Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 7D / b / Solution

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.