Ay~ written for Yx+~ --Yx, Equation (5.51) gives

A~y:= f t-xT(t) (--~-- l)~dt,

(5.52)

where i is a positive integer. Moreover, if ~ becomes infinitely small we have the differential form 1

diyx-i f t-xT(t) (log--~-)

i

(5.53)

360

M.A.B. DEAKIN

The second chapter repeats work on Equation (5.8), and the third some of the work already noticed in connection with his investigation of 1810 [48]. In particular, there is another "close encounter" with the inverse integral as we know it today in his Section 34. Further on (his Section 40), the work on Equation (5.24) reappears. It is clear from all this that LAPLACE, although anticipated by EULER and LAGRANGEproduced a systematic body of theory that went far beyond anything those investigators had produced. It is clear that by 1782 LAPLACE was using transform methods, often involving the kernel e -sx, to produce results of considerable depth and difficulty.

6. Systematization by Laeroix LACROIX was a textbook writer par excellence, and in the best sense of that label. In 1800, he published his Traitd des diffdrences et des sdries [40] a sequel to his already extant work in two volumes Traitd du Calcul diffdrential et du Calcul integral (the capitalizations are those of the title page). In Chapter III of that work, LACROIXconsiders explicitly solutions of the form b

y ---- f V(u, x) du g

(6.1)

for linear differential equations. Already, he is aware, in his thorough bibliography, of the work of EULER in Volume II of the Institutiones Calculi Integralis (see Section 3 above). We find here a clear distinction between the use of definite integrals and EULER'S (1779) [27] use of indefinite integrals in the solution of partial differential equations, such as (LACROIX' notation) d2z

dxdx

--

az

(6.2)

in the form ay

z = f e"x+ -~ n4o(n) dn,

(6.3)

this integral being, in essence, an indefinite one, as the limits of n are arbitrary. Separated in space and in logic are the "Applications des form. f e -ux v du, f uXx v du, etc. gl l'integration des dquations aux diffdrenees et diffdrentielles", the subheading in the margin at paragraph 1134 of this work. These follow LAPLACE'S paper of 1782 and add little, unless one includes an attempt (attributed to PARSEVAL in the revision) to treat the two-dimensional wave equation

a2z

dt---5 - -

o2(a z

\ d x 2 ~- dy2] = 0 .

(6.4)

This proceeds via a double MELLIN Transform z = and direct substitution.

ff

mtnXpy4o(n, 4o) dn dp,

(6.5)

The Laplace Transform, 1737-1880

361

The revision, published in 1819 as the third volume of his Traitd du calcul diffdrential et du calcul intdgral (contenant un traitd des diffdrences et des sdries), gives a LAPLACETransform type of approach (now in Chapter VII): z =

ff

emt+"x+PYdp(n,p) dn dp,

(6.6)

and finds, readily enough, the obvious m 2 ---- aZ(n 2 + 192) to reach the solution z = f f co, V.~+~~+,x-py dp(n, p) dn dp + f f e -°' ~

+,x+,q 4)(n, p) dn dp.

(6.7)

By the time of the revised version, this too is banished (rightfully) to the section involving indefinite integrals. LACROIX' work is significant for two reasons. First, his texts were widely read. BOOLE, even had he not had the benefit of SPITZER'S [84] comments on LAPLACE'S priority, would probably have reached a similar conclusion, for we know [3] that he was greatly influenced by LACROIX. Thus, despite LACROIX' careful documenting of the earlier Eulerian contributions, LAPLACEwould probably have achieved the credit for the transform that now bears his name even without the intervention of SPITZER and POINCARI~(for the latter, see the sequel to this article). Second, the juxtaposition in LACROIX' treatise of the work of EULER and LAPLACEsets off starkly a vast difference in methodology in the two men's work. Whereas EULER (see the analysis beginning with Equation (3.6) above) chooses a form of solution and then seeks differential equations which it will solve, LAPLACE (see, as the prime example, the analysis beginning with Equation (5.8) above) is thoroughly modern in beginning with the differential (or difference) equation, and seeking to solve it. In a preliminary analysis [17] of this point, I commented of EULER'Sapproach: "This analysis, which appears to a modern reader to be presented backwards, exhibits a methodology which Euler used quite often". One may go further. My earlier study exhibits the production of particular integrals to second order linear differential equations by the Eulerian technique. LAPLACE (see, in particular, the analysis beginning with Equations (5.16), (5.17)) shows how these may be used in relatively general situations to construct general solutions of a previously given differential equation. This has no counterpart in EULER, and, although SPITZER nowhere adverts to this point, we may now view it, if we are so minded, as solid ground for awarding priority in the field to LAPLACE. The second edition of LACROIX' work gives references to PARSEVAL, FOURIER and POISSON, and also, for completeness one must presume, to works, not here relevant, by PRONY, AMPERE and PLANA.

