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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 f(x) = x^{n} is an even function if n is an even integer, and it is an odd function if n is an odd integer.
The concept of evenness or oddness is defined for functions whose domain and image both have an additive inverse. This includes additive groups, all rings, all fields, and all vector spaces. Thus, for example, a real-valued function of a real variable could be even or odd, as could a complex-valued function of a vector variable, and so on.
The examples are real-valued functions of a real variable, to illustrate the symmetry of their graphs.
Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x and -x in the domain of f:^{[1]}
or
Geometrically speaking, the graph face 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 |x|, x^{2}, x^{4}, cos(x), and cosh(x).
Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all x and -x in the domain of f:^{[2]}
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 x, x^{3}, sin(x), sinh(x), and erf(x).
A function's being odd or even does not imply differentiability, or even continuity. For example, the Dirichlet function is even, but is nowhere continuous. Properties involving Fourier series, Taylor series, derivatives and so on may only be used when they can be assumed to exist.
In signal processing, harmonic distortion occurs when a sine wave signal is sent through a memoryless nonlinear system, that is, a system whose output at time t only depends on the input at time t and does not depend on the input at any previous times. Such a system is described by a response function V_\text{out}(t) = f(V_\text{in}(t)). The type of harmonics produced depend on the response function f:^{[3]}
Note that this does not hold true for more complex waveforms. A sawtooth wave contains both even and odd harmonics, for instance. After even-symmetric full-wave rectification, it becomes a triangle wave, which, other than the DC offset, contains only odd harmonics.
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