Analyticity and Convergence of Series Solutions

Definition A second-order linear homogeneous equation $P(x)\,y'' + Q(x)\,y' + R(x)\,y = 0$ is normalized by dividing by its leading coefficient wherever $P(x) \ne 0$, which gives

The convergence theorem below is stated for the normalized coefficients $p$ and $q$, so ordinary and singular points are read from them. When $P$, $Q$ and $R$ are polynomials with no common factor, the singular points are exactly the zeros of $P$, real and nonreal, because at each of them $Q/P$ or $R/P$ has a pole.

Definition A function $f$ is analytic at $x_0$ if $f(x) = \sum c_n (x - x_0)^n$ for all $x$ in some open interval around $x_0$, with a series of positive radius. Polynomials, $e^x$, $\sin x$ and $\cos x$ are analytic at every point, and so are sums, products and compositions of analytic functions, and quotients at points where the denominator is not zero. The normalized equation has analytic coefficients at $x_0$ when $p$ and $q$ are both analytic there, which is exactly what makes $x_0$ an ordinary point. How far a Taylor series reaches is decided in the complex plane. A power series that converges for real $x$ with $\lvert x - x_0 \rvert < R$ also converges for every complex $x$ in the disk $\lvert x - x_0 \rvert < R$, and there its sum is analytic and finite. So the radius of the Taylor series of $f$ cannot exceed the distance from $x_0$ to a complex point where $f$ becomes infinite, such as a zero of the denominator of a rational function in lowest terms.

Definition For every power series $\sum a_n (x - x_0)^n$ there is a number $R$ with $0 \le R \le \infty$, its radius of convergence, such that the series converges absolutely for $\lvert x - x_0 \rvert < R$ and diverges for $\lvert x - x_0 \rvert > R$. At the endpoints $x_0 - R$ and $x_0 + R$ anything can happen, so each endpoint is tested on its own. When $\lvert a_{n+1}/a_n \rvert$ has a limit $L$, possibly infinite, the ratio test gives $R = 1/L$, read as $R = \infty$ when $L = 0$ and $R = 0$ when $L = \infty$. If infinitely many coefficients are zero, as in a series of even powers only, apply the ratio test to the nonzero terms instead.

Definition Let $x_0$ be an ordinary point of $y'' + p(x)\,y' + q(x)\,y = 0$, and suppose the Taylor series of $p$ and $q$ about $x_0$ both converge for $\lvert x - x_0 \rvert < R$, where $0 < R \le \infty$. Then for every choice of $a_0$ and $a_1$, the series solution with these initial coefficients converges at least for $\lvert x - x_0 \rvert < R$ and solves the equation there, and every solution on the interval $x_0 - R < x < x_0 + R$ is one of these series. When $p$ and $q$ are rational functions in lowest terms, the Taylor series of each converges exactly out to its nearest pole in the complex plane. So $R$ can be taken as the distance from $x_0$ to the nearest complex singularity, the nearest real or nonreal singular point of the normalized equation. A point $a + bi$ lies at distance $\sqrt{(a - x_0)^2 + b^2}$ from $x_0$. If $p$ and $q$ are polynomials, there is no singular point and $R = \infty$.

Definition The radius $R$ from the theorem is a guaranteed radius, a lower bound that holds for every solution at once. The exact radius of a particular solution is the radius of convergence of its own Taylor series about $x_0$. It can be larger than $R$, even infinite, because a solution need not be singular where a coefficient is, and in some equations every solution beats the guarantee. Neither number decides convergence at the endpoints. A solution can also exist far beyond its interval of convergence. The solution of an initial-value problem at $x_0$ exists on the whole largest open interval around $x_0$ that contains no real singular point, while its Taylor series about $x_0$ may converge on only part of that interval.