Search results
Results from the WOW.Com Content Network
The definitions for even and odd symmetry for complex-valued functions of a real argument are similar to the real case. In signal processing, a similar symmetry is sometimes considered, which involves complex conjugation. [4] [5] Conjugate symmetry:
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 drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.. In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). [1]
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]
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.
From this definition it follows immediately that: is a Hermitian function if and only if the real part of f {\displaystyle f} is an even function , the imaginary part of f {\displaystyle f} is an odd function .
A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. (See constructible polygon .) A regular n -sided polygon can be constructed with origami if and only if n = 2 a 3 b p 1 ⋯ p r {\displaystyle n=2^{a}3^{b}p_{1}\cdots p_{r}} for some r ∈ N ...
An odd function, such as an odd power of a variable, gives for any argument the negation of its result when given the negation of that argument. It is possible for a function to be neither odd nor even, and for the case f ( x ) = 0, to be both odd and even. [ 20 ]