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2m − 1)(2m − 3)(2m − 5) (8) as a particular solution of Legendre’s Equation. (2m + 3)(2m + 5)(2m + 7) a−m−1 may be taken at pleasure, and is usually taken as &c. m! 5. · · · (2m + 1) and z = Qm (x) where Qm (x) = + m! (2m + 3)(2m + 5) xm+5 (9) is a second particular solution of Legendre’s Equation, provided the series is convergent. Qm (x) is called a Surface Zonal Harmonic of the second kind. It is easily seen to be convergent if x < −1 or x > 1, and divergent if −1 < x < 1. Hence if m is a positive integer, z = APm (x) + BQm (x) (10) is the general solution of Legendre’s Equation if x < −1 or x > 1.

55 56 CHAPTER III. CONVERGENCE OF FOURIER’S SERIES. 33. The question of the convergence of a Fourier’s Series is altogether too large to be completely handled in an elementary treatise. We will, however, consider at some length one of the most important of the series we have obtained, namely 4 sin 3x sin 5x sin 7x sin x + + + + ··· , π 3 5 7 [v. (3) Art. dα + sin 3x 0 0 as n is indefinitely increased, is 1, provided that x lies between zero and π. dα . (1) 0 Then Sn = = 2 π 1 π π [sin α sin x + sin 2α sin 2x + sin 3α sin 3x + · · · + sin nα sin nx]dα 0 π [ cos(α − x) − cos(α + x) + cos 2(α − x) − cos 2(α + x) + · · · 0 + cos n(α − x) − cos n(α + x)] dα = − 1 π 1 π π [cos(α − x) + cos 2(α − x) + cos 3(α − x) + · · · + cos n(α − x)]dα 0 π [cos(α + x) + cos 2(α + x) + cos 3(α + x) + · · · + cos n(α + x)]dα.

2 and DEVELOPMENT IN TRIGONOMETRIC SERIES. 48 bm and am can be simplified a little. dx. dx. dα. dα. π −π π −π π (2) (3) π (4) and this development holds for all values of x between −π and π. The second member of (2) is known as a Fourier’s Series. EXAMPLES. 1. Obtain the following developments, all of which are valid from x = −π to x = π:— (1) ex = 2 sinh π 1 1 1 1 1 − cos x + cos 2x − cos 3x + cos 4x + · · · π 2 2 5 10 17 2 sinh π 1 2 3 4 + sin x − sin 2x + sin 3x − sin 4x + · · · . π 2 5 10 17 DEVELOPMENT IN TRIGONOMETRIC SERIES.