Solution
= Solution
At $z=a$, insert $w\sim(z-a)^r$. The coefficient of the most singular power gives the indicial equation
$$
(r-\alpha)(r-\alpha')=0.
$$
Thus the two local exponents are $\alpha,\alpha'$. Cyclically, the exponents at $b$ are $\beta,\beta'$ and those at $c$ are $\gamma,\gamma'$. The assumed nonintegral exponent differences give two independent Frobenius solutions without logarithmic resonance at each point.