Section 6.6 Even or odd period extension

Objective:

1. Definition of even and odd functions

2. Find the even or odd period extension of a function and its Fourier series

In previous section, we have the Fourier series \[ \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\text{cos}(\frac{n\pi x}{L})+b_{n}\text{sin}(\frac{n\pi x}{L})\right)\] But [latex]X_{n}=c_{n}\text{sin}(\frac{n\pi}{L}x)[/latex] is an eigenfunction for the boundary value problem [latex]X''+\lambda X=0[/latex], [latex]X(0)=0=X(L)[/latex]. The function does not have the cosine function. Actually, sine functions are odd functions. We want to be able to find Fourier series in a more efficient way so that we can reduce the integrations. 

Definition: A function is called an even function if [latex]f(-x)=f(x)[/latex] for any [latex]x[/latex]. A function is called an odd function if [latex]f(-x)=-f(x)[/latex]. 

Notice if [latex]f(x)[/latex] is an even function, then [latex]\int_{-L}^{L}f(x)dx=2\int_{0}^{L}f(x)dx[/latex] and if [latex]f(x)[/latex] is an odd function, then [latex]\int_{-L}^{L}f(x)dx=0[/latex]. 

Corollary: (a) Suppose that [latex]f(x)[/latex] and [latex]f'(x)[/latex] are piecewise continuous on [latex][-L,L)[/latex] and that [latex]f(x)[/latex] is an even periodic function with period [latex]2L[/latex]. Then [latex]f(x)\text{cos}(n\pi x/L)[/latex] is even and [latex]f(x)\text{sin}(n\pi x/L)[/latex] is odd. Thus the Fourier series of [latex]f(x)[/latex] is  \[ f(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\text{cos}(\frac{n\pi x}{L})\right)\]  where [latex]a_{n}=\frac{2}{L}\int_{0}^{L}f(x)\text{cos}(\frac{n\pi x}{L})dx[/latex], [latex]n=0,1,2,...[/latex]. This series is called Fourier cosine series.

(b) Suppose that [latex]f(x)[/latex] and [latex]f'(x)[/latex] are piecewise continuous on [latex][-L,L)[/latex] and that [latex]f(x)[/latex] is an odd periodic function with period [latex]2L[/latex]. Then [latex]f(x)\text{cos}(n\pi x/L)[/latex] is odd and [latex]f(x)\text{sin}(n\pi x/L)[/latex] is even. Thus the Fourier series of [latex]f(x)[/latex] is  \[ f(x)=\sum_{n=1}^{\infty}\left(b_{n}\text{sin}(\frac{n\pi x}{L})\right)\] [latex]b_{n}=\frac{2}{L}\int_{0}^{L}f(x)\text{sin}(\frac{n\pi x}{L})dx[/latex], [latex]n=1,2,3,...[/latex]This series is called Fourier sine series.

The above corollary reduce our workload and now we just need to get the even or odd extension if we know a function is even or odd. 

Definition: (a) Given a function [latex]f(x)[/latex] that is defined on [latex][0,L][/latex]. Let  \[ g(x)=\begin{cases} f(x) & 0\leq x \lt L\\ f(-x) & -L \lt x \lt 0 \end{cases}\] and [latex]g(x+2L)=g(x)[/latex]. The function [latex]g(x)[/latex] is an even function. [latex]g(x)[/latex] is called even period extension of [latex]f(x)[/latex].

(b) Given a function [latex]f(x)[/latex] that is defined on [latex](0,L])[/latex]. Let  \[h(x)=\begin{cases} f(x) & 0 \lt x \lt L\\ 0 & x=0,x=L\\ -f(-x) & -L \lt x \lt 0 \end{cases} \] and [latex]h(x+2L)=h(x)[/latex]. The function [latex]h(x)[/latex] is an odd function. [latex]h(x)[/latex] is called odd period extension of [latex]f(x)[/latex].

Example 1: Let [latex]f(x)=x-3[/latex], [latex]0 \lt x \lt 4[/latex]. Sketch the even and the odd extension of [latex]f(x)[/latex] for 3 periods.

Exercise 1: Let [latex]f(x)=4-x[/latex], [latex]0 \lt x \lt 2[/latex]. Sketch the even and the odd extension of [latex]f(x)[/latex] for 3 periods.

Example 2: Let  \[ f(x)=\begin{cases} 2 & 0 \lt x \lt 2\\ 1 & 2\leq x \lt 4. \end{cases} \] Sketch the even and the odd extension of [latex]f(x)[/latex] for 3 periods.

Exercise 2: Let  \[ f(x)=\begin{cases} 1 & 0 \lt x \lt 3\\ 3 & 3\leq x \lt 5. \end{cases} \] Sketch the even and the odd extension of [latex]f(x)[/latex] for 3 periods.

Example 3: Let  \[ f(x)=\begin{cases} x & 0 \lt x \lt 2\\ 1 & 2\leq x \lt 3. \end{cases} \] Sketch the even and the odd extension of [latex]f(x)[/latex] for 3 periods.

Exercise 3: Let  \[ f(x)=\begin{cases} -1 & 0 \lt x \lt 1\\ x & 1\leq x \lt 4. \end{cases} \] Sketch the even and the odd extension of [latex]f(x)[/latex] for 3 periods.

Example 4: Let [latex]f(x)=2-x^{2}[/latex], [latex]0 \lt x \lt 1[/latex]. Sketch the even and the odd extension of [latex]f(x)[/latex] for 3 periods.

Exercise 4: Let [latex]f(x)=x^{2}+1[/latex], [latex]0 \lt x \lt 2[/latex]. Sketch the even and the odd extension of [latex]f(x)[/latex] for 3 periods.