Difference between revisions of "Mittag-Leffler function"
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
(latex details) |
||
Line 16: | Line 16: | ||
$$ | $$ | ||
− | E _ \rho ( z) = \sum _ { k= } | + | E _ \rho ( z) = \sum _ {k=0} ^ \infty \frac{z ^ {k} }{\Gamma ( 1 + k / \rho ) } |
− | + | ,\ \ 1 \leq \rho < \infty . | |
− | \frac{z ^ {k} }{\Gamma ( 1 + k / \rho ) } | ||
− | ,\ \ | ||
− | 1 \leq \rho < \infty . | ||
$$ | $$ | ||
Line 26: | Line 23: | ||
$$ | $$ | ||
− | E _ \rho ( z ; \mu ) = \sum _ { k= } | + | E _ \rho ( z ; \mu ) = \sum _ {k=0} ^ \infty |
\frac{z ^ {k} }{\Gamma ( \mu + k / \rho ) } | \frac{z ^ {k} }{\Gamma ( \mu + k / \rho ) } | ||
Line 36: | Line 33: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Mittag-Leffler, "Sur la | + | <table> |
− | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> G. Mittag-Leffler, "Sur la représentation 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> | |
− | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> M.L. Cartwright, "Integral functions" , Cambridge Univ. Press (1962)</TD></TR> | |
− | + | </table> | |
− | |||
− |
Latest revision as of 08:18, 6 January 2024
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 représentation 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) |
[a1] | M.L. Cartwright, "Integral functions" , Cambridge Univ. Press (1962) |
Mittag-Leffler function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mittag-Leffler_function&oldid=47857