Harmonic Conjugate

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
:

abla^2 u = 0

.
and thus is harmonic. Now suppose we have a

v(x,y)

such that the Cauchy-Riemann equations are satisfied:

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 , of finding the curves that cross a given family of non-intersecting curves at Right Angle s.

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 Operator s.

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	CATEGORIES ABOUT HARMONIC CONJUGATE

	harmonic functions

	partial differential equations

Information About

Harmonic Conjugate