harmonic conjugates

harmonic conjugates

[här′män·ik ′kän·jə·gəts]
(mathematics)
Two points, P3 and P4, that are collinear with two given points, P1 and P2, such that P3 lies in the line segment P1 P2 while P4 lies outside it, and, if x1, x2, x3, and x4 are the abscissas of the points, (x3-x1)/ (x3-x2) = - (x4-x1)/(x4-x2).
A pair of harmonic functions, u and v, such that u + iv is an analytic function, or, equivalently, u and v satisfy the Cauchy-Riemann equations.
References in periodicals archive ?
Therefore, in projective geometry, the four points [A.sub.1], [A'.sub.1], O, [V.sub.1] and the four points [B.sub.1], [B'.sub.1], O, [V.sub.1] are harmonic conjugates, respectively.
In this study, based on the theory of harmonic conjugates in projective geometry in combination with the vanishing point and the centre of concentric circles, a calibration method that uses circles and line is proposed.
He continues with Abel's theorem, the gamma function, universal covering spaces, Cauchy's theorem for non-holomorphic functions and harmonic conjugates. The result is suitable for a course in complex analysis for students with experience in advanced calculus.