even and odd functions

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series.

Even functions

Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all real x:

f(−x) = f(x)

Geometrically, an even function is symmetric with respect to the y-axis. The designation even is due to the fact that the Taylor series of an even function includes only even powers. Examples of even functions are x², x⁴, cos(x), and cosh(x).

Odd functions

Again, let f(x) be a valued function of a real variable. Then f is odd if the following equation holds for all real x:

f(−x) = −f(x)

Geometrically, an odd function is symmetric with respect to the origin. The designation odd is due to the fact that the Taylor series of an odd function includes only odd powers. Examples of odd functions are x, x³, sin(x), and sinh(x).

Some facts

Basic properties

The only function which is both even and odd is the constant function which is identically zero (i.e., f(x) = 0 for all x).
In general, the sum of an even and odd function is neither even nor odd; e.g. x + x².
The sum of 2 even functions is even, and any constant multiple of an even function is even.
The sum of 2 odd functions is odd, and any constant multiple of an odd function is odd.
The product of 2 even functions is an even function.
The product of 2 odd functions is again an even function.
The product of an even function and an odd function is an odd function.
The derivative of an even function is odd.
The derivative of an odd function is even.

Series

The Taylor series of an even function includes only even powers.
The Taylor series of an odd function includes only odd powers.
The Fourier series of a periodic even function includes only cosine terms.
The Fourier series of a periodic odd function includes only sine terms.

Algebraic Structure

Any linear combination of even functions is even, and the even functions form a vector space over the reals. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. In fact, the vector space of all real-valued functions is the direct sum of the spaces of even and odd functions. In other words, every function can be written uniquely as the sum of an even function and an odd function:

f(x)=\frac{f(x)+f(-x)}{2}\,+\,\frac{f(x)-f(-x)}{2}

The even functions form a commutative algebra over the reals. However, the odd functions do not form an algebra over the reals.