= Square-root reduction of a regular-singular differential equation
{title2=$4zy''+2y'+y=0$}
With $s=\sqrt z$ and $Y(s)=y(s^2)$, the displayed <linear differential equation> becomes $Y_{ss}+Y=0$ away from zero. Its <Frobenius solutions> have exponents $0$ and $1/2$: $\cos\sqrt z=\sum_{n\geq0}(-1)^nz^n/(2n)!$ and $\sin\sqrt z=z^{1/2}\sum_{n\geq0}(-1)^nz^n/(2n+1)!$. The first is an entire function of $z$; the second needs a local square-root branch. This illustrates how a <regular singular point> can arise from a degenerate change of independent variable.
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