Download An Elementary Treatise on Fourier's Series and Spherical, by William Elwood Byerly PDF

By William Elwood Byerly

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Additional resources for An Elementary Treatise on Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, With Applications to Problems in Mathematical Physics

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Dz. dz. π −π π (3) and (1) holds good from z = −π to z = π. Replace z by its value in terms of x and (1) becomes  πx 2πx 3πx 1  b0 + b1 cos + b2 cos + b3 cos + · · · 2 c c c (4) πx 2πx 3πx  + a1 sin + a2 sin + a3 sin + · · · c c c The coefficients in (4) are the same as in (1), and (4) holds good from x = −c to x = c. Formulas (2) and (3) can be put into more convenient shape. dz = f (x) cos dx π −π π π −c c c bm = 1 mπx 1 mπλ f (x) cos dx = f (λ) cos dλ. c −c c c −c c c or c (5) In like manner we can transform (3) into c am = c mπx mπλ 1 1 f (x) sin f (λ) sin dx = dλ.

Art. 83, p. 78), the coefficients in y = a1 sin x + a2 sin 2x + a3 sin 3x + · · · + an sin nx (2) can be determined so that the curve represented by (2) will pass through any n arbitrarily chosen points of the curve y = f (x) (3) whose abscissas lie between 0 and π and are all different, and these coefficients will have but one set of values. For the sake of simplicity suppose that the n points are so chosen that their projections on the axis of X are equidistant. π = ∆x; then the co¨ordinates of the n points will be [∆x, f (∆x)], Call n+1 [2∆x, f (2∆x)], [3∆x, f (3∆x)], · · · [n∆x, f (n∆x)].

2) π 1 the second member of (2) reduces to , for 2 2 2 π 1 1 1 1 − + − + ··· 1 3 5 7 = 1 2 by (5) (b); and we see that the series represents the function completely for all values of π x between x = 0 and x = π except for x = and there it has a value which 2 π is the mean of the values approached by the function as x approaches from 2 opposite sides. EXAMPLES. Obtain the following developments:— (1) (2) (3) π2 4 π2 4 π2 − 3 sin x − sin 2x + − 3 1 1 2 3 3 π2 4 + − 3 sin 5x − · · · . 5 5 π3 6π π3 6π 2 − 3 sin x − − 3 sin 2x + x3 = π 1 1 2 2 3 6π π − 3 sin 4x + · · · .

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