8.1 Power series at an ordinary point
Theorem 8.1.1. If \(p\) and \(q\) are analytic in \(|t - t_0| < R\), every solution of \(y'' + py' + qy = 0\) is analytic there: \(y = \sum _{k\geq 0}a_k(t - t_0)^k\) with radius of convergence at least \(R\). The coefficients \(a_0 = y(t_0)\) and \(a_1 = y'(t_0)\) are arbitrary, and the others are determined by a recursion.
Sketch of proof. Take \(t_0 = 0\), \(p = \sum p_kt^k\), \(q = \sum q_kt^k\). Substituting \(y = \sum a_kt^k\) and equating coefficients of \(t^k\), \[ (k + 2)(k + 1)a_{k+2} = -\sum _{j=0}^k\bigl [(j + 1)a_{j+1}p_{k-j} + a_jq_{k-j}\bigr ] , \] which determines every \(a_k\) from \(a_0\) and \(a_1\). For convergence, fix \(r < R\); by the Cauchy estimates there is \(M\) with \(|p_k|, |q_k| \leq Mr^{-k}\). Define \(A_0 = |a_0|\), \(A_1 = |a_1|\), and \(A_{k+2}\) by the same recursion with \(p_k\), \(q_k\) replaced by \(Mr^{-k}\) and every sign made positive. Then \(|a_k| \leq A_k\) by induction. The \(A_k\) are the coefficients of the solution of the majorant equation \(Y'' = \frac {M}{1 - t/r}(Y' + Y)\), and a direct estimate of its recursion shows \(A_{k+1}/A_k \to 1/r\). So \(\sum a_kt^k\) converges for \(|t| < r\), and \(r < R\) was arbitrary. The series is then a genuine solution, since it may be differentiated term by term. □
The radius of convergence is controlled by the distance to the nearest singularity of \(p\) or \(q\) in the complex plane, even if that singularity is not on the real line. The equation \((1 + t^2)y'' + y = 0\) has \(q = 1/(1 + t^2)\), analytic on the whole real line, but its solutions at \(0\) have series converging only for \(|t| < 1\), the distance to \(\pm i\).
Example 8.1.2. Airy’s equation \(y'' = ty\) arises in optics near a caustic and in quantum mechanics near a turning point. With \(p = 0\) and \(q = -t\), analytic everywhere, the recursion is \[ (k + 2)(k + 1)a_{k+2} = a_{k-1} \quad (k \geq 1), \qquad a_2 = 0 . \] So the coefficients come in three interleaved chains, and the two solutions with \(y(0) = 1, y'(0) = 0\) and \(y(0) = 0, y'(0) = 1\) are \[ f(t) = 1 + \frac {t^3}{3 \cdot 2} + \frac {t^6}{6 \cdot 5 \cdot 3 \cdot 2} + \cdots , \qquad g(t) = t + \frac {t^4}{4 \cdot 3} + \frac {t^7}{7 \cdot 6 \cdot 4 \cdot 3} + \cdots , \] both entire. The Airy function \(\operatorname {Ai}(t) = \operatorname {Ai}(0)f(t) + \operatorname {Ai}'(0)g(t)\) is the combination that decays as \(t \to +\infty \); it oscillates for \(t < 0\) and decays super-exponentially for \(t > 0\), like \(\exp (-\frac 23t^{3/2})\).
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