Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 7E / b

Put

t = 1/ z

and

y (t) = w (1/ t)

. Since

w^{'} = - t^{2} y^{'}, w^{''} = t^{4} y^{''} + 2 t^{3} y^{'},

the transformed equation is

y^{''} + (\frac{2}{t} - \frac{p ( 1/ t )}{t ^{2}}) y^{'} + \frac{q ( 1/ t )}{t ^{4}} y = 0.

i

By the transformed equation, infinity is ordinary precisely when

\frac{2}{t} - \frac{p ( 1/ t )}{t ^{2}}, \frac{q ( 1/ t )}{t ^{4}}

are holomorphic at

t = 0

. Equivalently, as convergent Laurent expansions near infinity,

p (z) = \frac{2}{z} + O (z^{- 2}), q (z) = O (z^{- 4}) .

By the regular singular point at infinity criterion, infinity is regular singular precisely when

2 - \frac{p ( 1/ t )}{t}, \frac{q ( 1/ t )}{t ^{2}}

are holomorphic at

t = 0

, but the ordinary-point conditions fail. Equivalently,

p (z) = O (z^{- 1}), q (z) = O (z^{- 2})

with convergent expansions in

1/ z

, excluding the ordinary case.

Infinity is irregular singular when at least one of

2 - z p (z), z^{2} q (z)

fails to be holomorphic as a function of

1/ z

1/ z = 0

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