Even Function

An even function is a univariate function satisfying . Geometrically, such functions are symmetric about the -axis. Examples of even functions include 1 (or, in general, any constant function), , , , and .

For any univariate function whose domain is symmetric about 0, the even part is defined by

It is even, and is the odd part of . For a univariate polynomial, the even part is the sum of the monomials containing even powers of the variable.

An even function times an odd function is odd, while the sum or difference of two even functions is even. The product or quotient of two even functions is again even.

If a univariate function that is even is differentiable, then its derivative is an odd function. If an even function is integrable over a symmetric interval , where , then its integral over is twice its integral over the interval . Similarly, if an odd function is differentiable, then its derivative is an even function. If an odd function is integrable over , then its integral over is zero. For , the two one-sided improper integrals must each be finite.

Ostensibly, one can define a similar notion for multivariate functions by saying that such a function is even iff

Even so, such functions are unpredictable and very well may lose many of the desirable geometric properties possessed by univariate functions. For example, both and satisfy this identity while the constant slices and of and are odd and even, respectively. Differentiability and integrability properties are similarly unclear.

The maclaurin series of an even function contains only even powers of .