Antiholomorphic function
Encyclopedia
In mathematics
, antiholomorphic functions (also called antianalytic functions) are a family of function
s closely related to but distinct from holomorphic function
s.
A function of the complex variable z defined on an open set
in the complex plane
is said to be antiholomorphic if its derivative
with respect to z* exists in the neighbourhood of each and every point in that set, where z* is the complex conjugate
.
One can show that if f(z) is a holomorphic function
on an open set D, then f(z*) is an antiholomorphic function on D*, where D* is the reflection against the x-axis of D, or in other words, D* is the set of complex conjugates of elements of D. Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function. This implies that a function is antiholomorphic if and only if
it can be expanded in a power series in z* in a neighborhood of each point in its domain.
If a function is both holomorphic and antiholomorphic, then it is constant on any connected component
of its domain.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, antiholomorphic functions (also called antianalytic functions) are a family of function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s closely related to but distinct from holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s.
A function of the complex variable z defined on an open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
is said to be antiholomorphic if its derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
with respect to z* exists in the neighbourhood of each and every point in that set, where z* is the complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
.
One can show that if f(z) is a holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
on an open set D, then f(z*) is an antiholomorphic function on D*, where D* is the reflection against the x-axis of D, or in other words, D* is the set of complex conjugates of elements of D. Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function. This implies that a function is antiholomorphic if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it can be expanded in a power series in z* in a neighborhood of each point in its domain.
If a function is both holomorphic and antiholomorphic, then it is constant on any connected component
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
of its domain.