Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 7E (Further Complex Methods)

The partial-fraction decomposition isThe residues at $i$ and $-i$ are respectively $1/(2i)$ and $-1/(2i)$. If two paths from $0$ to $z$ have winding-number differences $n_{+}$ and $n_{-}$ about those poles, their integrals differ byLoops can wind around either pole independently, so the value changes by arbitrary integer multiples of $\pi$ and $F$ is multivalued.

Along the straight real segment,Consequently the complete set of values is

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.

On the positive real axis, whose path does not cross the left-displaced cut,Since $F_B^{\prime }(z)=1/(1+z^2)=O(|z{|}^{-2})$ near infinity, integrating this derivative along large arcs extends the same expansion uniformly there:Thereforeand hence