Difference between revisions of "Transcendental function"
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| − | In the narrow sense of the word it is a [[Meromorphic function|meromorphic function]] in the complex | + | {{TEX|done}} |
| + | In the narrow sense of the word it is a [[Meromorphic function|meromorphic function]] in the complex $z$-plane $\mathbf C$ that is not a [[Rational function|rational function]]. In particular, entire transcendental functions are of this type, that is, entire functions that are not polynomials (cf. [[Entire function|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|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 | + | 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|Riemann surface]] is non-compact. |
Important classes of transcendental functions consist of the frequently encountered special functions: the Euler [[Gamma-function|gamma-function]] and [[Beta-function|beta-function]], the [[Hypergeometric function|hypergeometric function]] and the [[Confluent hypergeometric function|confluent hypergeometric function]], and, in particular, its special cases, the spherical functions (cf. [[Spherical functions|Spherical functions]]), the [[Cylinder functions|cylinder functions]] and the [[Mathieu functions|Mathieu functions]]. | Important classes of transcendental functions consist of the frequently encountered special functions: the Euler [[Gamma-function|gamma-function]] and [[Beta-function|beta-function]], the [[Hypergeometric function|hypergeometric function]] and the [[Confluent hypergeometric function|confluent hypergeometric function]], and, in particular, its special cases, the spherical functions (cf. [[Spherical functions|Spherical functions]]), the [[Cylinder functions|cylinder functions]] and the [[Mathieu functions|Mathieu functions]]. | ||
Revision as of 19:10, 17 April 2014
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) |
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) |
How to Cite This Entry:
Transcendental function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transcendental_function&oldid=31831
Transcendental function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transcendental_function&oldid=31831
This article was adapted from an original article by L.D. KudryavtsevE.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article