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Dear Swapnil,
The Laplace equation in two independent variables has the form
The real and imaginary parts of a complex analytic function both satisfy the Laplace equation. That is, if z = x + iy, and if
then the necessary condition that f(z) be analytic is that the Cauchy-Riemann equations be satisfied:
where ux is the first partial derivative of u with respect to x.
It follows that
Therefore u satisfies the Laplace equation. A similar calculation shows that v also satisfies the Laplace equation.
Conversely, given a harmonic function, it is the real part of an analytic function, (at least locally). If a trial form is
then the Cauchy-Riemann equations will be satisfied if we set
This relation does not determine ψ, but only its increments:
The Laplace equation for φ implies that the integrability condition for ψ is satisfied:
and thus ψ may be defined by a line integral. The integrability condition and Stokes theorem implies that the value of the line integral connecting two points is independent of the path. The resulting pair of solutions of the Laplace equation are called conjugate harmonic functions. This construction is only valid locally, or provided that the path does not loop around a singularity. For example, if r and θ are polar coordinates and
then a corresponding analytic function is
However, the angle θ is single-valued only in a region that does not enclose the origin.
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