Transcendental function
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.
References
[1] | A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960) |
[2] | E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 2 |
[3] | S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian) |
Comments
References
[a1] | R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German) |
[a2] | C. Carathéodory, "Theory of functions of a complex variable" , 1 , Chelsea, reprint (1983) pp. 170 (Translated from German) |
[a3] | A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. 280 (Translated from Russian) |
Transcendental function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transcendental_function&oldid=31831