= Solution
Analytically continuing the <free-particle propagator> to imaginary time gives the <Gaussian heat kernel>
$$
K_0(x;\beta)=\left(\frac m{2\pi\hbar^2\beta}\right)^{1/2}\exp\left(-\frac{mx^2}{2\hbar^2\beta}\right).
$$
The two image endpoints have displacements $2rL$ and $2rL-2q$ from the starting point. Inserting their kernels into the <thermal trace of an interval image kernel> gives
$$
\boxed{Z=\left(\frac m{2\pi\hbar^2\beta}\right)^{1/2}\int_0^L dq\sum_{r\in\mathbb Z}\left[e^{-2m(rL)^2/(\hbar^2\beta)}-e^{-2m(rL-q)^2/(\hbar^2\beta)}\right]}.
$$
The prefactor comes from the normalization of the <free-particle propagator>; it cannot be dropped from a thermal trace.
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