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Fully analytic formulas, which do not involve any numerical integration, arederived for the discretized influence functionals of a very extensiveassortment of spectral distributions. For Feynman integrals derived using theTrotter splitting and Strang splitting, we present general formulas for thediscretized influence functionals in terms of proper integrals of the bathresponse function. When an analytic expression exists for the bath responsefunction, these integrals can almost always be evaluated analytically. In caseswhere these proper integrals cannot be integrated analytically, numericallycomputing them is much faster and less error-prone than calculating thediscretized influence functionals in the traditional way, which involvesnumerically calculating integrals whose bounds are both infinite. As anexample, we present the analytic discretized influence functional for a bathresponse function of the form \alpha(t)=\sum_{j}^{K}p_{j}e^{\Omega_{j}t}, whichis a natural form for many spectral distribution functions (including the verypopular Lorentz-Drude/Debye function), and for other spectral distributionfunctions it is a form that is easily obtainable by a least-squares fit .Evaluating our analytic formulas for this example case is much faster andeasier to implement than numerically calculating the discretized influencefuncitonal in the traditional way. In the appendix we provide analyticexpressions for p_{j} and \Omega_{j} for a variety of spectral distributionforms, and as a second example we provide the analytic bath response functionand analytic influence functionals for spectral distributions of the formJ(\omega)\propto\omega^{s}e^{-(\omega/\omega_{c})^{q}}. The value of theanalytic expression for this bath response function extends beyond its use forcalculating Feynman integrals.

Analytic influence functionals for numerical Feynman integrals in most open quantum systems