## 28. Transcendental functions.

All functions of \(x\) which are not rational or even algebraical are called *transcendental* functions. This class of functions, being defined in so purely negative a manner, naturally includes an infinite variety of whole kinds of functions of varying degrees of simplicity and importance. Among these we can at present distinguish two kinds which are particularly interesting.

**E. The direct and inverse trigonometrical or circular functions.** These are the sine and cosine functions of elementary trigonometry, and their inverses, and the functions derived from them. We may assume provisionally that the reader is familiar with their most important properties.^{1}

## 29. F. Other classes of transcendental functions.

Next in importance to the trigonometrical functions come the exponential and logarithmic functions, which will be discussed in Chs. IX and X. But these functions are beyond our range at present. And most of the other classes of transcendental functions whose properties have been studied, such as the elliptic functions, Bessel’s and Legendre’s functions, Gamma-functions, and so forth, lie altogether beyond the scope of this book. There are however some elementary types of functions which, though of much less importance theoretically than the rational, algebraical, or trigonometrical functions, are particularly instructive as illustrations of the possible varieties of the functional relation.

- The definitions of the circular functions given in elementary trigonometry presuppose that any sector of a circle has associated with it a definite number called its
*area*. How this assumption is justified will appear in Ch. VII.↩︎ - See Chs. IV and V for explanations as to the precise meaning of this phrase.↩︎

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