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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/2022/paper-2/4/a/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 / 2022 / Paper 2 / 4 / a / Solution by

Codex  0 2026-10-18

Specify the branch before defining a signed jump. Choose

ϕ < ar g z < ϕ + 2 π

, and let

F_{+}

be the value on the side with argument tending to

ϕ

from above, while

F_{-}

is the value on the other side, with argument tending to

ϕ + 2 π

from below. The entire function

f

has the same limiting value on both sides, so

F_{+} (r) = f (r e^{i ϕ}) r^{- α} e^{- i α ϕ}, F_{-} (r) = e^{- 2 πi α} F_{+} (r) .

Thus the jump of a fractional power across a branch cut is

F_{+} - F_{-} = (1 - e^{- 2 πi α}) f (r e^{i ϕ}) r^{- α} e^{- i α ϕ} .

Equivalently the ratio of the limiting values is

F_{-} / F_{+} = e^{- 2 πi α}

wherever

f

is nonzero. Reversing the definition of the jump reverses its sign. Choosing a different branch multiplies both limits by the corresponding common phase.

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