Read e-book online An Elementary Treatise on Fourier's Series and Spherical, PDF

By William Elwood Byerly

ISBN-10: 1116151464

ISBN-13: 9781116151466

It is a pre-1923 old copy that was once curated for caliber. caliber coverage used to be carried out on every one of those books in an try to get rid of books with imperfections brought via the digitization technique. notwithstanding we've got made top efforts - the books can have occasional mistakes that don't bog down the studying event. We think this paintings is culturally vital and feature elected to convey the booklet again into print as a part of our carrying on with dedication to the upkeep of revealed works around the world. this article refers back to the Bibliobazaar variation.

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Read or Download An Elementary Treatise on Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, With Applications to Problems in Mathematical Physics PDF

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

Example text

X2 = c2 4c2 πx 1 2πx 1 3πx 1 4πx − 2 cos − 2 cos + 2 cos − 2 cos + ··· 3 π c 2 c 3 c 4 c from x = −c to x = c. 1 + ec πx 2(1 − ec ) 2πx sin + 2 sin 2 2 c +π c c + 4π 2 c c 3(1 + e ) 3πx 4(1 − ec ) 4πx + 2 sin + 2 sin + ··· , 2 c + 9π c c + 16π 2 c 1 ec − 1 ec + 1 πx ec − 1 2πx ex = 2c − 2 cos + 2 cos 2 2 2 c c +π c c + 4π 2 c c e +1 3πx − 2 cos + ··· c + 9π 2 c (4) ex = 2π from x = 0 to x = c. (5) f (x) = 4c 1 1 πx 3πx 5πx − 2 sin + 2 sin + ··· sin π2 c 3 c 5 c from x = 0 to x = c, where f (x) = x from x = 0 to x = c c and f (x) = c − x from x = to x = c.

2) x= 2c πx 1 2πx 1 3πx 1 4πx sin − sin + sin − sin + ··· π c 2 c 3 c 4 c from x = −c to x = c. x= πx 3πx 5πx 7πx c 4c 1 1 1 cos − + 2 cos + 2 cos + 2 cos + ··· 2 π2 c 3 c 5 c 7 c from x = −c to x = c. (3) x2 = 2c2 π3 π2 π2 4 2πx 4 πx π 2 − 3 sin − sin + − 2 1 1 c 2 c 3 3 π2 4πx π2 4 5πx − sin + − 3 sin + ··· 4 c 5 5 c sin 3πx c from x = 0 to x = c. x2 = c2 4c2 πx 1 2πx 1 3πx 1 4πx − 2 cos − 2 cos + 2 cos − 2 cos + ··· 3 π c 2 c 3 c 4 c from x = −c to x = c. 1 + ec πx 2(1 − ec ) 2πx sin + 2 sin 2 2 c +π c c + 4π 2 c c 3(1 + e ) 3πx 4(1 − ec ) 4πx + 2 sin + 2 sin + ··· , 2 c + 9π c c + 16π 2 c 1 ec − 1 ec + 1 πx ec − 1 2πx ex = 2c − 2 cos + 2 cos 2 2 2 c c +π c c + 4π 2 c c e +1 3πx − 2 cos + ··· c + 9π 2 c (4) ex = 2π from x = 0 to x = c.

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. Since sin mx = − sin(−mx) the ordinate corresponding to any value of x between −π and zero in the sine curve; will be the negative of the ordinate corresponding to the same value of x with the positive sign. In other words the curve y = a1 sin x + a2 sin 2x + a3 sin 3x + · · · (1) is symmetrical with respect to the origin.

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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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