Fuchs' theorem

Remarks

The coefficients come from the recurrence of the Frobenius method. For $r = r_1$, the factor $F(r_1 + k)$ in front of $a_k$ never vanishes for $k \ge 1$, because $r_1$ has the largest real part. Convergence on the whole disc follows by comparing the recurrence with a majorant series built from Cauchy estimates for the coefficients of $P$ and $Q$. The second solution comes from reduction of order, which is where the logarithm enters.

Example: for the Bessel equation of order $0$, $z^2 y'' + z y' + z^2 y = 0$, the indicial equation is $r^2 = 0$, so case 3 applies. The solutions are $J_0(z) = \sum_{m \ge 0} \frac{(-1)^m}{(m!)^2} \left( \frac{z}{2} \right)^{2m}$ and a second solution with a term $J_0(z) \log z$.

If $z_0$ is an ordinary point, then $P(z_0) = Q(z_0) = 0$, the exponents are $0$ and $1$, and the theorem gives two analytic solutions, with radius of convergence at least $R$. The hypothesis that $z_0$ is at worst a regular singular point is needed: at the irregular point $0$ of $z^2 y'' + y' = 0$, the solution $\int e^{1/z} dz$ has no such form. For real equations, the same holds on $0 < \abs{x - x_0} < R$ with $(x - x_0)^r$ replaced by $\abs{x - x_0}^r$.

Going deeper

For a student

The theorem is the basic local result for linear equations in the complex domain. It explains the behaviour of Bessel, Legendre, Laguerre and hypergeometric functions near their singular points. Going around $z_0$ once multiplies $(z - z_0)^r$ by $e^{2 \pi i r}$ and adds $2 \pi i$ to the logarithm. So the theorem computes the local monodromy of the equation from the exponents. At an irregular singular point, solutions are described instead by asymptotic expansions in sectors, which leads to the Stokes phenomenon.