In mathematics, a function on the complex plane is antiholomorphic at a point if its derivative with respect to z* exists, where here, z* is the complex conjugate. If the function is antiholomorphic at every point of some subset of the complex plane, then it is antiholomorphic on that set.
If f(z) is a holomorphic function, then f(z*) is an antiholomorphic function.
A function
is antianalytic if f(z*) is an analytic function (i.e. holomorphic).
The local representation of analytic functions by means of power series shows that being antianalytic in a neighbourhood of a complex number a is the same condition as the existence of a power series in z* − a.
, antiholomorphicfunctions (also called antianalytic functions) are a family of function In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range).
A function defined on an open set In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U.
One can show that if f(z) is a holomorphic function Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point.
Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point.
All polynomial functions in z with complex coefficients are holomorphic on C, and so are the trigonometric functions of z and the exponential function.
A complex analytic function of several complex variables is defined to be analytic and holomorphic at a point if it is locally expandable (within a polydisk, a cartesian product of disks, centered at that point) as a convergent power series in the variables.
Feeds |
Contact