By William Elwood Byerly

ISBN-10: 1116151464

ISBN-13: 9781116151466

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

**Example text**

45 DEVELOPMENT IN TRIGONOMETRIC SERIES. 46 29. Although any function can be expressed both as a sine series and as a cosine series, and the function and either series will be equal for all values of x between zero and π, there is a decided difference in the two series for other values of x. Both series are periodic functions of x having the period 2π. If then we let y equal the series in question and construct the portion of the corresponding curve which lies between the values x = −π and x = π the whole curve will consist of repetitions of this portion.

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)].

Int. Cal. 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)].

### An Elementary Treatise on Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, With Applications to Problems in Mathematical Physics by William Elwood Byerly

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