Special functions and series in applications
Series solutions give rise to named special functions that appear across physics and engineering. Knowing their properties is as useful as knowing sine and cosine.
Why special functions?
Sine, cosine, and exponentials are solutions of simple differential equations. When the equation is more complex, its solutions get names too: Bessel, Legendre, Hermite, Laguerre, Airy. Each has known properties, tables, and software implementations, making them as practical as elementary functions.
Hermite polynomials
Hermite's equation \(y'' - 2xy' + 2ny = 0\) has polynomial solutions \(H_n(x)\) for integer \(n\): \(H_0 = 1\), \(H_1 = 2x\), \(H_2 = 4x^2 - 2\). Multiplied by \(e^{-x^2/2}\), they give the quantum harmonic oscillator's states, which model vibrating molecules.
Orthogonal polynomials
Legendre, Hermite, and Laguerre polynomials are each orthogonal with respect to a weight function on an interval. Like sines in Fourier series, they form bases for expanding other functions, used in numerical integration (Gaussian quadrature) and approximation.
Using series for approximation
Truncating a series solution gives a polynomial approximation valid near the expansion point. For the pendulum equation \(\theta'' + \frac{g}{L}\sin\theta = 0\), replacing \(\sin\theta\) by its series \(\theta - \frac{\theta^3}{6}\) gives corrections to the period at larger swings.
Recurrences for special functions
Special functions satisfy recurrence relations that make computation efficient. Legendre polynomials obey \((n + 1)P_{n+1} = (2n + 1)xP_n - nP_{n-1}\), so each is computed from the previous two without recomputing series.
Generating functions
A generating function packs a whole family into one expression. For Legendre polynomials, \(\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum P_n(x)t^n\). This formula arises from expanding the gravitational or electric potential of a point mass, explaining why Legendre polynomials appear in multipole expansions.
Numerical evaluation
Software libraries evaluate special functions using series near small arguments, asymptotic formulas for large arguments, and recurrences in between. Engineers rarely compute them by hand, but understanding their behaviour, such as oscillation and decay, guides modelling.
Common mistakes
Using a truncated series far from its expansion point. Confusing different normalisation conventions for special functions between books and software. Forgetting weight functions in orthogonality relations.
Key takeaways
Special functions are named solutions of important differential equations. Hermite, Legendre, and Laguerre polynomials are orthogonal families useful for expansions. Recurrences and generating functions organise their computation. Truncated series give local approximations, such as corrections to the pendulum period.
Flashcard
A named solution of an important ODE
Check your understanding
Hermite polynomials are most associated with: