8.2 Regular singular points: the method of Frobenius
Definition 8.2.1. A singular point \(t_0 = 0\) is regular if \(tp(t)\) and \(t^2q(t)\) are analytic at \(0\). Writing \(tp(t) = \sum _kP_kt^k\) and \(t^2q(t) = \sum _kQ_kt^k\), the equation is \(t^2y'' + t\bigl (tp\bigr )y' + \bigl (t^2q\bigr )y = 0\), and its indicial polynomial is \[ F(r) = r(r - 1) + P_0r + Q_0 . \]
The model is the Euler equation \(t^2y'' + P_0ty' + Q_0y = 0\), whose solutions are \(t^r\) with \(F(r) = 0\). Frobenius’s idea is that near a regular singular point the solutions behave like those of the Euler equation, multiplied by power series.
Theorem 8.2.2 (Frobenius). Let \(0\) be a regular singular point, with \(tp\) and \(t^2q\) analytic for \(|t| < R\), and let \(r_1\), \(r_2\) be the roots of the indicial equation, labelled so that \(\Real r_1 \geq \Real r_2\). Then on \(0 < t < R\) there is a solution \[ y_1 = t^{r_1}\sum _{k=0}^\infty a_kt^k, \qquad a_0 = 1 , \] and a second, independent solution of the following form:
- (i)
- if \(r_1 - r_2\) is not an integer, \(y_2 = t^{r_2}\sum _kb_kt^k\), \(b_0 = 1\);
- (ii)
- if \(r_1 = r_2\), \(y_2 = y_1\log t + t^{r_1}\sum _{k\geq 1}c_kt^k\);
- (iii)
- if \(r_1 - r_2 = N\) is a positive integer, \(y_2 = Cy_1\log t + t^{r_2}\sum _kd_kt^k\), \(d_0 = 1\), where the constant \(C\) may be zero.
All the series converge for \(|t| < R\).
Proof. The first solution. Substituting \(y = \sum _ka_kt^{k+r}\) into \(t^2y'' + t(tp)y' + (t^2q)y = 0\) and collecting powers \(t^{k+r}\), \[ F(r + k)a_k = -\sum _{j=0}^{k-1}\bigl [(j + r)P_{k-j} + Q_{k-j}\bigr ]a_j \qquad (k \geq 1), \] with \(F(r)a_0 = 0\) for \(k = 0\). With \(r = r_1\) and \(a_0 = 1\), the recursion determines every \(a_k\) provided \(F(r_1 + k) \neq 0\) for \(k \geq 1\), and this holds because the only roots of \(F\) are \(r_1\) and \(r_2\), and \(r_2 \neq r_1 + k\) since \(\Real r_2 \leq \Real r_1\). Convergence follows by a majorant argument like that of Theorem 8.1.1, using \(|F(r_1 + k)| \geq ck^2\) for large \(k\). In case (i) the same argument works with \(r = r_2\), since \(r_2 + k \neq r_1\) for every \(k \geq 1\).
The form of the second solution. By reduction of order, \(y_2 = y_1v\) with \(v' = W/y_1^2\), and by Abel’s identity \(W = \exp (-\int p) = t^{-P_0}h(t)\) with \(h\) analytic and \(h(0) \neq 0\). Since the roots satisfy \(r_1 + r_2 = 1 - P_0\), and \(y_1^2 = t^{2r_1}g(t)\) with \(g\) analytic and \(g(0) = 1\), \[ v' = t^{-P_0 - 2r_1}\frac {h(t)}{g(t)} = t^{-1-(r_1 - r_2)}\sum _kc_kt^k , \] with \(c_0 \neq 0\). Integrating term by term, every power \(t^{k - 1 - (r_1 - r_2)}\) integrates to a power, except the one with \(k = r_1 - r_2\), which exists only when \(r_1 - r_2\) is a non-negative integer \(N\) and integrates to \(c_N\log t\). Hence \(v = c_N\log t + t^{-(r_1 - r_2)}\times (\text {power series})\), and \(y_2 = y_1v\) has the stated form: in case (ii), \(N = 0\) and \(c_0 \neq 0\), so the logarithm is always present; in case (iii) its coefficient \(c_N\) may vanish. □
The proof shows where logarithms come from: they appear exactly when the Euler-type behaviour \(t^{r_2}\) would collide with the series of the first solution. In practice the coefficients of the second solution are found by substituting the stated form, or by differentiating the first solution with respect to \(r\) (the method of Frobenius in its original form).
Example 8.2.3. For \(2t^2y'' + 3ty' - (1 + t)y = 0\), \(tp = \frac 32\) and \(t^2q = -\frac 12(1 + t)\), so \(P_0 = \frac 32\), \(Q_0 = -\frac 12\), and \(F(r) = r(r - 1) + \frac 32r - \frac 12 = \frac 12(2r - 1)(r + 1)\). The roots \(\frac 12\) and \(-1\) differ by \(\frac 32\), case (i). Substituting \(y = \sum a_kt^{k+r}\) into the equation and collecting powers of \(t\), the recursion is \((2(k + r) - 1)(k + r + 1)a_k = a_{k-1}\), so \[ r = \tfrac 12:\ a_k = \frac {a_{k-1}}{k(2k + 3)}, \qquad r = -1:\ a_k = \frac {a_{k-1}}{k(2k - 3)} , \] and both series converge for all \(t\), since the coefficients are entire.
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