This is the first in a series of books that will try to show how and why special functions arise in many applications of mathematics. The elementary transcendental functions, such as the exponential, its inverse, the logarithm, and the trigonometric functions, form part of the working tools of all mathematicians and most users of mathematics. There was a time when knowledge of some of the higher transcendental functions was almost as widespread. For example, there were a surprisingly large number of books written on elliptic functions in the last half of the nineteenth century, and esoteric facts about Bessel functions and Legendre functions were regularly set as tripos problems. However, knowledge of these functions and the few other very useful special functions is no longer as widespread, and it has even been possible for important special functions to arise in applications and be studied for twenty-five years or more without any of the people studying them being aware that some of the results they rediscovered were found about a hundred years earlier. This has occurred in the last forty years with what are called 3–j symbols. These functions occur when studying the decomposition of the direct product of two irreducible representations of SU(2).