Equivalently, is conjugate to in if and only if and satisfy the Cauchy–Riemann equations in As an immediate consequence of the latter equivalent definition, if is any harmonic function on the function is conjugate to for then the Cauchy–Riemann equations are just and the symmetry of the mixed second order derivatives, Therefore, a harmonic function admits a conjugated harmonic function if and only if the holomorphic function has a primitive in in which case a conjugate of is, of course, So any harmonic function always admits a conjugate function whenever its domain is simply connected, and in any case it admits a conjugate locally at any point of its domain. There is an operator taking a harmonic function u on a simply connected region in to its harmonic conjugate v. This is well known in applications as the Hilbert transform; it is also a basic example in mathematical analysis, in connection with singular integral operators. Conjugate harmonic functions are also one of the simplest examples of a Bäcklund transform, in this case linear; more complex transforms are of interest in solitons and integrable systems. Geometrically u and v are related as having orthogonal trajectories, away from the zeroes of the underlying holomorphic function; the contours on which u and v are constant cross at right angles. In this regard, u+iv would be the complex potential, where u is the potential function and v is the stream function.
Examples
For example, consider the function Since and it satisfies and is thus harmonic. Now suppose we have a such that the Cauchy–Riemann equations are satisfied: and Simplifying, and which when solved gives Observe that if the functions related to u and v were interchanged, the functions would not be harmonic conjugates, since the minus sign in the Cauchy–Riemann equations makes the relationship asymmetric. The conformal mappingproperty ofanalytic functions gives rise to a geometric property of harmonic conjugates. Clearly the harmonic conjugate of x is y, and the lines of constant x and constant y are orthogonal. Conformality says that contours of constant u and v will also be orthogonal where they cross. That means that v is a specific solution of the orthogonal trajectory problem for the family of contours given by u : the question, going back to the mathematics of the seventeenth century, of finding the curves that cross a given family of non-intersecting curves at right angles.
Harmonic conjugate in geometry
There is an additional occurrence of the term harmonic conjugate in mathematics, and more specifically in projective geometry. Two points A and B are said to be harmonic conjugates of each other with respect to another pair of points C, D if the cross ratio = –1.
