Harmonic conjugate

In mathematics, a function defined on some open domain is said to have as a conjugate a function if and only if they are respectively real and imaginary part of a holomorphic function f(z) of the complex variable That is, v is conjugated to u if f(z): = u(x,y) + iv(x,y) is holomorphic on Ω. As a first consequence of the definition, they are both harmonic real-valued functions on Ω. Moreover, the conjugate of u, if it exists, is unique up to an additive constant. Also, u is conjugate to v if and only if v is conjugate to − u.

Equivalently, v is conjugate to u in Ω if and only u and v satisfy the Cauchy-Riemann equations in Ω. As an immediate consequence of the latter equivalent definition, if u is any harmonic function on the function u_y is conjugate to − u_x, for then the Cauchy-Riemann equations are just Δu = 0 and the symmetry of the mixed second order derivatives, u_xy = u_yx. Therefore an harmonic function u admits a conjugated harmonic function if and only if the holomorphic function g(z): = u_x(x,y) − iu_y(x,y) has a primitive f(z) in Ω, in which case a conjugate of u 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 R² to its harmonic conjugate v (putting e.g. v(x₀)=0 on a given x₀ in order to fix the indeterminacy of the conjugate up to constants). This is well known in applications as (essentially) the Hilbert transform; it is also a basic example in mathematical analysis, in connection with singular integral operators. 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 thus is harmonic. Now suppose we have a v(x,y) 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 mapping property of analytic functions (at points where the derivative is not zero) 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 equally contours of constant u(x,y) and v(x,y) will also be orthogonal where they cross (away from the zeroes of f′(z)). That means that v is a specific solution of the orthogonal trajectory problem for the family of contours given by u (not the only solution, naturally, since we can take also functions of v): 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.

There is an additional occurrence of the term harmonic conjugate in mathematics, and more specifically in 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 (ABCD) = −1, where (ABCD) is the cross-ratio of points A, B, C, D.

External links

• Harmonic Ratio