7. Fourier and Poisson

The contributions of FOURIER and POISSON are important, but indirect. They relate mainly to attempts to solve the heat equation under various boundary conditions, and have been well summarized by GRATTAN-GUINNESS [30, 31], so that only a brief resume is required here.

362

M.: A, B. DEAKIN

':

FOURIER'S work on heat spanned the years 1805, when his first draft of a monograph on the propagation of heat was prepared, to 1822 when he published the final version, Thdorie analytique de la chaleur [28]. Relevant to our purpose is the one-dimensional heat equation, which we would now write •

t~/Y

_

8t

~2V -

-

~X 2'

(7.1)

appropriate to the conduction of heat in a linear body such as a bar. When the bar has finite length; the solution is possible as a FOURIER Series ......

v

~ (ar cos rx + bl sin rx) e ~m.

(7.2)

r=0

Both LAPLACE and POrSSON considered the case in which the bar is infinite, in which case, as we now know, an integrai analogue of Equation (7.2) is appropriate. An analogy with the formula for the coefficients of the FOURIER Series led FOURIER to the integral theorem OO

OO

- ~ F(x) = of dq cos qx of dx F(x) cos qx

(7.3)

(FouRIER'S Equation (e) of his Chapter IX). By modern standards, the notation is clumsy, and it is also true that the ~¢arious proofs offered lack rigor as we would now require it Indeed the question o f a rigorous proof Of the theorem which we would now write 1

~

co

F(x) = ~ J -~f F(q) cos p(q -- x) dq dp

(7.4)

stood over mathematics for a long time. • Hors coneours, PRINGSHEIM [75] attributes Equation (7.4) to CAUCHY, but despffe the poor notation; FOURIER had clearly stated the theorem. His proof [28, § 359] proceeds via a heuristic n o t unknown in present-day texts of engineering mathematics, and despite the fact that later authors such as PRINGSHEIMfound it deficient, and despite the controversy (for which see GRATTAN-GUINNESS'accounts [30,31]) that such work involved at its inception, no serious doubt was cast on the correctness of the result. For example, RIEMAN~ (see Section 17 below) showed no hesitation in using it to invert, in essence, the MELLIN Transform. , As will be made clearer in Section 10 below, LIOUVILLEalso was greatly influenced by this work. One of his studies comes closest to modern LAPLACETransform theory when he attempts (unsuccessfully) to discuss the passage from discrete to continuous spectra involved in the extension of the FOURIER Series to the FOURIER Integral [54]. This study, among others, owes a debt to PolssoN whom LIOtrVlLLE clearly admired, to the point of transcribing, with acknowledgement, w h o l e p a g e s from one of his papers [74]. ~ It .is also relevant to make explicit at this point that, despite the claim by PETgOVA [64], neither FOURIER nor POISSON made any use whatsoever of the LAPLACE Transform in the solution of differential equations. (A somewhat similar

The Laplace Transform, 1737-1880

363

remark applies to CAUCHY whose very real contributions to the theory will be discussed in the next section.) POISSON, however, comes close to the LAPLACE Transform at one point of his discussion of heat [74, Section 17]. Here the equations

p = f e-hYf(y) dy, 0

(7.5)

