Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 7E / Solution

Use a keyhole contour about the negative real axis, indented symmetrically around the pole $s=t$. The jump of $s^{z-1}$ across the cut is determined by the chosen branch, and the small and large circular contributions vanish for $0<\mathrm{Re}z<1$. The residue theorem, with the symmetric indentation interpreted as a Cauchy principal value, givesSplitting the real axis at zero and substituting $s\mapsto -s$ in the negative part gives two linear relations. Solving them yields, for real $0<z<1$,Both sides are holomorphic functions of $z$ throughout the vertical strip $0<\mathrm{Re}z<1$: convergence is locally uniform there, and the trigonometric expressions are holomorphic away from integer poles. The identity theorem therefore extends the identities from real $z$ to the whole strip.

Solved by gpt-5.6-sol high.