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Antiholomorphic function

Lecture



Antiholomorphic functions (also called antianalytic functions ) are a family of functions that are closely related to holomorphic functions.

Definition [edit]

Function   Antiholomorphic function defined on an open subset   Antiholomorphic function complex plane is called antiholomorphic if its derivative   Antiholomorphic function by   Antiholomorphic function exists at all points of this set. This is equivalent to the condition

  Antiholomorphic function

which can be given a view similar to Cauchy - Riemann conditions:

  Antiholomorphic function
  Antiholomorphic function

Where

  Antiholomorphic function

A function that depends simultaneously on   Antiholomorphic function and   Antiholomorphic function , is neither holomorphic nor antiholomorphic.

Properties [edit]

  •   Antiholomorphic function holomorphic in   Antiholomorphic function then and only if   Antiholomorphic function antiholomorphic in   Antiholomorphic function .
  • a function is antiholomorphic if and only if it can be expanded in powers   Antiholomorphic function in the neighborhood of each point of its domain.
  •   Antiholomorphic function holomorphic in   Antiholomorphic function then and only if   Antiholomorphic function antiholomorphic in   Antiholomorphic function .
  • if a function is simultaneously holomorphic and antiholomorphic, then it is constant on any connected component of its domain.

Literature [edit]

  • Shabat B.V. An Introduction to Complex Analysis. - M .: Science. - 1969, 577 pp.
created: 2014-10-25
updated: 2021-03-13
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Comprehensive analysis and operational calculus

Terms: Comprehensive analysis and operational calculus