From Pythagoras to Fourier and From Geometry to Nature

DOI: https://doi.org/10.55060/b.p2fg2n.ch005.220215.008

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Consider in the plane *x*, *y* the circle *C*, centered at the origin *O* and with radius *r*. We want to construct a function *harmonic* regular in *C*, which takes on assigned values on the boundary *∂C*. It is therefore necessary to solve the problem:

It is usual to introduce the polar coordinates (*x* = *ρ* cos *φ*, *y* = *ρ* sin *φ*) and to translate the problem (5.1) into the equivalent one (with a simplified notation):

For the determination of these coefficients there are several methods: separation of variables, transforms or computation of coefficients by using identities of trigonometric series. Here we will operate in a “formal” way which can be verified. Assuming ∀*ρ* < *r* that we can derive *twice* by series, substituting (5.3) into (5.2) we get:

These are all equations of the Euler type, with the peculiarity of claiming regular solutions even for *ρ* = 0. From:

From Equation (5.4) for *ρ* = *r* (as a consequence of the claimed regularity) it follows that:

We thus formally arrive at the expression of the solution:

The appropriate checks can be carried out on this expression.

Let us start with some definitions necessary for the understanding of what follows.

The Gamma function is the extension of the factorial to non-integer values of the number *n* ∈ ℕ^{+}. For *x* ≠ −*n* it is defined as:

In fact, we have:

The Bessel functions of the first kind *J _{n}*, together with those of the second kind

We get the explicit expression of *J _{n}* in the form:

One of the well known applications of the Bessel functions [1] is related to the separation of variables in the partial differential equation representing the heat equation for a circular plate.

In fact, denoting by *B* a circular domain of radius *r* = 1 centered at the origin, by *∂B* the relevant boundary, by *κ* a constant representing the known diffusivity and by *f* (*x*,* y*) ∈ *C*^{0}(*B*) the initial temperature, the solution:

Another well known application of the Bessel functions [1] is related to the separation of variables in the partial differential equation representing the free vibrations of a circular membrane (drumhead). Denoting by *B* a circular domain of radius *r* = 1 centered at the origin, by *∂B* the relevant boundary, by *τ* denotes the tension and *μ* the density) and by *f*(*x*,* y*) ∈ *C*^{0}(*B*) the initial displacement, the solution

Moreover, the eigenvalues of a vibrating circular membrane are related to the zeros of the Bessel functions, since the relevant elementary frequencies are given by: