= 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 $\alpha'p^\mu\tau$ rather than the more usual $2\alpha'P^\mu\tau$,
$$
\boxed{P^\mu=\frac12p^\mu}.
$$
With $\alpha_{-n}=\alpha_n^*$ for a real embedding,
$$
\boxed{J^{\mu\nu}=\frac12(x^\mu p^\nu-x^\nu p^\mu)
-\frac{i\alpha'}2\sum_{n=1}^\infty\frac1n
(\alpha_{-n}^\mu\alpha_n^\nu-\alpha_{-n}^\nu\alpha_n^\mu)}.
$$
The first term is orbital angular momentum and the second is the contribution of the <string oscillator>[string oscillators].
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