Fourier s series

Fourier s Series. Let/(x) be a function which is finite in the interval from x = — c to x = + c and whose graph has finite arc length in that interval. Then, for any value of x between —c and c. 

Carslaw, H. S., Introduction to the Theory of Fourier s Series and Integrals (third edition), Dover Publications, New York (1952). 

Byerle, W. E., Fourier s Series, Ginn Co., Boston, 1893 (Dover, New York,... 

Fourier s series and spherical harmonics. Boston Ginn and Co. Caldwell, D. H. and Babbitt, H. E. 

W. E. Byerly, Fourier s Series and Spherical Harmonics," pp. 151, 159,198, Ginn and Company, Boston, 1893. 

In view of the fact that the terms of Fourier s series are all periodic we may say that Fourier s series is an artificial way of representing the propagation or progression of any physical quality by a series of waves or vibrations. It is only a mathematical fiction, says Helmholtz, admirable because it renders calculation easy, but not necessarily corresponding with anything in reality. ... 

Assuming Fourier s series to be valid between the limits x = + 7r and x = - 7r, we shall now proceed to find values for the coefficients A0, av a2,..., bv b2,..., which will make the series true. 

There are several graphic methods for evaluating the coefficients of- a Fourier s series. See J. Perry, Electrician, 28, 362,1892 W. B. Woodhouse, the same journal, 46, 987,1901 or, best of all, O. Henrici, Phil. Mag. [5], 38, 110, 1894, when the series is used to express the electromotive force of an alternating current as a periodic function of the time. 

Up to the present, the values of the variable in Fourier s series only extend over the range + tv. The integration may however be extended so as to include all values of x between any limits whatever. 

Itos sometimes convenient to refer to the following alternative way of writing Fourier s series ... 

An adequate discussion of the conditions of convergency of Fourier s series must be omitted. W. E. Byerly s An Elementary Treatise on Fourier s Series, etc., is one of the best practical works on the use of Fourier s integrals in mathematical physics. J. Fourier s pioneer work Thiorie analytiquc de la Chaleur, Paris, 1822, is perhaps as modern as any other work on this subject see also W. Williams, Phil. Mag. [5], 42, i25,1896 Lord Kelvin s Collected Papers and Riemann-Weber s work (Z.c.), etc. 

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