  This monograph offers a toolbox of mathematical techniques, which have been
effective and widely applicable in information-theoretic analysis. The first
tool is a generalization of the method of types to Gaussian settings, and then
to general exponential families. The second tool is Laplace and saddle-point
integration, which allow to refine the results of the method of types, and are
capable of obtaining more precise results. The third is the type class
enumeration method, a principled method to evaluate the exact random-coding
exponent of coded systems, which results in the best known exponent in various
problem settings. The fourth subset of tools aimed at evaluating the
expectation of non-linear functions of random variables, either via integral
representations, or by a refinement of Jensen's inequality via
change-of-measure, by complementing Jensen's inequality with a reversed
inequality, or by a class of generalized Jensen's inequalities that are
applicable for functions beyond convex/concave. Various application examples of
all these tools are provided along this monograph.
