 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/14e/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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

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