= Exponential reduction of a confluent hypergeometric equation with equal parameters
{title2=$zY''+(b-z)Y'-bY=0$}
Putting $Y=e^zV$ in the displayed <Kummer differential equation> gives $zV''+(z+b)V'=0$. Thus $V'=Cz^{-b}e^{-z}$ on a chosen branch, yielding $Y=e^z[A+B\int z^{-b}e^{-z}dz]$. For $b=1/2$ and $z=x/2$, one solution is $e^{x/2}$ and a normalized second <Frobenius solution> is $\sqrt{x}\sum_{n\geq0}x^n/(2n+1)!!$. The analytic factors in both solutions have infinite <radius of convergence>, since their coefficient ratios tend to zero. The second solution still has a square-root branch at zero; an entire series factor does not remove a nonintegral Frobenius exponent.
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