1. 12.3 Fourier Cosine and Sine Series Odd function Even function (a) The product of two even functions is even. (b) The product of two odd functions is even. (c) The product of an even function and an odd function is odd. (d) The sum ( difference) of two even functions is even. (e) The sum ( difference) of two odd functions is odd. Properties of Even/Odd Functions If f(x) is an odd function on the interval (-p, p ) Case1: Odd If f(x) is an even function on the interval (-p, p ) Case2: Even

2. 12.3 Fourier Cosine and Sine Series Fourier cosine Series The Fourier series of an even function on the interval ( -p, p) is the cosine series Fourier sine Series The Fourier series of an odd function on the interval ( -p, p) is the sine series Expand f(x) = x, -2 < x< 2, in a Fourier series. Example the series converges to the function on ( -2, 2) and the periodic extension (of period 4)

3. Example: Expand in a Fourier series 12.3 Fourier Cosine and Sine Series Fourier sine Series The Fourier series of an odd function on the interval ( -p, p) is the sine series is odd on the interval.

4. Gibbs phenomenon Example: 15 terms 25 terms 125 terms Remark: 1) The graphs has spikes near the discontinuities at 2) This behavior near a point of discontinuity does not smooth out but remains, even when the value n is taken to be large. 3) This behavior is known as the Gibbs phenomenon

5. Half-Range Expansions All previous examples the function defined on ( -p, p) we are interested in representing a function that is defined on an interval (0, L) This can be done in many different ways 1 2 The cosine and sine series obtained in this manner are known as half-range expansions. Half-Range Expansions

6. All previous examples the function defined on ( -p, p) we are interested in representing a function that is defined on an interval (0, L) This can be done in many different ways 1 2 3

7. 2L-periodic even extension 2L-periodic odd extension L-periodic extension Half-Range Expansions Example: Expand (a)in a cosine series, (b)in a sine series, (c)in a Fourier series