1. 12.2 Fourier Series Trigonometric Series is orthogonal on the interval [ -p, p]. In applications, we are interested to expand a function f(x) defined on [-p, p] as a linear combination Fourier series of the function f Fourier coefficients of f

2. 12.2 Fourier Series Example: Fourier series Expand in a Fourier series

3. 12.2 Fourier Series Example: Fourier series Expand in a Fourier series

4. Convergence of a Fourier Series f(x) is piecewise continuous on the interval [-p,p]; if f(x) is continuous except at a finite number of points in the interval and have only finite discontinuities at these points. piecewise continuous Theorem 12.2.1 Conditions for Convergence piecewise continuous on [-p,p] is a point of continuity. is a point of discontinuity. denote the limit of f at x from the right and from the left

5. 12.2 Fourier Series Example: Expand in a Fourier series Remark:

6. Sequence of Partial Sums Example: 15 terms 25 terms

7. Sequence of Partial Sums Example: 15 terms 25 terms 125 terms 1000 terms MATHEMATICA Plot[0.5+Sum[ (1-(-1)^n)*Sin[n x]/(n Pi),{n,1,1000}],{x,-Pi,Pi}];

8. Periodic Extension Example: Consider the funciion Periodic extension of the function f

9. 12.2 Fourier Series Example: Consider the function Periodic extension of the function f

10. 12.2 Fourier Series Example: Consider the function Periodic extension of the function a Fourier series not only represents the function on the interval ( -p, p) but also gives the periodic extension of f outside this interval. 2p is the fundamental period

11. Periodic Extension Example: Consider the funciion (A) (B) (C) Which one represents FS(x) ?