= Euler differential operator
{c}
{title2=$D=x\,d/dx$}
The <Euler differential operator> differentiates in logarithmic position: if $r=\log x$ and $V(r)=v(e^r)$, then $Dv=V'$. In particular $D(x^n)=nx^n$, so a polynomial $p(D)$ acts on the power law $x^n$ by multiplication by $p(n)$. This converts a <Cauchy-Euler equation> into a constant-coefficient <linear differential equation>. Repeated roots give additional logarithmic factors, as ordinary constant-coefficient equations give polynomial factors multiplying exponentials.
Back to article page
