5: Non-sinusoidal Harmonics and Special Functions
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To the pure geometer the radius of curvature is an incidental characteristic-like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely incidental to the grin. Physics is concerned with interrelatedness such as the interrelatedness of cats and grins. In this case the "cat without a grin" and the "grin without a cat" are equally set aside as purely mathematical phantasies.
~ Sir Arthur Stanley Eddington (1882-1944)
In this chapter we provide a glimpse into generalized Fourier series in which the normal modes of oscillation are not sinusoidal. For vibrating strings, we saw that the harmonics were sinusoidal basis functions for a large, infinite dimensional, function space. Now, we will extend these ideas to non-sinusoidal harmonics and explore the underlying structure behind these ideas. In particular, we will explore Legendre polynomials and Bessel functions which will later arise in problems having cylindrical or spherical symmetry.
The background for the study of generalized Fourier series is that of function spaces. We begin by exploring the general context in which one finds oneself when discussing Fourier series and (later) Fourier transforms. We can view the sine and cosine functions in the Fourier trigonometric series representations as basis vectors in an infinite dimensional function space. A given function in that space may then be represented as a linear combination over this infinite basis. With this in mind, we might wonder
- Do we have enough basis vectors for the function space?
- Are the infinite series expansions convergent?
- What functions can be represented by such expansions?
In the context of the boundary value problems which typically appear in physics, one is led to the study of boundary value problems in the form of Sturm-Liouville eigenvalue problems. These lead to an appropriate set of basis vectors for the function space under consideration. We will touch a little on these ideas, leaving some of the deeper results for more advanced We note that the above determination of vector components for finite dimensional spaces is precisely what we had done to compute the Fourier coefficients using trigonometric bases. Reading further, you will see how this works. courses in mathematics. For now, we will turn to function spaces and explore some typical basis functions, many which originated from the study of physical problems. The common basis functions are often referred to as special functions in physics. Examples are the classical orthogonal polynomials (Legendre, Hermite, Laguerre, Tchebychef) and Bessel functions. But first we will introduce function spaces.
- 5.3: Fourier-Legendre Series
- In the last chapter we saw how useful Fourier series expansions were for solving the heat and wave equations. Here we will investigate partial differential equations in higher dimensions and find that problems with spherical symmetry may lead to the series representations in terms of a basis of Legendre polynomials.
- 5.4: Gamma Function
- A function that often occurs in the study of special functions is the Gamma function. We will need the Gamma function in the next section on Fourier-Bessel series.
- 5.5: Fourier-Bessel Series
- Bessel functions arise in many problems in physics possessing cylindrical symmetry such as the vibrations of circular drumheads and the radial modes in optical fibers. They also provide us with another orthogonal set of basis functions.
Thumbnail: Plot of Bessel function of the first kind,\(J_α(x)\) for integer orders \(α = 0,\, 1,\, 2\). (Public Domain; Inductiveload via Wikipedia)