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If a real function has a domain that is self-symmetric with respect to the origin, it may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the even part (or the even component) and the odd part (or the odd component) of the function, and are defined by = + (), and = ().
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).
Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric -tensors on a vector space is isomorphic to the space of homogeneous polynomials of degree on . Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.
Examples include even and odd functions in calculus, symmetric groups in abstract algebra, symmetric matrices in linear algebra, and Galois groups in Galois theory. In statistics, symmetry also manifests as symmetric probability distributions, and as skewness—the asymmetry of distributions. [16]
Schematic representation of the Dirac delta function by a line surmounted by an arrow. ... the delta function is an even distribution (symmetry), ... if n is odd and ...
The function is Hermitian if and only if the Fourier transform of is real-valued. Since the Fourier transform of a real signal is guaranteed to be Hermitian, it can be compressed using the Hermitian even/odd symmetry.
Wave functions with eigenvalue + under a parity transformation are even functions, while eigenvalue corresponds to odd functions. [1] However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than ± 1 {\displaystyle \pm 1} .
The odd graph = (,) In the mathematical field of graph theory, the odd graphs are a family of symmetric graphs defined from certain set systems. They include and generalize the Petersen graph. The odd graphs have high odd girth, meaning that they contain long odd-length cycles but no short ones.