Conjugate Functions

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Conjugate Functions

 

Two functions u(x, y) and v(x, y) of the two variables x and y are said to be conjugate if in some region D they satisfy the Cauchy-Riemann equations

Under certain conditions—for example, if u and v have continuous first-order partial derivatives— u and v are the real and imaginary parts, respectively, of some analytic function f(x + iy). They satisfy Laplace’s equation in D:

That is, they are harmonic functions. If a harmonic function, such as u(x, y), is specified in a simply connected region D, the conjugate harmonic function v(x, y) and, consequently, the analytic function f(x + iy) are thereby uniquely defined (to within a constant term).

As an example, let us consider the function

where ø = arg(x + iy). If this function is harmonic within some circle ǀx + iyǀ = r < R, then its conjugate function is

and

The values of the conjugate functions on the circle r = 1 are periodic functions of ø and can be expanded in the conjugate trigonometric series

and

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Babenko, "On conjugate functions," Doklady Akademii Nauk SSSR, vol.
Riesz on conjugate functions," Matematicheskii Sbornik, vol.
Shawe-Taylor, "Sparse semi-supervised learning using conjugate functions," Journal of Machine Learning Research, vol.
STECH KIN Best approximations and differential properties of two conjugate functions (Russian) Trudy Moskov.