Difference between revisions of "Mittag-Leffler function"
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| + | An entire function $ E _ \rho ( z) $ | ||
| + | of a complex variable $ z $, | ||
| + | introduced by G. Mittag-Leffler [[#References|[1]]] as a generalization of the exponential function: | ||
| + | |||
| + | $$ | ||
| + | E _ \rho ( z) = \sum _ { k= } 0 ^ \infty | ||
| + | |||
| + | \frac{z ^ {k} }{\Gamma ( 1 + k / \rho ) } | ||
| + | ,\ \ | ||
| + | 1 \leq \rho < \infty . | ||
| + | $$ | ||
Since the Mittag-Leffler function and the more general functions of Mittag-Leffler type | Since the Mittag-Leffler function and the more general functions of Mittag-Leffler type | ||
| − | + | $$ | |
| + | E _ \rho ( z ; \mu ) = \sum _ { k= } 0 ^ \infty | ||
| + | |||
| + | \frac{z ^ {k} }{\Gamma ( \mu + k / \rho ) } | ||
| + | ,\ \ | ||
| + | \mu , \rho \in \mathbf C , | ||
| + | $$ | ||
are widely used in integral representations and transforms of analytic functions, their properties, in particular asymptotic properties, have been studied in great detail (see [[#References|[2]]], [[#References|[3]]]). | are widely used in integral representations and transforms of analytic functions, their properties, in particular asymptotic properties, have been studied in great detail (see [[#References|[2]]], [[#References|[3]]]). | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène" ''Acta Math.'' , '''29''' (1905) pp. 101–181</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène" ''Acta Math.'' , '''29''' (1905) pp. 101–181</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.L. Cartwright, "Integral functions" , Cambridge Univ. Press (1962)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.L. Cartwright, "Integral functions" , Cambridge Univ. Press (1962)</TD></TR></table> | ||
Revision as of 08:00, 6 June 2020
An entire function $ E _ \rho ( z) $
of a complex variable $ z $,
introduced by G. Mittag-Leffler [1] as a generalization of the exponential function:
$$ E _ \rho ( z) = \sum _ { k= } 0 ^ \infty \frac{z ^ {k} }{\Gamma ( 1 + k / \rho ) } ,\ \ 1 \leq \rho < \infty . $$
Since the Mittag-Leffler function and the more general functions of Mittag-Leffler type
$$ E _ \rho ( z ; \mu ) = \sum _ { k= } 0 ^ \infty \frac{z ^ {k} }{\Gamma ( \mu + k / \rho ) } ,\ \ \mu , \rho \in \mathbf C , $$
are widely used in integral representations and transforms of analytic functions, their properties, in particular asymptotic properties, have been studied in great detail (see [2], [3]).
References
| [1] | G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène" Acta Math. , 29 (1905) pp. 101–181 |
| [2] | M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian) |
| [3] | A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian) |
Comments
References
| [a1] | M.L. Cartwright, "Integral functions" , Cambridge Univ. Press (1962) |
How to Cite This Entry:
Mittag-Leffler function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mittag-Leffler_function&oldid=16180
Mittag-Leffler function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mittag-Leffler_function&oldid=16180
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article