There are several subjects in analysis that are frequently used in applied mathematics, theoretical physics and engineering sciences, such as complex variable, ordinary differential equations, special functions, asymptotic methods, integral transforms and distribution theory. However, for graduate students or upper-level undergraduate students who are not going to specialize in these areas, there is no need for them to study these subjects in great depth. Instead, it would probably be more beneficial for them to have an introduction to these topics so that when the need arises, they know what approach to take. With this in mind, this set of lecture notes has been written for a one-semester course. Sufficient details have also been included to make it sufficiently adaptable for self-study. There are in total six chapters with each covering only a few topics. Furthermore, the chapters are all self-contained. The prerequisites for the readers of this book are advanced calculus, a first course in ordinary differential equations and elementary complex variable.