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...

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

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...

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...

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

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...

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...

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

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

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

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...

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

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...

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...

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...

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)...

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...

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...

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...

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...

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

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...

then F(Λ) is an analytic function of z. If α is a non-zero complex number and αΛ is the lattice obtained by multiplying each element of Λ by α, then F(αΛ)...

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...