Harmonic conjugate

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For geometric conjugate points, see Projective harmonic conjugates.

In mathematics, the harmonic conjugate of a harmonic real-valued function of two variables u(x,y), is a function v(x,y) such that v is harmonic and u and v satisfy the Cauchy-Riemann equations, that is, the complex-valued function u(x,y) + iv(x,y) = f(z) is analytic. The harmonic conjugate (when it exists, in a given connected region) is unique up to addition of a constant to v.

For example, consider the function

u(x,y)=e^x \sin y. \,

Since

{\partial u \over \partial x } = e^x \sin y, {\partial^2 u \over \partial x^2} = e^x \sin y

and

{\partial u \over \partial y} = e^x \cos y, {\partial^2 u \over \partial y^2} = - e^x \sin y,

it satisfies

\nabla^2 u = 0.\,

and thus is harmonic. Now suppose we have a v(x,y) such that the Cauchy-Riemann equations are satisfied:

{\partial u \over \partial x} = {\partial v \over \partial y} = e^x \sin y\,

and

{\partial u \over \partial y} = -{\partial v \over \partial x} = e^x \cos y.\,

Simplifying,

{\partial v \over \partial y} = e^x \sin y

and

{\partial v \over \partial x} = -e^x \cos y

which when solved gives

v = -e^x \cos y. \!\;

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.

Another formulation of the harmonic conjugate is given by the theory of the Hilbert transform. Estimates for it form an important topic in mathematical analysis, typical of the theory of singular integral operators.

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 collinear points C, D if (ABCD) = −1, where (ABCD) is the cross-ratio of points A, B, C, D.

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