An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics . ue of xbetween — tt and zero in the cosine curve will be the same as the ordinatebelonging to the corresponding positive value of x. In other words the curve 1/ = ^bo + bi cos X -\- b2 cos 2x + b^ cos 3x -\- • • • (2) is symmetrical with respect to the axis of Y. If then f(x) =^ — /(— x), that is if /(a?) is an odd function the sine seriescorresponding to it will be equal to it for all values of x between — tt and tt,except perhaps for the va
An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics . ue of xbetween — tt and zero in the cosine curve will be the same as the ordinatebelonging to the corresponding positive value of x. In other words the curve 1/ = ^bo + bi cos X -\- b2 cos 2x + b^ cos 3x -\- • • • (2) is symmetrical with respect to the axis of Y. If then f(x) =^ — /(— x), that is if /(a?) is an odd function the sine seriescorresponding to it will be equal to it for all values of x between — tt and tt,except perhaps for the value x = 0 for which the series will necessarily bezero. If f(x) =f(—x), that is if f(x) is an even function the cosine series cor-responding to it will be equal to it for all values of x between x = — tt andX = TT, not excepting the value a = 0. As an example of the difference between the sine and cosine developmentsof the same function let us take the series for. x. sin 2x , sin ox y = 2 [sii TT 4 r , cos , COS // = t; — - COS X H -, h ?in 4. + 3^ ? 5^ [v. Art. 26(rt) and Art. 2S(«)]. (3) represents the curve (3)(4). / and (4) the curve Chap. II.] FOUMEKs SERIES. 47 Both coincide with y = x from x = 0 to x = tt, (3) coincides witliy =z X from x = — ir to .i; = tt , and neither coincides with y z=. x forvalues of x less than — tt or greater than tt. Moreover (3), in addition tothe continuous portions of the locus represented in the figure, gives the iso-lated points (— 7r,0) (7r,0) (37r,0) &c. 30. We have seen that if f(x) is an odd function its development in silfeseries holds for all values of x from — tt to tt , as does the development off(x) in cosine series if f{x) is an even function. Thus the developments of Art. 26(^0> ^it- 26 Exs. (2), (4), (6); Art. 28(^<)Art. 28 Exs. (3), (7), (9) are valid for all values of x between — ir and tt. Any function of x can be developed into a Trigonometric series to which itis equal for all va
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