INFINITESIMAL CALCULUS Accordingly But when li is infinitely small, we may, by analogy (see 21), assume
Hence <j>(x + a) = e aD <p(x) , as before. 86. Again, as in 84, representing the symbol c u by E, we may write ef(x) = Ef(x). Also, if A prefixed to any function of x denote the operation of taking the increment of that function when x receives the incre ment unity, we have /(*+!) -/(aO-A/fc). Accordingly E/(.r) = (l + A)/(x). And, by the index law, we have A 2 /(a-) + &c. Adopting the notation <t>(x)<= U X) <j>(x + K) = u x +K, &c., this leads to the following fundamental theorem of the calculus of finite dif ferences m Again, since A = E 1 , we have A. n u x = (E - l}"u x Hence, in like manner, n(n 1) . , , > A. n u x = u x+a - mi x+n . j + v - - u x+n - 2 + ... + ( - l)"t* . J. * t For example, A Vj" 1 = (x + 7t) m - n(x + n - l) m +...+(- l)"x m . Again, if A"0 m represent the value of A"a" when a:=0, we have , The numbers represented by the symbol A n O m , called the differences of the powers of zero, are of frequent occurrence in analysis, and their values can be readily tabulated from this series. 87. Again, since in which we suppose D x to operate on u only, and D 2 on v only, we infer that -!) d?v d*--u d"v This is Leibnitz s theorem, given in 27. This result can be extended to the nth differential of the product of any number of functions. 88. More generally, if y(x) represent any function of x, and if f(x) be any rational function, and we suppose D t operates on u only, and D 2 on ^(x) only, AVB have In like manner the equation
can be established. These expansions form the basis of Hargreave s well-known memoir on the " Solution of Differential Equations" (Philosophical Transactions, 1848). Hargreave observes that on mere inspection of these results it is apparent that if D be substituted for x, and - x for I), the former equation transforms into the latter. Hence, in any differential equation and in its symbolical solution, if the foregoing substitutions be made we shall obtain another form, accompanied with its symbolical solution. This principle was applied by Hargreave to the solution of several classes of differential equations. 89. Again, if in Leibnitz s theorem we make v = e ax , we get
1.2 Hence we readily infer that, if /() represent any function involv ing only positive integral powers of a, we shall have f(D + a)u = c Again, if this bo transformed by assuming e x =y, we have ^ = y t UnC and / d dii f d f d TS/ . . ( -j- )tt = -.- I -r- }u= I 2/-r- ) Dt*. axj dx dy J dyj Hence the foregoing result may be exhibited as follows : This may be written f(xD + a)u = x~ "f(xD}x a u . 90. The interpretation of negative and fractional powers of a symbol of operation is a subject necessarily suggested by the intro duction of such symbols. We pass over all allusion to the case of fractional powers, as no satisfactory theory for their interpretation has as yet been arrived at. The interpretation of an integer negative power of a symbol is easily established, and is in all cases of the nature of an inverse problem. For instance let IT be a symbol of operation such that 7T = V , then, if v be given and u unknown, we may write u = Tr~ l v , and the problem contained in the inverse symbol of operation will be answered when, by any process, we have determined u so as to satisfy the equation TTU = V, or irir- 1 v = v. In other words, we define the inverse symbol ir" 1 to be that which the direct operation ir simply annuls ; and this is in accordance with the analogy of ordinary algebra. For example, since D/(a?) =f (x), we write D - f (x) =f(x) , and the symbol D - * is equivalent to an integration. In like manner D ~ " is equivalent to n successive integrations. Similarly the symbol (D + a)" n is regarded as the inverse of the symbol (D + a)", i.e., such that We now proceed to investigate how far the equation holds for inverse symbols. We have already seen that when n is a positive integer (D + a}"u = e~ ax T)"e a:c u = v > suppose ; . . u (D + a)~"v. Moreover from the equation D n e* u = e ax v , u=e- ax ~D- n e lx v . ( D -t- a) * "v = e - ax D ~ "e a *v , we get or Consequently Hence we infer that the symbolic equation also holds for negative powers of D. 91. In general, since we have Again |D + <}> (z) J- u {D + (,r and in general D where n is an integer. From this we conclude that in all intcrpretable cases we have /{ D + (ff) ]- = c-* ( - r) /(D)c* ( r) . The results here given have been generalized and extensively em ployed in the integration of differential equations by Boole. See Philosophical Transactions, 1844 ; also Boole s Differential Equa tions, chapter xvii. 92. We conclude this short account of symbolic methods by applying them to establish one or two well-known formulae. It has been shown already ( 84) that we may write (C AD - l)0(x) = <f(x + h)- <f>(x) . Hence - 1) - 1 -[<j>(x+ h) -
