  Fully analytic formulas, which do not involve any numerical integration, are
derived for the discretized influence functionals of a very extensive
assortment of spectral distributions. For Feynman integrals derived using the
Trotter splitting and Strang splitting, we present general formulas for the
discretized influence functionals in terms of proper integrals of the bath
response function. When an analytic expression exists for the bath response
function, these integrals can almost always be evaluated analytically. In cases
where these proper integrals cannot be integrated analytically, numerically
computing them is much faster and less error-prone than calculating the
discretized influence functionals in the traditional way, which involves
numerically calculating integrals whose bounds are both infinite. As an
example, we present the analytic discretized influence functional for a bath
response function of the form \alpha(t)=\sum_{j}^{K}p_{j}e^{\Omega_{j}t}, which
is a natural form for many spectral distribution functions (including the very
popular Lorentz-Drude/Debye function), and for other spectral distribution
functions it is a form that is easily obtainable by a least-squares fit .
Evaluating our analytic formulas for this example case is much faster and
easier to implement than numerically calculating the discretized influence
funcitonal in the traditional way. In the appendix we provide analytic
expressions for p_{j} and \Omega_{j} for a variety of spectral distribution
forms, and as a second example we provide the analytic bath response function
and analytic influence functionals for spectral distributions of the form
J(\omega)\propto\omega^{s}e^{-(\omega/\omega_{c})^{q}}. The value of the
analytic expression for this bath response function extends beyond its use for
calculating Feynman integrals.
