12 3 Fourier Cosine and Sine Series Odd
- Slides: 7
12. 3 Fourier Cosine and Sine Series Odd function Even function Properties of Even/Odd Functions (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. Case 1: Odd If f(x) is an odd function on the interval (-p, p ) Case 2: Even If f(x) is an even function on the interval (-p, p )
12. 3 Fourier Cosine and Sine Series Fourier cosine Series Fourier sine Series The Fourier series of an even function on the interval ( -p, p) is the cosine series The Fourier series of an odd function on the interval ( -p, p) is the sine series Example Expand f(x) = x, -2 < x< 2, in a Fourier series. the series converges to the function on ( -2, 2) and the periodic extension (of period 4)
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 Example: Expand in a Fourier series is odd on the interval.
Gibbs phenomenon 25 terms 125 terms Example: 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
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 Half-Range Expansions The cosine and sine series obtained in this manner are known as half-range expansions.
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 3
Half-Range Expansions L-periodic extension Example: Expand (a) in a cosine series, (b) in a sine series, (c) in a Fourier series 2 L-periodic even extension 2 L-periodic odd extension
