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In the narrow sense of the word it is a [[Meromorphic function|meromorphic function]] in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t0936301.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t0936302.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t0936303.png" />, the trigonometric functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t0936304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t0936305.png" />, the hyperbolic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t0936306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t0936307.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t0936308.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t0936309.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363010.png" /> and, respectively, an essential singularity or a limit of poles at infinity; of this type, e.g., are the trigonometric functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363012.png" />, the hyperbolic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363014.png" />, and the gamma-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363015.png" />. The definition of transcendental functions given above can be extended to meromorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363016.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363018.png" />, of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363019.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363020.png" /> has two transcendental branch points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093630/t09363022.png" />. An analytic function is transcendental if and only if its [[Riemann surface|Riemann surface]] is non-compact.
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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
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