Chebyshev's equation

Some special functions come from physics; Chebyshev's come from the question of how to approximate a function by a polynomial as evenly as possible. Chebyshev's equation is the differential equation whose polynomial solutions, the Chebyshev polynomials, are the champions of uniform approximation — the polynomials that wiggle with perfectly equal amplitude across an interval.

It is (1 - x^2) y'' - x y' + n^2 y = 0 on -1 <= x <= 1, with regular singular points at x = +1 and x = -1, like Legendre's. A power-series solution truncates to a polynomial exactly when n is a non-negative integer, giving the Chebyshev polynomial of the first kind T_n(x). The striking fact is a hidden change of variables: setting x = cos(theta) turns the whole equation into the trivial d^2y/d(theta)^2 + n^2 y = 0, so T_n(cos theta) = cos(n theta). The famous wiggling polynomial is just a cosine wearing an algebraic disguise.

Because cos(n theta) oscillates between +1 and -1 with equal-height ripples, T_n is the polynomial of its degree that stays closest to zero on [-1, 1] in the worst-case (uniform) sense. This 'equal-ripple' property makes Chebyshev polynomials the backbone of numerical analysis: Chebyshev interpolation avoids the wild oscillations (Runge's phenomenon) that plague equally-spaced polynomial fits, Chebyshev nodes are the optimal sampling points, and Chebyshev expansions underlie spectral methods and the design of electronic filters. Here the 'equation' is the gateway, but the payoff is overwhelmingly computational rather than physical.