Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 14E (Further Complex Methods)

The identity theorem states that if two holomorphic functions on a connected open set $\mathcal{D}$ agree on a subset having an accumulation point in $\mathcal{D}$, then they agree throughout $\mathcal{D}$. In particular, agreement on any nonempty open subset is sufficient.

An analytic continuation of $f_1$ from ${\mathcal{D}}_1$ to ${\mathcal{D}}_2$ is a holomorphic function $f_2$ on ${\mathcal{D}}_2$ whose restriction to the nonempty overlap ${\mathcal{D}}_1\cap {\mathcal{D}}_2$ agrees with $f_1$.

Suppose $f_2$ and ${\tilde{f}}_2$ are two such continuations. Both equal $f_1$ on ${\mathcal{D}}_1\cap {\mathcal{D}}_2$, soon that nonempty open set. Since ${\mathcal{D}}_2$ is connected, the identity theorem givesThus analytic continuation to ${\mathcal{D}}_2$ is unique.

For $\mathrm{Im}z>0$, close the $t$-contour in the upper half-plane. Jordan lemma removes the large semicircle, and the only enclosed pole is the order-$n$ pole at $t=z$. By the residue at a pole of order n,The residue theorem therefore gives

To continue the integral itself, move the $t$-contour downward locally as $z$ approaches and crosses the real axis, always keeping the pole above the contour. This analytic continuation by contour deformation defineswhere ${\mathcal{C}}_z$ passes below $z$. Closing ${\mathcal{C}}_z$ upward continues to enclose the pole, even when $\mathrm{Im}z\le 0$, sothere as well. The right-hand side is an entire function, so it is the unique analytic continuation of $F$ to the whole complex plane.

If $\mathrm{Im}z>0$, the same upper-half-plane contour calculation givesIf $\mathrm{Im}z<0$, the pole lies below the real integration contour. Closing upward encloses no pole, so Jordan lemma and the Cauchy integral theorem giveHence at a real point $x$ the upper and lower boundary values satisfyThus crossing from the upper half-plane to the lower half-plane produces the negative of this jump.

An analytic continuation must be holomorphic, hence continuous, across every point where it is defined. The undeformed real-axis formula for $G$ has the nonzero jump above, while the continuation constructed in part (c) retains $2\pi i^n{e}^{iz}/(n-1)!$ below the axis. Therefore $G$ cannot be the analytic continuation of $F$.