even and odd functions
Definition
Let $f$ be a function^{} from $\mathbb{R}$ to $\mathbb{R}$. If $f(x)=f(x)$ for all $x\in \mathbb{R}$, then $f$ is an even function^{}. Similarly, if $f(x)=f(x)$ for all $x\in \mathbb{R}$, then $f$ is an odd function.
Although this entry is mainly concerned with functions $\mathbb{R}\to \mathbb{R}$, the definition can be generalized to other types of function.
Notes
A real function is even if and only if it is symmetric about the $y$axis. It is odd if and only if symmetric about the origin.
Examples

1.
The function $f(x)=x$ is odd.

2.
The function $f(x)=x$ is even.

3.
The sine and cosine functions are odd and even, respectively.
Properties

1.
The only function that is both even and odd is the function defined by $f(x)=0$ for all real $x$.

2.
A sum of even functions is even, and a sum of odd functions is odd. In fact, the even functions form a real vector space, as do the odd functions.

3.
Every real function can be expressed in a unique way as the sum of an odd function and an even function.

4.
From the above it follows that the vector space^{} of real functions is the direct sum^{} of the vector space of even functions and the vector space of odd functions. See the entry direct sum of even/odd functions (example) (http://planetmath.org/DirectSumOfEvenoddFunctionsExample).)

5.
Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function.

(a)
If $f$ is an even function, then the derivative^{} ${f}^{\prime}$ is an odd function.

(b)
If $f$ is an odd function, then the derivative ${f}^{\prime}$ is an even function.
(For a proof, see the entry derivative of even/odd function (proof) (http://planetmath.org/DerivativeOfEvenoddFunctionProof).)

(a)

6.
Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function. Then there exist smooth functions $g,h:\mathbb{R}\to \mathbb{R}$ such that
$$f(x)=g({x}^{2})+xh({x}^{2})$$ for all $x\in \mathbb{R}$. Thus, if $f$ is even, we have $f(x)=g({x}^{2})$, and if $f$ is odd, we have $f(x)=xh({x}^{2})$. ([1], Exercise 1.2)

7.
The Fourier transform^{} of a real even function is purely real and even. The Fourier transform of a real odd function is purely imaginary and odd.
References
 1 L. Hörmander, The Analysis^{} of Linear Partial Differential Operators I, (Distribution^{} theory and Fourier Analysis), 2nd ed, SpringerVerlag, 1990.
Title  even and odd functions 

Canonical name  EvenAndOddFunctions 
Date of creation  20130322 13:34:19 
Last modified on  20130322 13:34:19 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  12 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 26A06 
Related topic  HermitianFunction 
Defines  even function 
Defines  odd function 