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The Helmholtz Equation and Its Solution 

An equation of the form
is known as a Helmholtz equation. It may have solutions for only discrete values of λ. Those values of λ would be called eigenvalues. Generally the relevant values of λ are positive.
Suppose ψ is a function of the polar coordinates (r, θ). Furthermore let us assume there is a separation of variables; i.e., ψ(r, θ)=R(r)Θ(θ).
The Laplacian ∇² for polar coordinates
With separation of variables the Helmholtz Equation becomes
The next step in the solution is to multiply through by r² to get
The LHS of the above is a function only of r and the RHS is a function only of θ. Therefore their common value must be a constant, say N.
This means that Θ(θ) is a solution to the equation
The solution of this equation is
where A and θ_{0} are constants. In order for this to be a valid solution it must be that
This requires that √N(2π)=n(2π), where n is an integer. Thus N=n²; i.e., the constant N is the square of an integer.
With a proper orientation of the polar coordinate system θ_{0} can be made equal to zero.
The function R(r) must satisfy
This is closely related to the Bessel equation
To transform the radial equation into the Bessel equation let ρ=√λr. Then
Thus the radial equation is transformed into
This has the solution
where J_{n}(ρ) is a Bessel function of index n. There are two types of Bessel functions.
The general solution to the Helmholtz equation is then
where A and B are constants and J_{n}(.) and K_{n}(.) are Bessel functions of the first and second type, both of index n, an integer.
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