Power series

A power series is an infinite series of the form

\sum_{n=0}^{\infty}a_{n}(x-c)^{n}=a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+\cdots

in which the coefficients aₙ are constants. Number c is known as the centre of the series.

Radius of convergence

A power series converges absolutely for |x − c| < R and diverges for |x − c| > R; the number R is the radius of convergence (the endpoints |x − c| = R must be examined separately). When the ratio test applies, the radius follows directly from the coefficients:

The exponential series e^x = Σxⁿ/n! converges for every real x (R = ∞), whereas the geometric series 1/(1 − x) = Σxⁿ has radius R = 1.

Power series solutions of differential equations

Linear differential equations whose coefficients vary with x rarely admit closed-form solutions. When the coefficients are analytic at a point, one can still seek a solution in the form of a power series about that point: substituting the series into the equation and equating the coefficients of like powers of x turns the differential equation into recurrence relations for the coefficients. The free constants left by the recurrence are exactly the arbitrary constants of the equation, and the resulting series is an exact solution on its interval of convergence.

The role of such series among the approaches to differential equations is discussed in the article Differential equation.

Worked example: the Airy equation

The Airy equation

y″ − xy = 0

is named after George Biddell Airy (1801–1892), who encountered it in 1838 while studying the intensity of light near a caustic. It has no solution expressible as a finite combination of elementary functions, so it is the standard test case for series methods.

Seek a solution about the ordinary point x = 0 as a power series y = Σaₙxⁿ. Substituting the series for y″ and for xy, then reindexing so that both run over the same powers of x, gives

Equating the coefficients of xⁿ on the two sides of y″ − xy = 0:

The coefficient of x⁰ gives a₂ = 0, and for n ≥ 1 each coefficient is fixed by the one three places earlier: a₃ = a₀/6, a₄ = a₁/12, a₅ = 0, a₆ = a₀/180, a₇ = a₁/504, and so on. The two free constants a₀ and a₁ therefore generate two linearly independent solutions:

Each series converges for every x (the ratio test gives R = ∞). Up to normalisation these two solutions are the Airy functions Ai and Bi:

y''-xy=0\qquad\Longrightarrow\qquad y=C_{1}\operatorname{Ai}(x)+C_{2}\operatorname{Bi}(x)

The power series thus solve the Airy equation exactly, in open form, on the whole real line. The Airy functions appear in quantum mechanics, optics and elsewhere whenever solutions change from oscillatory to exponential behaviour.

Ordinary points, singular points and the Frobenius method

Write a linear second-order equation as y″ + p(x)y′ + q(x)y = 0. A point x₀ is ordinary when both p and q are analytic there; the power-series method above applies directly. If at least one of p or q has a pole at x₀, the point is singular; it is a regular singular point when (x − x₀)p(x) and (x − x₀)²q(x) are still analytic there.

Around a regular singular point the Frobenius method seeks a solution with a shifted leading power, y = x^r Σaₙxⁿ; equating the lowest power of x fixes the exponent r through an indicial equation. For example, the Bessel equation of order ν,

x^{2}y''+x\,y'+(x^{2}-\nu^{2})\,y=0;\qquad r^{2}-\nu^{2}=0\ \Rightarrow\ r=\pm\nu;\qquad y=C_{1}J_{\nu}(x)+C_{2}Y_{\nu}(x)

has the indicial equation r² − ν² = 0, whose roots r = ±ν usually give two independent series solutions (Bessel functions of the first kind). When the roots coincide or differ by an integer, the second independent solution, the Bessel function of the second kind Y_ν, contains a logarithmic term. The full theory is developed in standard textbooks on ordinary differential equations.

Further reading

• Power series, Wikipedia
• Power Series, Wolfram MathWorld
• Frobenius method, Wikipedia