Fourier Series

only be continuous if fx 0 and this is satisfied by both of the two examples. If x 0ispermitted to be a point of discontinuity, then an odd function may possess a jumpdiscontinuity at this point but the left-hand and right-hand limits are not arbitrary and mustsatisfy the requirement that the function is odd.CombinationsThe concept of evenness and oddness plays an important role in the way that we understandfunctions. Any function on −a,a can be written as the sum of an odd and an evenfunction in a very simple way,fx 1 fx fx2 1 fx f−x 2 1fx − f−x2 Even function Odd function .Example 1Consider fx e x defined upon −a,a. Using the description abovefx 1 fx f−x 2 1fx − f−x2 e x 1 2 ex e −x 1 2 ex − e −x coshx sinhxand we already know that the hyperbolic functions coshx and sinhx are even and oddrespectively.Example 2Suppose that fx and gx are an even and odd function respectively on −a,a and may becontinuous or piecewise continuous, thena−afx gx dx 0.Write the integral asa−afx gx dx 0−afx gx dx 0afx gx dx I 1 I 2 .Then3