Extension of Fouriers series

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. 

This development (17) is true from + tt to - tt. From (16), therefore, 

The coefficients a and h are the same as in (17), and consequently (19) holds good throughout the range + c. From (18), we have  

Hence the rule Any arbitrary function, whose period ranges from — c to + c, so that T — 2c, can be represented as a series of trigonometrical functions with periods T, T,. . 

Examples.—(1) From (19), show that the sine series, from x=0 to x=c, is 

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