= Radial reduction and growth at infinity
{title2=$u''+2u'/r-u=g\iff y''-y=rg,\quad y=ru$}
For $r>0$, the substitution $y=ru$ removes the first derivative in the three-dimensional <radial Laplacian>, since $u''+2u'/r=y''/r$. Bounded $u$ at zero implies $y(0)=0$. Bounded $u$ at infinity means $y=O(r)$; it does not require bounded $y$. For example, constant forcing $g=1$ has the bounded particular solution $u=-1$, whose reduced function is $y=-r$. A <Dirichlet half-line Green function for d2 minus 1> can still represent this solution when its integral converges, but the resulting reduced function has the growth demanded by the forcing.
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