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In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function is an even function if n is an even integer, and it is an odd function if n is an odd integer. Evenness and oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups, all rings, all fields, and all vector spaces. Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable, and so on. The given examples are real functions, to illustrate the symmetry of their graphs. Let f be a real-valued function of a real variable. Then f is even if the following equation holds for all x such that x and −x are in the domain of f: or equivalently if the following equation holds for all such x: Geometrically, the graph of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis. Examples of even functions are: The absolute value cosine hyperbolic cosine Gaussian function Again, let f be a real-valued function of a real variable. Then f is odd if the following equation holds for all x such that x and −x are in the domain of f: or equivalently if the following equation holds for all such x: Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin. Examples of odd functions are: The sign function The identity function sine hyperbolic sine The error function If a function is both even and odd, it is equal to 0 everywhere it is defined.

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