Causal Green function for a repeated-root Euler operator ($G(x,\xi )=H(x-\xi )x^r{\xi }^{1-r}\log (x/\xi ),r=(1-\alpha )/2$)

For $L=y^{\prime \prime }+\alpha y^{\prime }/x+(\alpha -1{)}^{2}y/(4x^2)$ 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 $L_{x}G=\delta (x-\xi )$. 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 $x\downarrow 0$. 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 $\alpha \ge -1$ the solution is unique. Causality selects the displayed kernel even when the bare singular-endpoint conditions are insufficient.