Transcendental function

In the narrow sense of the word it is a meromorphic function in the complex $z$-plane $\mathbf C$ 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 $e^z$, the trigonometric functions $\sin z$, $\cos z$, the hyperbolic functions $\sinh z$, $\cosh z$, and the function $1/\Gamma(z)$, where $\Gamma(z)$ 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 $\mathbf C$ and, respectively, an essential singularity or a limit of poles at infinity; of this type, e.g., are the trigonometric functions $\tan z$, $\operatorname{cotan}z$, the hyperbolic functions $\tanh z$, $\coth z$, and the gamma-function $\Gamma(z)$. The definition of transcendental functions given above can be extended to meromorphic functions $f(z)$ in the space $\mathbf C^n$, $n\geq2$, of several complex variables $z=(z_1,\ldots,z_n)$.

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 $\ln z$ has two transcendental branch points $z=0$ and $z=\infty$. 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)
[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)