 Indicial equation

An indicial equation determines candidate leading powers in a Frobenius solution near a regular singular point. Fornear $x=0$, suppose $p(x)=xP(x)$ and $q(x)=x^{2}Q(x)$ extend analytically to zero. Substituting $y=x^r(a_0+a_{1}x+\cdots )$ with $a_0\ne 0$ and collecting the lowest power givesIts roots are the indicial exponents. Subsequent powers determine the coefficient recurrence; the leading polynomial alone does not decide whether the second solution contains a logarithm.

For the Cauchy-Euler equation $x^2{y}^{\prime \prime }+x{y}^{\prime }+{\mu }^{2}y=0$, $I(r)=r^2+{\mu }^2$. If $\mu >0$, its complex roots $\pm i\mu$ give real solutions $\cos (\mu \log x)$ and $\sin (\mu \log x)$ on $x>0$, exhibiting log-periodic oscillation.^[ref]