In the narrow sense of the word it is a meromorphic function in the complex -plane that is not a rational function. In particular, entire transcendental functions are of this type, that is, entire functions that are not polynomials (cf. Entire function), e.g. the exponential function , the trigonometric functions , , the hyperbolic functions , , and the function , where is the Euler gamma-function. Entire transcendental functions have one essential singularity, at infinity. The proper meromorphic transcendental functions are characterized by the presence of a finite or infinite set of poles in the finite plane and, respectively, an essential singularity or a limit of poles at infinity; of this type, e.g., are the trigonometric functions , , the hyperbolic functions , , and the gamma-function . The definition of transcendental functions given above can be extended to meromorphic functions in the space , , of several complex variables .
In the broad sense of the word a transcendental function is any analytic function (single-valued or many-valued) the calculation of whose values requires, apart from algebraic operations over the arguments, a limiting process in some form or other. Typical for a transcendental function is the presence of either a singularity that is not a pole or an algebraic branch point; e.g. the logarithmic function has two transcendental branch points and . An analytic function is transcendental if and only if its Riemann surface is non-compact.
Important classes of transcendental functions consist of the frequently encountered special functions: the Euler gamma-function and beta-function, the hypergeometric function and the confluent hypergeometric function, and, in particular, its special cases, the spherical functions (cf. Spherical functions), the cylinder functions and the Mathieu functions.
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Transcendental function. L.D. KudryavtsevE.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Transcendental_function&oldid=11541