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Math Notes

University Mathematics

TheoremComplete

Fuchs' Theorem

Fuchs' theorem provides the complete classification of singularities for linear ODEs and guarantees the existence of Frobenius-type solutions near regular singular points.

Statement

Theorem9.4Fuchs' theorem

Consider the nn-th order linear ODE:

y(n)+p1(x)y(n−1)+⋯+pn(x)y=0.y^{(n)} + p_1(x)y^{(n-1)} + \cdots + p_n(x)y = 0.

A singular point x0x_0 is regular if and only if (x−x0)kpk(x)(x-x_0)^k p_k(x) is analytic at x0x_0 for each k=1,…,nk = 1, \ldots, n.

At a regular singular point x0x_0, there exist nn linearly independent solutions of the form:

yj(x)=(x−x0)rj∑m=0∞am(j)(x−x0)m⋅(ln⁡(x−x0))kj,y_j(x) = (x - x_0)^{r_j} \sum_{m=0}^{\infty} a_m^{(j)} (x - x_0)^m \cdot (\ln(x-x_0))^{k_j},

where r1,…,rnr_1, \ldots, r_n are roots of the indicial equation, and kj∈{0,1,…,n−1}k_j \in \{0, 1, \ldots, n-1\} (logarithmic terms appear only when indicial roots differ by integers). These series converge for 0<∣x−x0∣<R0 < |x - x_0| < R, where RR is the distance to the nearest other singular point.

The Indicial Equation for nn-th Order

Definition9.6Generalized indicial equation

For the nn-th order equation at a regular singular point x0x_0, write (x−x0)kpk(x)=∑m=0∞qk,m(x−x0)m(x-x_0)^k p_k(x) = \sum_{m=0}^{\infty} q_{k,m}(x-x_0)^m. The indicial equation is:

I(r)=r(r−1)⋯(r−n+1)+q1,0r(r−1)⋯(r−n+2)+⋯+qn,0=0.I(r) = r(r-1)\cdots(r-n+1) + q_{1,0} r(r-1)\cdots(r-n+2) + \cdots + q_{n,0} = 0.

This is a polynomial of degree nn in rr. Its roots determine the leading behavior (x−x0)rj(x-x_0)^{r_j} of each solution.

Proof Outline

Proof

The proof proceeds in several stages.

Step 1: Formal series construction. Substitute y=(x−x0)r∑am(x−x0)my = (x-x_0)^r \sum a_m (x-x_0)^m into the ODE. The lowest-order term gives the indicial equation I(r)=0I(r) = 0. Higher-order terms give recurrences:

I(r+m)am=−∑k=1n∑j=0m−1qk,m−j (r+jn−k)falling aj,I(r + m) a_m = -\sum_{k=1}^{n} \sum_{j=0}^{m-1} q_{k,m-j}\, \binom{r+j}{n-k}_{\text{falling}} \, a_j,

where (rk)falling=r(r−1)⋯(r−k+1)\binom{r}{k}_{\text{falling}} = r(r-1)\cdots(r-k+1). Since I(r1+m)≠0I(r_1 + m) \neq 0 for m≥1m \geq 1 when r1r_1 does not differ from another root by a positive integer, this determines all ama_m uniquely.

Step 2: Convergence. The key analytic step uses the method of majorants. Since the coefficient functions are analytic in ∣x−x0∣<R|x - x_0| < R, they are bounded by geometric series. One constructs a majorant ODE whose Frobenius series has coefficients ∣am∣≤Am|a_m| \leq A_m and converges. The comparison is facilitated by taking Am=C⋅Km/m!A_m = C \cdot K^m / m! for suitable constants C,KC, K.

Step 3: Logarithmic cases. When r1−r2=N∈Z≥0r_1 - r_2 = N \in \mathbb{Z}_{\geq 0}, the recurrence at m=Nm = N involves I(r2+N)=I(r1)=0I(r_2 + N) = I(r_1) = 0, potentially causing a division by zero. The resolution requires introducing ln⁡(x−x0)\ln(x - x_0) terms. Differentiation with respect to rr at r=r2r = r_2 produces the logarithmic solution. ■\blacksquare

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Fuchsian Equations

Definition9.7Fuchsian equation

An ODE is Fuchsian if all its singular points (including the point at infinity) are regular. Equivalently, it is Fuchsian on the Riemann sphere P1\mathbb{P}^1.

A Fuchsian equation with singular points at a1,…,ak,∞a_1, \ldots, a_k, \infty has the form:

y′′+(∑j=1kAjx−aj)y′+(∑j=1k(Bj(x−aj)2+Cjx−aj))y=0,y'' + \left(\sum_{j=1}^{k} \frac{A_j}{x - a_j}\right)y' + \left(\sum_{j=1}^{k}\left(\frac{B_j}{(x-a_j)^2} + \frac{C_j}{x - a_j}\right)\right)y = 0,

with constraints from the regularity condition at ∞\infty (the Fuchs relation): ∑(rj,1+rj,2)=k−1\sum (r_{j,1} + r_{j,2}) = k - 1 where rj,1,rj,2r_{j,1}, r_{j,2} are the exponents at each singular point.

RemarkRiemann's count

For a Fuchsian equation with k+1k+1 singular points (including ∞\infty), the exponents satisfy the Fuchs relation: the sum of all 2(k+1)2(k+1) exponents equals (k+1)(k+1−1)/2−1=k(k+1)/2−1(k+1)(k+1-1)/2 - 1 = k(k+1)/2 - 1 for second-order equations. The Riemann scheme (or P-symbol) encodes the singular points and their exponents, classifying all Fuchsian equations.