Show that the function (10×2 = 20)

a) u = x4 − 6x2 y2 + y is a solution of the two-dimensional Laplace equation.

b) u = sin w ct sin wx is a solution of the one-dimensional wave equation.

Dear Swapnil,

The Laplace equation in two independent variables has the form

[edit]Analytic functions

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 u_x 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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