The standard integrals of the previous chapter are of fairly limited use, so this chapter develops some much more widely applicable techniques. These are integration by substitution, integration by parts and integration by partial fractions.

This appendix collects a review of the calculus and analysis in one and several variables that the reader should be familiar with. Notions of convergence, continuity, differentiability and integrability are recalled here.

Here we collect all the function spaces, their norms, and properties that appear throughout the text. Some of these the reader should be familiar with, while others will be new. The use of these spaces is particularly essential in the topics of approximation theory and partial differential equations.

We prove that the singularities occurring from the form of the propagator are a simple technical nuisance, and can be removed in the limit, provided we accept to deal with distributions rather than with functions.

This Chapter covers optional Calculus topics such as the Fubini Theorem, Riemann Integration, the Fundamental Theorem of Calculus, Integration by Parts, Mean Value Theorem, and Absolute Continuity.

With many physical applications already on the table, in this chapter, we return to some of the simplified ones and re-complexify them.These problems require more sophisticated, and incomplete, solutions.Instead of finding the position of the bob for the simple pendulum, we find the period of motion for the ``real" pendulum.Instead of the classical harmonic oscillator, with its familiar solution, we study the period of the relativistic harmonic oscillator, and find that in the high energy limit, a mass attached to a spring behaves very differently from its non-relativistic counterpart.