= Analytic solution at a maximal integer Fuchsian exponent
{title2=$F'+(R/z+B(z))F=0$}
Suppose the <matrix> $B$ is <holomorphic> at zero and $0$ is an <eigenvalue> of $R$. Let $m$ be the largest nonnegative integer for which $-m$ is an <eigenvalue> of $R$. Seek $F=z^m\sum_{n\geq0}h_nz^n$, with $h_0\in\ker(mI+R)$ nonzero. The <Frobenius method> gives $((m+n)I+R)h_n=-\sum_{j=0}^{n-1}B_jh_{n-1-j}$ for $n\geq1$, where scalar terms multiply the identity. Every left-hand <matrix> is invertible by the choice of $m$. Its inverse is $O(1/n)$, so a scalar convergent majorant proves convergence of the series. This yields a nonzero analytic solution; it is necessarily nonconstant when $m>0$, but can be constant when $m=0$. For the displayed sign convention, indicial exponents are eigenvalues of $-R$.
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