--oo

q = f ehYf(y) dy 0

are introduced (the second slightly misprinted), h being a constant with positive real part. Physical conditions imposed on the problem of heat flow have led POISSON to the equation

e-#Yd[e¢Yf(l -~- y)] = eeYd[e-eef(l -- y)]

(7.6)

and to a similar one for which/3, l are replaced by --/3', --I respectively. He multiplies both sides of Equation (7.6) by e -by and integrates from 0 to cxa to produce

(h q-/3)

f

e-hYf(l + y) dy ---- (h --/3) ~ e-hYf(l -- y) dy

0

(7.7)

0

and as

e-hYf(l--{- y) dy = eht (p -- /e-hYf(y)dy) 0

(7.8)

0

and

he can express the relation (7.7) in terms of integrals involving f(y). A similar calculation is carried out for the analogue of Equation (7.6) and the net result is a pair of equations in p, q, which he solves in the form ~0(h) P -- ~(h)'

~(--h) q ---- ~b(--h)"

(7.10)

where (b, ~p are explicitly given in terms of h, fl, fl', l and the function f. He now sets h = g -k z I/------]-and uses the above work to investigate the value of OO

f cos

(y -- x) zf(y) dy

--oo

via the complex exponential. The ultimate purpose of this investigation was t o examine the relation between the heat problem for a finite bar of length 2l and an infinite bar and to justify the use of FOURIER Series solutions, whose acceptability was questioned at the time (see GRATTAN-GuINNESS [30]). The first aim was not satisfactorily accom, plished and the second could have been done much more simply.

364

M.A.B. DEAKIN

8. Theoretical Developments by Cauchy CAUCHY'S contributions to the mathematics required for the modern version of the LAPLACETransform are extensive. In the first place, we have his work on the Calculus of Residues; second, there are his extensive analyses and applications of the FOURIER Transform; finally, it is necessary to mention his work on Operational (or Symbolic) Methods. Much of this is covered, briefly but thoroughly, in the study of CAUCHY'S work by FREUDENTHAL [29]. Inter alia, this deals with his studies of contour integrals and the others, with which these become intertwined, of the FOURIER Integral Theorem. It is only necessary here to deal in detail with a few of the most directly relevant papers. The first of these is a memoir of 1823 on the integration of linear partial differential equations. This paper begins with the FOURIER Integral Theorem (in the multidimensional case) and follows this immediately with the complex form. A version of the one-dimensional real form occurs as Equation (8) of his PremiOre partie, and in an unnumbered equation between his Equations (47) and (48), the modern form of the theorem is given. (This antedates by about four years the statement quoted by PRINGSHEIM [75].) It is the use of the complex version of the Fourier Transform that is most important for the purposes of this study. However, § IV of the paper makes use of the real version in solving systems of linear partial differential equations, and this study is taken up again in the second part where quite general equations are studied. Here, however, it is the complex Fourier Transform that is pressed into service. He writes the differential equation, in an unknown q~ as a4 _ a24 a"4 V4 = Vo4 + V, ~ -5 v 2 - ~ + ... + vm at" - 0

(8.1)

where the variable t is that corresponding to the highest derivative (of order in) and the operators 70, V1, etc. contain only the other n independent variables x, y, z . . . . . He now supposes initial conditions given for t ---- 0 in the form e4>

4~ = fo(X, y, z, ...), - ~ = f l ( x , y, z . . . . ),

etc.

(8.2)

and seeks a solution ~b of the form

+=

r i ;s

--cx)

ill r

--~

... Toe~'(~-~) ~/--Ye~(y-o f_--f

is #

Xfo(be, ~'. . . . ) do~ d# d{3 d~, . . . . -5 ( M - - 2) similar terms.

(8.3)

(I have slightly abbreviated CAUCHV'Snotation.) Here the limits be',/z" pertaining to the variable be are arbitrary, but for the restriction be' < x < #", and so on. To is supposed to be a function of t and the parameters o~, {3. . . . . The T i

The Laplace Transform, 1737-1880

365

satisfy eJT,. [1 e t i It=o = l 0

if i = j otherwise

(8.4)

for O --< i