Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 7D (Further Complex Methods)

For $w^{\prime \prime }+P(z)w^{\prime }+Q(z)w=0$, a finite point $z_0$ is regular singular whenextend analytically to $z_0$. Inspection shows that $a,b,c$ satisfy these conditions. There are no other finite singularities; with the Fuchs relation on the six exponents, the point at infinity is ordinary in the corresponding sphere description.

At $z=a$, insert $w\sim (z-a{)}^r$. The coefficient of the most singular power gives the indicial equationThus the two local exponents are $\alpha ,{\alpha }^{\prime }$. Cyclically, the exponents at $b$ are $\beta ,{\beta }^{\prime }$ and those at $c$ are $\gamma ,{\gamma }^{\prime }$. The assumed nonintegral exponent differences give two independent Frobenius solutions without logarithmic resonance at each point.

The Papperitz symbol isIt records the three regular singular points and the two characteristic exponents at each.

Set $a=0$, $b=1$, $c=\infty$, $\alpha =\beta =0$. Comparing the $w^{\prime }$ coefficient givesand comparison of the $w$ coefficient gives the exponents $A,B$ at infinity. Hence