12 2 Fourier Series Trigonometric Series Fourier coefficients

12. 2 Fourier Series Trigonometric Series Fourier coefficients of f is orthogonal on the interval [ -p, p]. Fourier series of the function f In applications, we are interested to expand a function f(x) defined on [-p, p] as a linear combination

12. 2 Fourier Series Example: Expand in a Fourier series

12. 2 Fourier Series Example: Expand in a Fourier series

Convergence of a Fourier Series piecewise continuous 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. 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

12. 2 Fourier Series Example: Remark: Expand in a Fourier series

Sequence of Partial Sums 25 terms 15 terms Example:

MATHEMATICA Plot[0. 5+Sum[ (1 -(-1)^n)*Sin[n x]/(n Pi), {n, 1, 1000}], {x, -Pi, Pi}]; 25 terms 15 terms Example: 1000 terms 125 terms Sequence of Partial Sums

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

12. 2 Fourier Series Example: Consider the function Periodic extension of the function f

12. 2 Fourier Series Example: Consider 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. 2 p is the fundamental period Periodic extension of the function

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