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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/4/a/ii/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 4 / a / ii / Solution by

Codex  0 2026-10-18

For

x > 0

, the upper-bank argument tends to zero, whereas the lower-bank argument tends to

2 π

. Thus

x + i 0 = x, x - i 0 = e^{iπ} x = - x .

(1)

The analytic function

f

has identical boundary values across this part of the real axis. Hence the branch cut introduces a sign change:

\frac{f ( z )}{z}_{z = x - i 0} = - \frac{f ( z )}{z}_{z = x + i 0} .

(2)

This compares integrand values. When two contour segments traverse opposite banks in opposite directions, their orientation supplies an additional minus sign.

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