Students in their second year of studying applied mathematics are the target audience for this introduction to Laplace transforms and the Fourier series. It is uncommon to treat Laplace transforms at a level that is reasonably easy and to provide a large number of examples. Students majoring in mathematics typically do not encounter this subject matter until much later in their degree program, but students majoring in applied mathematics and engineering require an early introduction. In addition to being suitable for use as a textbook, it will also be of interest to physicists and engineers as additional reading material.