I would like to evaluate
$$F_1 [f] \equiv \int \cdots \int da_0 da_1 \cdots db_1 db_2 \cdots \exp \left[ - \frac{1}{\hbar} \int _{0} ^{U} \left( \frac{m \dot{x}^2}{2} + i f(u) x(u) \right) du \right]$$
, where $f$ is an arbitrary function, $i$ is imaginary number, $x$ is expanded as
$$x(u) = a_0 + \sum _{n = 1}^\infty \left( a_n \cos \left( \frac{2 \pi n u}{\beta} \right) + b_n \sin \left( \frac{2 \pi n u}{\beta}\right) \right)$$
and other symbols denote constants.
According to Statistical Mechanics a set of lectures, Richard P. Feynman, (p.83), this can easily be evaluated and, to do that, the equation
$$ \frac{1}{2a^2} + \sum _{n = 1}^\infty \frac{\cos (n \theta)}{n^2 + a^2} = \frac{\pi}{2a} \frac{\cosh (\pi - |\theta|) a}{\sinh (\pi a)} $$
is usuful.
But I don't have any idea how to calculate the integral. Could anyone give me a hint?