= Causal Green function for a repeated-root Euler operator
{title2=$G(x,\xi)=H(x-\xi)x^r\xi^{1-r}\log(x/\xi),\quad r=(1-\alpha)/2$}
For $L=y''+\alpha y'/x+(\alpha-1)^2y/(4x^2)$ on $x>0$, the repeated <indicial root> is $r=(1-\alpha)/2$ and the homogeneous solutions are $x^r,x^r\log x$. Their <Wronskian> is $x^{-\alpha}$. The displayed <causal Green function> is continuous at each source $\xi>0$, has a unit jump in its first <derivative>, and satisfies $L_xG=\delta(x-\xi)$. It is zero near the origin, so its value and first derivative there vanish. Those origin conditions alone fail to give uniqueness when $\alpha<-1$: then both homogeneous solutions and their first derivatives vanish as $x\downarrow0$. For forcing $f(x)=x$, source integration from zero converges when $\alpha>-5$ and gives $y_p=4x^3/(\alpha+5)^2$. All zero-origin-data solutions in $-5<\alpha<-1$ are $y_p+x^r(C_1+C_2\log x)$; for $\alpha\geq-1$ the solution is unique. Causality selects the displayed kernel even when the bare singular-endpoint conditions are insufficient.
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