the two terms add together and the value of the integral is 7. Hence the definite integral1d sin q cos qe is a function of æ qx equal to 1 if the variable x has any value included between 1 and -1; and the same function is nothing for every other value of x not included between the limits 1 and -1. 358. We might deduce also from the transformation of series into integrals the properties of the two expressions* the first (Art. 350) is equivalent to e* when x is positive, and to e when x is negative. The second is equivalent to e ̄* if x is positive, and to e if x is negative, so that the two integrals have the same value, when x is positive, and have values of contrary sign when x is negative. One is represented by the line eeee (fig. 19), the other by the line eeee (fig. 20). which we have arrived at (Art. 226), gives immediately the integral x 2 [ dq sin qr sin qu; which expression" is equivalent to sin a, if æ 1-q 2 is included between 0 and 7, and its value is 0 whenever a ex ceeds π. 1 At the limiting values of x the value of this integral is; Riemann, § 15. 2 Cf. Riemann, § 16. 3 The substitutions required in the equation are for x, dq for 2, q for i ах .a π We then have sin x equal to a series equivalent to the above integral for values of x between 0 and π, the original equation being true for values of x between 0 and a. [A. F.] 359. The same transformation applies to the general equation ‡ π 4 (u) = sin u fdu $ (u) sin u + sin 2u fdu 4 (u) sin 2u+ &c. πφ Making u == =2, denote 4 (u) or 4 (2) by f(x), and introduce into n \n the analysis a quantity q which receives infinitely small incre we find dq sin qæ fdæ ƒ (a) sin qr. The integral with respect to u qx is taken from u = 0 to uπ, hence the integration with respect to x must be taken from x=0 to x = nπ, or from x nothing to x infinite. We thus obtain a general result expressed by the equation dx 4 π ƒ (x) = [” dq sin qæ ["dæ ƒ (a) sin qæ ........ (e), for which reason, denoting by Q a function of q such that we have f (u) = fdqQ sin qu an equation in which ƒ(u) is a given function, we shall have Q=2fduf (u) sin qu, the integral being taken from u nothing to u infinite. We have already solved a similar problem (Art. 346) and proved the general equation 1 π F(x) = ["dq cos qœ which is analogous to the preceding. 360. To give an application of these theorems, let us suppose f(x)=x, the second member of equation (e) by this substitution 1 is equivalent to du sin u u”, the integral being taken from u nothing to u infinite. denoting the last integral by v, taken from u nothing to u infinite, we have as the result of two successive integrations the term α' μν. We must then have, according to the condition expressed by the equation (e), +πα = μνα or μν = {π; thus the product of the two transcendants and from these two equations we might also conclude the following', 1 dqe-=√, which has been employed for some time. 361. By means of the equations (e) and (e) we may solve the following problem, which belongs also to partial differential analysis. What function Q of the variable q must be placed under 1 The way is simply to use the expressions e+cos √12+ √-1 sin√-12, transforming a and b by writing y for u and recollecting that√√11+/-1 √2 Cf. § 407. [R. I. E.] the integral sign in order that the expression fdq2e g may be equal to a given function, the integral being taken from q nothing to q infinite'? But without stopping for different consequences, the examination of which would remove us from our chief object, we shall limit ourselves to the following result, which is obtained by combining the two equations (e) and (€). and 1 TF (x) = ["dq cos qx ["d.F (2) cos qz. If we took the integrals with respect to a from ∞ to +∞, the result of each integration would be doubled, which is a necessary consequence of the two conditions and ƒ (a) =—ƒ(− a) and F(a) = F (− a). We have therefore the two equations πf(x)=dq sin qx | daf (a) sin qz, TF (x) = [" dq cos qx ["daF (2) cos qz. -8 We have remarked previously that any function (x) can always be decomposed into two others, one of which F(x) satisfies the condition F(x) = F (− x), and the other f(x) satisfies the condition f(x)=-f(x). We have thus the two equations daF (a) sin qz, and 0 0 = ["dif (a) cos q1, 1 To do this write +x-1 in f(x) and add, therefore 2 fq cos qx dq = f (x √ = 1) +ƒ ( − x √ −1), therefore Q== fix [ƒ (x√ = 1) + ƒ ( − x √−1)] cos qx dx. π Again we may subtract and use the sine but the difficulty of dealing with imaginary quantities recurs continually. [R. L. E.] whence we conclude π [F(x) + ƒ (w)] = w¢ (x) = ["*"dq sin qæ [** dxf (a) sin qu qu 1 or lastly', $ (x) = = [ "d. (a) ["dq cos q (x − a )... π -8 ...(E). The integration with respect to q gives a function of x and a, and the second integration makes the variable a disappear. Thus the function represented by the definite integral fdqcosq (x − a) has the singular property, that if we multiply it by any function (a) and by da, and integrate it with respect to a between infinite limits, the result is equal to π (a); so that the effect of the integration is to change a into x, and to multiply by the number π. 362. We might deduce equation (E) directly from the theorem 1 Poisson, in his Mémoire sur la Théorie des Ondes, in the Mémoires de l'Académie des Sciences, Tome 1., Paris, 1818, pp. 85-87, first gave a direct proof of the theorem in which is supposed to be a small positive quantity which is made equal to 0 after the integrations. Boole, On the Analysis of Discontinuous Functions, in the Transactions of the Royal Irish Academy, Vol. xxI., Dublin, 1848, pp. 126-130, introduces some analytical representations of discontinuity, and regards Fourier's Theorem as unproved unless equivalent to the above proposition. Deflers, at the end of a Note sur quelques intégrales définies &c., in the Bulletin des Sciences, Société Philomatique, Paris, 1819, pp. 161-166, indicates a proof of Fourier's Theorem, which Poisson repeats in a modified form in the Journal Polytechnique, Cahier 19, p. 454. The special difficulties of this proof have been noticed by De Morgan, Differential and Integral Calculus, pp. 619, 628. An excellent discussion of the class of proofs here alluded to is given by Mr J. W. L. Glaisher in an article On sin co and cos∞, Messenger of Mathematics, Ser. 1., Vol. v., pp. 232-244, Cambridge, 1871. [A. F.] |