Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/53

INFINITESIMAL CALCULUS 43 The limit of the latter integral, as already seen, is 0(0) log Also, whenever 4>(~) tends to a definite limiting value <(co ) when ; is infinite, we have in this case. Again, whenever _i__ i s zero, we have In the latter form this result is called Frullani s theorem, having been communicated by Frullani to Plana in 1821, and subsequently published in Mem. del. Soc. Jtal., 1828. These results, though limited as to their generality, contain many particular integrals under them. For example, since c~ ax becomes when x = >, and 1 when x = 0, we have / ff-ax _ g-bx f) dx log . x a Again, when x = 0, taii-^aa 1 ) becomes 0; and when =<, tan- 1 ;/ = . Consequently we have tan ^ tan - Also, from the periodic character of cos z, it is readily seen that vanishes when Hence dz = log . In like manner we have Frullani s theorem has attracted considerable attention recently, and many remarkable applications, both in single and multiple in tegrals, have been given by Mr Elliott, Mr Leudesdorf, and others, chiefly in the Proceedings of the London Matlmnatical Society 1876, 1877, 1878. 143. The consideration of singular definite integrals furnishes a method for the calculation of the general value of a definite integral when its principal value is known. Thus, if A be the general value and B the principal value of /x / /0 - X" > where /(a;) is supposed to become infinite for the values Ji x v x. 2 . . . x n of x, then the difference A - B, from the preceding investigation, will consist of the sum of the singular definite inte grals /^!-ne rxi+e f(x]dx, / f(x)dx, . . . -. SXl + Vlt Consequently if f lt / 2 . . . /, as before, denote the limiting values of (x-Xj)f(x), (x-x. 2 )f(x), . . . (x-x n )f(x], when x = x lt x = x, 2 , . . . x^x,,, respectively, we shall have lo 1+ ... +,,lo5. Accordingly, in order that the definite integral / f(x)dx should J.C have a finite and determinate value it is necessary that the quan tities/^ /jj . . . / should each be evanescent. When the limits X and x are + oo and - oo , to the value of A - B here given we must add the term / log -^ , provided xf(x), as v x becomes infinitely great, tends to a definite limiting value/. 144. For example, if /A !L be a rational algebraic function, then F(a;) /+ fM ^~clx has a finite and determinate value, pro- oo -t 1 ( x ) vided (1) the equation F(.r) = has no real roots, and (2) the de- grec of a- in the denominator F(o-) exceeds that of the numerator f(x] by two at least. For the former condition implies that ^ J F(a ) does not become infinite for any real finite value of a 1 , and it follows from the latter condition that -Vrr-r becomes evanescent when a: 1- (x) becomes infinite. In order to find the value of / wT"^ i 1 this case, we sup- ./ 00 * (&) pose ~(x - a) 2 + i 2 to represent the partial fraction corresponding to a pair of conjugate roots a /& of the equation F(.) = 0; then, as we have shown that the general and the principal values of the definite integral are the same in this case, we may write Km. T_ (x-af+P"" 1 But we have already seen that -i- - 1 - . i /~ e ~ (x - a)dx i /" e dx IT - -^ 7^ = 0, and / _ - =_ . Consequently "(T^T* +. where B 1; & t , B 2 , 1. 2 , . . B,,, b n represent the constants corre sponding to the n pairs of imaginary roots in the equation F(a-) = 0. As an example, let us consider the definite integral -+x m dx where m and n are positive integers, of which n is the greater. By aid of the theory of equations it can be shown without difficulty that are respectively equal to sin sin 30 sin (2??,- where Consequently we have 2 = _[sin + sin 30 + . b,i < . . + x- m dx Accordingly Hence it follows immediately that sin . TT By a corresponding investigation it can be shown that / a; 2m rfa: TT 2m + 1 1 _ V 2n ^ O^ c t 9,, V J. tc ftltf fllif These results are readily transformed into / 1_J =^L_ , and / ^ = T cot7r, JQ 1+x sin air y 1 - x where is less than unity. A few simple examples are added. /-<*> 7. (1) To show that^ (T -^-^^ is equal to -J