---
tag: 6EZ400WM94
kind: statement
role: theorem
name: Fuchs' theorem
msc: [34M03, 34A25]
uses: [H96HXG6CA0, W8ZWXC2HQS, 0MZB06ST2B, 0N288P2TB5, 02QGGD5G87]
about: [H96HXG6CA0, W8ZWXC2HQS]
sources:
  - url: https://en.wikipedia.org/wiki/Fuchs%27s_theorem
same_as:
  wikidata: Q5506929
  wikipedia: Fuchs's theorem
---

## Statement

Let $R > 0$ and $z_0 \in \C$, and let $D = \set{z : 0 < \abs{z - z_0} < R}$.
Let $p, q$ be functions such that $P(z) = (z - z_0)\, p(z)$ and $Q(z) = (z - z_0)^2 q(z)$ are analytic on $\abs{z - z_0} < R$.
Let $r_1, r_2$ be the roots of the indicial equation $r (r - 1) + P(z_0)\, r + Q(z_0) = 0$, labelled so that $\operatorname{Re} r_1 \ge \operatorname{Re} r_2$.
Fix a branch of $\log (z - z_0)$ on $D$ minus a ray from $z_0$, and put $(z - z_0)^r = e^{r \log (z - z_0)}$.
Then the equation $y'' + p(z) y' + q(z) y = 0$ has two linearly independent solutions $y_1, y_2$ there, of the following form, where all the [[power series|0MZB06ST2B]] converge on $\abs{z - z_0} < R$.

1. $y_1 = (z - z_0)^{r_1} \sum_{k \ge 0} a_k (z - z_0)^k$ with $a_0 = 1$.
2. If $r_1 - r_2 \notin \Z$, then $y_2 = (z - z_0)^{r_2} \sum_{k \ge 0} b_k (z - z_0)^k$ with $b_0 = 1$.
3. If $r_1 = r_2$, then $y_2 = y_1 \log (z - z_0) + (z - z_0)^{r_1} \sum_{k \ge 1} b_k (z - z_0)^k$.
4. If $r_1 - r_2$ is a positive integer, then $y_2 = c\, y_1 \log (z - z_0) + (z - z_0)^{r_2} \sum_{k \ge 0} b_k (z - z_0)^k$ with $b_0 = 1$ and a constant $c$, which may be $0$.

## Remarks

The coefficients come from the recurrence of the [[Frobenius method|W8ZWXC2HQS]].
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|02QGGD5G87]] 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|0N288P2TB5]] 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 $\abs{x - x_0}^r$.

## Explanations

### L1 human

The theorem says that the Frobenius method always works at a regular singular point.
There is always one solution of the form $z^r$ times a convergent power series.
The second solution has the same form, unless the two exponents differ by a whole number; then a logarithm may appear.
The series converge as far as the coefficients of the equation stay nice.

### L2 human

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|C8HD62AB6Q]] 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|TTGF8KCA8C]].