Even, odd, and periodic functions

The concepts of oddness, evenness, and periodicity are closely related to the Fourier series.

Definition A function ${\phi(x)}$ that is defined for ${-\infty<x<\infty}$ is called periodic if there is a number ${p>0}$ such that

$\displaystyle \phi(x)=\phi(x+p) \hspace{0.5cm} \mathrm{for\hspace{3pt}all}\hspace{3pt} x \ \ \ \ \ (1)$

p is called a period of ${\phi(X)}$ . A periodic function has the following properties:

If ${\phi(x)}$ has period ${p}$ , then ${\phi(x+np)=\phi(x)}$ for all ${x}$ and for all integers ${n}$ .
The sum of two functions of period ${p}$ has period ${p}$ .
If ${\phi(x)}$ has period ${p}$ , then ${\int_{a}^{a+p}\phi(x)}$ does not depend on ${a}$ .

For a function defined on the interval ${-l<x<l}$ , its periodic extension is

$\displaystyle \phi_{per}(x)=\phi(x-2lm)\hspace{3pt} \mathrm{for}\hspace{3pt} -l+2lm<x<l+2lm \ \ \ \ \ (2)$

for all integers ${m}$ . This definition does not specify what the periodic extension is at the endpoints ${x = l + 2lm}$ . In fact, the function ${\phi(x)}$ may have jump discontinuities at the endpoints if the the one-sided limits ${\phi(l^-)}$ and ${\phi(-l^+)}$ both exist but not equal.

An even function is a function the satisfy the equation

$\displaystyle \phi(x)=\phi(-x) \ \ \ \ \ (3)$

This mean that the graph ${y=\phi(x)}$ is symmetric with respect to the y axis.

An odd function is a function the satisfy the equation

$\displaystyle \phi(-x)=-\phi(x) \ \ \ \ \ (4)$

This mean that the graph ${y=\phi(x)}$ is symmetric with respect to the origin.

A monomial ${x^n}$ is an even function if ${n}$ is even and is an odd function if n is odd. The functions cos ${x}$ , cosh ${x}$ , and any function of ${x^2}$ are even functions. The functions sin ${x}$ , tan ${x}$ , and sinh ${x}$ are odd functions. In fact, the products of functions follow the usual rules: even × even = even, odd × odd = even, odd × even = odd. The sum of two odd functions is again odd, and the sum of two evens is even.

But the sum of an even and an odd function can be anything. Proof: Let ${f(x)}$ be any function at all defined on ${(-l,l)}$ . Let ${\phi(x)=\frac{1}{2}[\phi(x)+\phi(-x)]}$ and ${\psi(x)=\frac{1}{2}[\psi(x)-\psi(-x)]}$ . Then we easily check that ${f(x)=\phi(x)+\psi(x)}$ , that ${\phi(x)}$ is even and that ${\psi(x)}$ is odd. The functions ${\phi}$ and ${\psi}$ are called the even and odd parts of ${f}$ , respectively. If ${p(x)}$ is any polynomial, its even part is the sum of its even terms, and its odd part is the sum of its odd terms.

Integration and differentiation change the parity (evenness or oddness) of a function. That is, if ${\phi(x)}$ is even, then both ${\frac{d\phi}{dx}}$ and ${\int_0^x\phi(s)ds}$ are odd. If ${\phi(x)}$ is odd, then its derivative and integral are even.

The graph of an odd function ${\phi(x)}$ must pass through the origin since ${\phi(0) = 0}$ follows directly from (4) by putting ${x = 0}$ . The graph of an even function ${\phi(x)}$ must cross the ${y}$ axis horizontally, ${\phi'(x)=0}$ , since the derivative is odd (provided the derivative exists).

The concepts of oddness, evenness, and periodicity have the following relationships with the boundary conditions:

${u(0,t)=u(l,t)=0}$ : Dirichlet BCs corresponding to the odd extension
${u_x(0,t)=u_x(l,t)=0}$ : Nuemann BCs corresponding to the even extension
${u(l,t)=u_(-l,t), u_x(l,t)=u_x=(-l,t)}$ : Periodic BCs corresponding to the periodic extension

Even, odd, and periodic functions

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Even, odd, and periodic functions

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