Even Function


A univariate function f(x) is said to be even provided that f(x)=f(-x). Geometrically, such functions are symmetric about the y-axis. Examples of even functions include 1 (or, in general, any constant function), |x|, cosx, x^2, and e^(-x^2).

An even function times an odd function is odd, while the sum or difference of two nonzero functions is even if and only if each summand function is even. The product or quotient of two even functions is again even.

If a univariate even function is differentiable, then its derivative is an odd function; what's more, if an even function is integrable, then its integral over a symmetric interval I=[-a,a], a in R union {infty}, is precisely the same as twice the integral over the interval [0,a]. Similarly, if an odd function is differentiable, then its derivative is an even function while the integral of such a function over a symmetric interval I is identically zero.

Ostensibly, one can define a similar notion for multivariate functions f(x_1,x_2,...,x_n) by saying that such a function is even if and only if


Even so, such functions are unpredictable and very well may lose many of the desirable geometric properties possessed by univariate functions. For example, both f(x,y)=sin(xy) and g(x,y)=cos(xy) satisfy this identity while the constant slices x=C_1 and y=C_2 of f and g are odd and even, respectively. Differentiability and integrability properties are similarly unclear.

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

See also

Odd Function

Portions of this entry contributed by Christopher Stover

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Cite this as:

Stover, Christopher and Weisstein, Eric W. "Even Function." From MathWorld--A Wolfram Web Resource.

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