Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 / 1 / c / Solution

Orthogonality of the cosine modes gives ${\int }_0^{\pi }\cos (n\sigma )d\sigma =0$ and ${\int }_0^{\pi }\cos (n\sigma )\cos (m\sigma )d\sigma =\pi {\delta }_{nm}/2$. Since the supplied expansion uses ${\alpha }^{\prime }{p}^{\mu }\tau$ rather than the more usual $2{\alpha }^{\prime }{P}^{\mu }\tau$,With ${\alpha }_{-n}={\alpha }_n^{\ast }$ for a real embedding,The first term is orbital angular momentum and the second is the contribution of the string oscillators.