Transcendental Function

transcendental function

In mathematics, a function not expressible as a finite combination of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and extracting a root. Examples include the functions log x, sin x, cos x, e^x and any functions containing them. Such functions are expressible in algebraic terms only as infinite series. In general, the term transcendental means nonalgebraic. See also transcendental number.

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Transcendental Function 

an analytic function that is not algebraic. Simple examples of transcendental functions are exponential functions, trigonometric functions, and logarithmic functions.

If transcendental functions are treated as functions of a complex variable, a characteristic feature of a transcendental function is the existence of at least one singularity other than poles and branch points of finite order (seeSINGULAR POINT). For example, e², cos z, and sin z have the essential singularity z = ∞, and In z has branch points of infinite order at z = 0 and z = ∞.

The foundations of the general theory of transcendental functions are provided by the theory of analytic functions. Special transcendental functions are studied in the corresponding disciplines—for example, the theory of hypergeometric functions, the theory of elliptic functions, or the theory of Bessel functions.

REFERENCE

Whittaker, E. T., and G. N. Watson. Kurs sovremennogo analiza, 2nd ed., parts 1–2, Moscow, 1969. (Translated from English.)