mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size...

engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis...

such a way that the operations of multiplication and taking the inverse are analytic functions of the matrix entries. Thus SO(n) is for each n a Lie group...

multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so...

convergent on a half-plane, that may give rise to an L-function via analytic continuation. The Riemann zeta function is an example of an L-function, and some...

that holomorphic functions are analytic and analytic complex functions are complex-differentiable. In particular, holomorphic functions are infinitely...

the matrix valued analogue csgn ⁡ ( A ) {\displaystyle \operatorname {csgn} (A)} . Although the sign function is not analytic, the matrix function is well...

In mathematics, a matrix (pl.: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows...

matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always exists when treated as a function of...

functional analysis, a linear function is a linear map. In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or...

quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to ordered...

publication Perspectives in 1970. The matrix helps a company to allocate resources and is used as an analytical tool in brand marketing, product management...

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers...

sign function or signum function (from signum, Latin for "sign") is a function that has the value −1, +1 or 0 according to whether the sign of a given...

domain of a given function, for example by continuity or by analytic continuation. This means that it is not worthy to explicitly define the domain of a function...

orthogonal matrix. For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertible complex analytic functions...

complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's...

f ( A ) = r ( A ) . {\displaystyle f(A)=r(A).} Thus, the analytic function of the matrix A can be expressed as a matrix polynomial of degree less than...

Roger A.; Johnson, Charles R. (1990). Matrix analysis. Cambridge: Cambridge Univ. Press. p. 411. ISBN 9780521386326. For analytic functions of matrices...

n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space)...

In linear algebra, a circulant matrix is a square matrix in which all rows are composed of the same elements and each row is rotated one element to the...

meaning of this superscript is inconsistent with the others. Most special functions are considered as a function of a complex variable. They are analytic; the...

Gaussian functions are analytic, and their limit as x → ∞ is 0 (for the above case of b = 0). Gaussian functions are among those functions that are elementary...

Riemannian manifold) plurisubharmonic functions can be defined in full generality on complex analytic spaces. A function f : G → R ∪ { − ∞ } , {\displaystyle...

objects that figure prominently in their function theory, by taking τ to be the period matrix with respect to a canonical basis for its first homology group...

from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series. Zonal spherical functions have...

derivative of a scalar function with respect to a matrix, known as the gradient matrix, which collects the derivative with respect to each matrix element...

linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A  is symmetric ⟺ A = A T . {\displaystyle A{\text{ is symmetric}}\iff...

invertible matrix (non-singular, non-degenarate or regular) is a square matrix that has an inverse. In other words, if some other matrix is multiplied...

definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential...