1 in the right-hand member replace x by; then -M 1+x -m dx; 1 in the second integral of the right-hand member replace a by-; SECTION 5.-Definite Integrals involving Impossible Quantities.— Cauchy's Method of Evaluating Definite Integrals. 102.] In our researches into the properties of definite integrals, contained in the preceding sections of the present Chapter, both the element-function and the limits of integration have been taken to be possible quantities. These quantities have been thus restricted, that there might be less risk of error in a subject of so delicate a kind, and because it was expedient to confine within the narrowest bounds the circumstances under which the elementfunction might become infinite or discontinuous; this latter being an event which is in all cases to be excluded, or at least to be separately considered. Impossible quantities however as well as possible are continuous; and we propose now to extend the theory, and to evaluate definite integrals into which impossible quantities enter, either in the limits or in the element-functions. We shall hereby be led also to a process of evaluation devised by PRICE, VOL. II. R M. Cauchy*; and which is of so great generality, that it includes almost all the known values of these functions. As a definite integral is the sum of a series, of which in this case some of the terms may be possible, and others may be impossible; so for the determinateness of the sum of such a series. it is requisite that it should be convergent; and thus that none of the terms should be infinite, in either the possible or the impossible part of the series. And consequently for no value of the variable within the range of integration must the element-function become infinite. When this is the case and the sum of the series is determined, that sum will consist of two parts, one of which is possible and the other impossible; and these will severally be equal to the corresponding parts of the definite integral. Thus, if then [**{ƒ'(x) + √=1¢'(x)} dx = A + √√−1B, N If however within the range of integration a value of the variable occurs, either possible or impossible, for which the elementfunction becomes infinite, then we may have recourse to the method explained in Art. 89: we may divide the integral into two parts, and take the range of one part from the lower limit to within an infinitesimal of the value of the variable for which the element-function is infinite; and take the range of the other from within an infinitesimal of that value to the superior limit; if these two infinitesimal parts are equal, so that in the symbols of Art. 89, μv = 1, the resulting sum will be the principal value of the integral; (see Art. 91). In this process however a part of the original definite integral is excluded or lost; viz., that within an infinitesimal range above and below the value of the variable for which the element-function is infinite. This part may be either finite or even zero; consequently it must be determined, and added to the other two integrals. It is called the correction for infinity or for discontinuity. Instances of it will occur in the sequel. 103.] The method of M. Cauchy depends on the application of * The original memoir of Cauchy, in which the method was explained, was read to the French Institute on Aug. 22, 1814; and is contained in the first volume of "Mémoires des Savans Étrangers, Paris, 1827. the theory of double integration, which has been explained in Art. 99, to the following identical equation. Let u = f(z), where z = (x, y); then which is an identical equation, as both members express the same quantity, whether x and y are possible or impossible, and whatever is the form of f. Let both sides be multiplied by dx dy; and then let them be the element-functions of a double integration with respect to x and y; the limits of a being ax and x, and those of y being y, and yo; these four limits being independent of x and y, so that the order of integration may be inverted, if it is necessary. Also let us in the first place assume that the element-function does not become infinite or discontinuous for any values of the variables within the ranges of integration. Then we have which is an absolute identity, and may be called M. Cauchy's equation, as it is the basis of his method of evaluating definite integrals of it. For the first general application of this equation, let and effecting in each member the integration which stands first, we have ƒ^"^ { ƒ (x + y √ = 1) = ƒ (x + y √ = 1)} dr = √ = 1 f'"* {ƒ ̃′(x + y √ = 1) −ƒ′(x + y√−1)} dy. (127) Let us apply this equation to some examples; and for a first application let f'(z) = e−az2; ··· ƒ'(x + y√ −1) = e−a (x2-v2) (cos 2 axy -√-1 sin2axy); (128) then from (127), we have eayn e-ax2 {cos 2 axy1- √−1 sin 2 a xy„} dx = √=1 { -e-ax2 In • ["heav2 { cos 2 ax。 y — √=1 sin 2ax,y} dy}; (129) yo from which two equations may be formed by taking separately the possible and the impossible parts. Moreover, to simplify further, let xoyo = 0; then (132) is the same result as (118), Art. 101. 104.] For another application of (126), let z = x (a+y√ −1); and effecting in each member the integration which stands first, we have [* * { (a + y„√ = 1) ƒ' (ax + x¥„√—1) In application of this equation let us suppose ƒ'(z) = zm-1e"; x1 = ∞, Y1 = b, xo=yo = 0; Yn then f'(z) does not become infinite for any value of its subjectvariable within the range of integration, and ƒ'(∞) = 0 = ƒ′(0). Consequently from (135), (a + b√ −1)m | 20m-1e−(a+b √/−1) * dx = √ = [® (ax)TM-1e-a2 d. ax ; 0 ·· (a+b√=1)TM √ ® 2m-le ̄(a+b√/−1)* dx = xm-1e-dx. (136) The integral in the right-hand member is evidently a function of m only; let us abbreviate the notation, and anticipating the notation of the seventh section, let us denote it by r(m); so that m-le-* dx = r(m); Also let a k cos a, b = k sin a; then (136) becomes (137) |