 Semicircle integral from a slit-plane square root

ID: semicircle-integral-from-a-slit-plane-square-root

For

a > 0

, choose the branch

F (z) = z - a z + a

that satisfies

F (z) \sim z

at infinity and whose cut is the interval

[- a, a]

. Its boundary values are

F_{\pm} (x) = \pm i a^{2} - x^{2}

. Comparing the integral around the two sides of the cut with the coefficient of

z^{- 1}

F (z) = z - \frac{a ^{2}}{2 z} + O (z^{- 3})

gives

\int_{- a}^{a} a^{2} - x^{2} d x = \frac{π a ^{2}}{2} .

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