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.
is an example of an even function.
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:: p. 11
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.
The composition of an even function and an odd function is even.
The composition of any function with an even function is even (but not vice versa).
Every function may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the even part and the odd part of the function; if one defines
then is even, is odd, and
where g is even and h is odd, then and since
For example, the hyperbolic cosine and the hyperbolic sine may be regarded as the even and odd parts of the exponential function, as the first one is an even function, the second one is odd, and
Further algebraic propertiesEdit
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 functions is the direct sum of the subspaces of even and odd functions. This is a more abstract way of expressing the property in the preceding section.
The space of functions can be considered a graded algebra over the real numbers by this property, as well as some of those above.
The even functions form a commutative algebra over the reals. However, the odd functions do not form an algebra over the reals, as they are not closed under multiplication.
The integral of an odd function from −A to +A is zero (where A is finite, and the function has no vertical asymptotes between −A and A). For an odd function that is integrable over a symmetric interval, e.g. , the result of the integral over that interval is zero; that is
The integral of an even function from −A to +A is twice the integral from 0 to +A (where A is finite, and the function has no vertical asymptotes between −A and A. This also holds true when A is infinite, but only if the integral converges); that is
In signal processing, harmonic distortion occurs when a sine wave signal is sent through a memory-less 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 . The type of harmonics produced depend on the response function f:
When the response function is even, the resulting signal will consist of only even harmonics of the input sine wave;
The fundamental is also an odd harmonic, so will not be present.
When it is asymmetric, the resulting signal may contain either even or odd harmonics;
Simple examples are a half-wave rectifier, and clipping in an asymmetrical class-A amplifier.
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.
^Berners, Dave (October 2005). "Ask the Doctors: Tube vs. Solid-State Harmonics". UA WebZine. Universal Audio. Retrieved 2016-09-22. To summarize, if the function f(x) is odd, a cosine input will produce no even harmonics. If the function f(x) is even, a cosine input will produce no odd harmonics (but may contain a DC component). If the function is neither odd nor even, all harmonics may be present in the output.
^Proakis, John G.; Manolakis, Dimitri G. (1996), Digital Signal Processing: Principles, Algorithms and Applications (3 ed.), Upper Saddle River, NJ: Prentice-Hall International, ISBN 9780133942897, sAcfAQAAIAAJ
Gelfand, I. M.; Glagoleva, E. G.; Shnol, E. E. (2002) , Functions and Graphs, Mineola, N.Y: Dover Publications