40. If where and а + а ̧x + α ̧2x2 + ........=ƒ(x), = - 2 [** (F (u) + F (v)} {ƒ (u) +ƒ(v) } d0 — A ̧a, u = xe√(-1) and v = xe-√(−1) ̧ Apply the above formula to express the sum of the 41. If the sum of the series a ̧+a1x+αx2+ 2 ... can be expressed in a finite form, then the sum of the series a2 + a 2x2 + a 2x2 + ...... can be expressed by a definite integral. Prove this, and hence shew that the sum of the squares of the coefficients of the terms of the expansion of (1+x)" when n is a positive whole number, may be expressed by 0 cos cx dx 2 1 + x2 П + 21+01+0°) π ¿ (sin 2x) cos x dx = [2 4 (cos2x) cos x dx. (Liouville's Journal de Mathématiques, Vol. XVIII. page 168.) 250 CHAPTER XIII. EXPANSION OF FUNCTIONS IN TRIGONOMETRICAL SERIES. 301. THE subject we are about to introduce is one of the most remarkable applications of the Integral Calculus, and although in an elementary work like the present, only an imperfect outline can be given of it, yet on account of the novelty of the methods, and the importance of the results, even such an outline may be of service to the student. For fuller information we may refer to the Differential and Integral Calculus of Professor De Morgan. The subject is also frequently considered in the writings of Poisson, for example, in his Traité de Mécanique, Vol. 1. pp. 643-653; in his Traité de la Chaleur; and in different Memoirs in the Journal de l'Ecole Polytechnique. The student may also consult a Memoir by Professor Stokes, in the 8th Vol. of the Cambridge Philosophical Transactions, a Memoir by Sir W. Hamilton, in the 19th Vol. of the Transactions of the Royal Irish Academy, and a Memoir by Professor Boole, in the 21st Vol. of the same Transactions. 302. It is required to find the values of the m constants 1⁄4 ̧, a‚‚ Ã ̧‚...... Ã, so that the expression 29 may coincide in value with an assigned function of x when x Let f(x) denote the assigned function of x, then we have by hypothesis the following m equations from which the constants are to be determined, ƒ(0) = A ̧ sin 0 + A, sin 20 + A, sin 30 +..............+ A„ sin m✪, 2 ...... f(20) = A, sin 20+ A, sin 40 + A ̧ sin 60 +......+ Am sin 2m0, 2 3 f(m0)=A ̧sinm0+А, sin 2m0+A, sin 3m0+......+Amsinmmo. Multiply the first of these equations by sin re, the second by sin 2r0, the last by sin mre; then add the results. The coefficient of A, on the second side will then be sin re sin se + sin 2r0 sin 280 +......+ sin mre sin mse; we shall now shew that this coefficient is zero if s be different from r, and equal to (m + 1) when s is equal to r. First suppose s different from r. coefficient is equal to the series Now twice the above cos (rs) + cos 2 (r − s) 0 + + cos m (r − s) 0, diminished by the series ...... cos (r+s) + cos 2 (r+s) 0 +......+ cos m (r+s) 0. The sum of the first series is known from Trigonometry to This expression vanishes when rs is an odd number, and is equal to -1 when r s is an even number. The sum of the second series can be deduced from that of the first by changing the sign of s; hence this sum vanishes when r+s is an odd number, and is equal to -1 when r+s is an even number. Thus when s is different from r, the coefficient of A, is zero. When s is equal to r, the coefficient becomes sin re+ sin 2r0 +......+ sin2 mre, - † {cos 2r0 + cos 4r0 +......+ cos 2mr0}. And by the method already used it will be seen that the sum of the series of cosines is -1; thus the coefficient of A, is (m + 1). A, Hence we obtain = 2 m+ 1 {sin ref(0) + sin 2r0 ƒ (20)+......+ sin mr✪ ƒ (m0)}, and thus by giving to r in succession the different integral values from 1 to m, the constants are determined. Now suppose m to increase indefinitely, then we have ultimately And as f(x) now coincides in value with the expression for an infinite number of equidistant values of x between O and π, we may write the result thus where the symbol Σ indicates a summation to be obtained by giving to n every positive integral value. 303. The theorem and demonstration of the preceding article are due to Lagrange; we have given this demonstra |