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 $$\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\text{cos}(\frac{n\pi x}{L})+b_n\text{sin}(\frac{n\pi x}{L}))$$ But $X_n=c_n\text{sin}(\frac{n\pi }{L}x)$ is an eigenfunction for the boundary value problem $X^{″}+\lambda X=0$, $X(0)=0=X(L)$. 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 $f(-x)=f(x)$ for any $x$. A function is called an odd function if $f(-x)=-f(x)$. 

Notice if $f(x)$ is an even function, then ${\int }_{-L}^{L}f(x)dx=2{\int }_0^{L}f(x)dx$ and if $f(x)$ is an odd function, then ${\int }_{-L}^{L}f(x)dx=0$. 

Corollary: (a) Suppose that $f(x)$ and $f^{\prime }(x)$ are piecewise continuous on $[-L,L)$ and that $f(x)$ is an even periodic function with period $2L$. Then $f(x)\text{cos}(n\pi x/L)$ is even and $f(x)\text{sin}(n\pi x/L)$ is odd. Thus the Fourier series of $f(x)$ is  $$f(x)=\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\text{cos}(\frac{n\pi x}{L}))$$  where $a_n=\frac{2}{L}{\int }_0^{L}f(x)\text{cos}(\frac{n\pi x}{L})dx$, $n=0,1,2,...$. This series is called Fourier cosine series.

(b) Suppose that $f(x)$ and $f^{\prime }(x)$ are piecewise continuous on $[-L,L)$ and that $f(x)$ is an odd periodic function with period $2L$. Then $f(x)\text{cos}(n\pi x/L)$ is odd and $f(x)\text{sin}(n\pi x/L)$ is even. Thus the Fourier series of $f(x)$ is  $$f(x)=\sum _{n=1}^{\mathrm{\infty }}(b_n\text{sin}(\frac{n\pi x}{L}))$$ $b_n=\frac{2}{L}{\int }_0^{L}f(x)\text{sin}(\frac{n\pi x}{L})dx$, $n=1,2,3,...$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 $f(x)$ that is defined on $[0,L]$. Let  $$g(x)=\{\begin{array}{ll}f(x)& 0\le x<L\\ f(-x)& -L<x<0\end{array}$$ and $g(x+2L)=g(x)$. The function $g(x)$ is an even function. $g(x)$ is called even period extension of $f(x)$.

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

Example 1: Let $f(x)=x-3$, $0<x<4$. Sketch the even and the odd extension of $f(x)$ for 3 periods.

Exercise 1: Let $f(x)=4-x$, $0<x<2$. Sketch the even and the odd extension of $f(x)$ for 3 periods.

Example 2: Let  $$f(x)=\{\begin{array}{ll}2& 0<x<2\\ 1& 2\le x<4.\end{array}$$ Sketch the even and the odd extension of $f(x)$ for 3 periods.

Exercise 2: Let  $$f(x)=\{\begin{array}{ll}1& 0<x<3\\ 3& 3\le x<5.\end{array}$$ Sketch the even and the odd extension of $f(x)$ for 3 periods.

Example 3: Let  $$f(x)=\{\begin{array}{ll}x& 0<x<2\\ 1& 2\le x<3.\end{array}$$ Sketch the even and the odd extension of $f(x)$ for 3 periods.

Exercise 3: Let  $$f(x)=\{\begin{array}{ll}-1& 0<x<1\\ x& 1\le x<4.\end{array}$$ Sketch the even and the odd extension of $f(x)$ for 3 periods.

Example 4: Let $f(x)=2-x^2$, $0<x<1$. Sketch the even and the odd extension of $f(x)$ for 3 periods.

Exercise 4: Let $f(x)=x^2+1$, $0<x<2$. Sketch the even and the odd extension of $f(x)$ for 3 periods.