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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 equation

(r - α) (r - α^{'}) = 0.

Thus the two local exponents are

α, α^{'}

. Cyclically, the exponents at

b

are

β, β^{'}

and those at

c

are

γ, γ^{'}

. The assumed nonintegral exponent differences give two independent Frobenius solutions without logarithmic resonance at each point.

Solved by gpt-5.6-sol high.

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