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, 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.
Even functions
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 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:
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 composition of an even function and an odd function is even.
The composition of any function with an even function is even.
Even–odd decomposition
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 and then is even, is odd, and Conversely, if where is even and 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 properties
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 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.
Analytic properties
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. In the following, properties involving derivatives, Fourier series, Taylor series, and so on suppose that these concepts are defined of the functions that are considered.
The integral of an odd function from −A to +A is zero. 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 ; that is
Series
The Maclaurin series of an even function includes only even powers.
The Maclaurin series of an odd function includes only odd powers.
The Fourier series of a periodic even function includes only cosine terms.
The Fourier series of a periodic odd function includes only sine terms.
The Fourier transform of a purely real-valued even function is real and even.
The Fourier transform of a purely real-valued odd function is imaginary and odd.
Harmonics
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;
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.
Generalizations
Multivariate functions
Even symmetry: A function is called even symmetric if: Odd symmetry: A function is called odd symmetric if:
Complex-valued functions
The definitions for even and odd symmetry for complex-valued functions of a real argument are similar to the real case but involve complex conjugation. Even symmetry: A complex-valued function of a real argument is called even symmetric if: Odd symmetry: A complex-valued function of a real argument is called odd symmetric if:
The definitions of odd and even symmetry are extended to N-point sequences as follows: Even symmetry: A N-point sequence is called even symmetric if Such a sequence is often called a palindromic sequence; see also Palindromic polynomial. Odd symmetry: A N-point sequence is called odd symmetric if Such a sequence is sometimes called an anti-palindromic sequence; see also Antipalindromic polynomial.
