Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 7E / b / Solution

Let $\Gamma$ be any closed curve in $\mathbb{C}\setminus B$. Since the arc $B$ joins $-i$ to $i$ without meeting $\Gamma$, those two points lie in the same component of $\mathbb{C}\setminus \Gamma$. Their winding numbers about $\Gamma$ are therefore equal. The residue theorem givesThus every closed-path period vanishes. Integrals from $0$ to $z$ along any two paths avoiding $B$ agree, so $F_B$ is single-valued. Notice that this argument uses cancellation of the two residues; simple connectedness of the slit domain is not required.

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