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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 14E / c / Solution

When

s

is a nonpositive integer,

t^{s - 1}

is meromorphic and the Hankel contour may be shrunk to a small circle about zero. The integral is then computed by the residue at zero. This gives

Li_{0} (z) = \frac{z}{1 - z},

Li_{- 1} (z) = \frac{z}{( 1 - z ) ^{2}},

Li_{- 2} (z) = \frac{z ( 1 + z )}{( 1 - z ) ^{3}} .

These also follow by repeatedly applying

z d / d z

to the geometric series.

Solved by gpt-5.6-sol high.

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