Mittag-Leffler theorem
The Mittag-Leffler theorem on expansion of a meromorphic function (see , ) is one of the basic theorems in analytic function theory, giving for meromorphic functions an analogue of the expansion of a rational function into the simplest partial fractions. Let be a sequence of distinct complex numbers,
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and let be a sequence of rational functions of the form
![]() | (1) |
so that is the unique pole of the corresponding function
. Then there are meromorphic functions
in the complex
-plane
having poles at
, and only there, with given principal parts (1) of the Laurent series corresponding to the points
. All these functions
are representable in the form of a Mittag-Leffler expansion
![]() | (2) |
where is a polynomial chosen in dependence of
and
so that the series (2) is uniformly convergent (after the removal of a finite number of terms) on any compact set
and
is an arbitrary entire function.
The Mittag-Leffler theorem implies that any given meromorphic function in
with poles
and corresponding principal parts
of the Laurent expansion of
in a neighbourhood of
can be expanded in a series (2) where the entire function
is determined by
. G. Mittag-Leffler gave a general construction of the polynomials
; finding the entire function
relative to a given
is sometimes a more difficult problem. To obtain (2) it is possible to apply methods of the theory of residues (cf. Residue of an analytic function, see also –).
A generalization of the quoted theorem, also due to Mittag-Leffler, states that for any domain of the extended complex plane
, any sequence
of points
all limit points of which are in the boundary
, and corresponding principal parts (1), there is a function
, meromorphic in
, having poles at
, and only there, with the given principal parts (1). In this form the Mittag-Leffler theorem generalizes to open Riemann surfaces
(see ); for the existence of meromorphic functions on compact Riemann surfaces with given singularities see Abelian differential; Differential on a Riemann surface; Riemann–Roch theorem. The Mittag-Leffler theorem is also true for abstract meromorphic functions
,
, with values in a Banach space
(see ).
Another generalization of the Mittag-Leffler theorem states that for any sequence ,
,
, and corresponding functions
![]() |
that are entire functions of the variable , there is a single-valued analytic function
having singular points at
, and only there, and with principal parts
(see ).
For analytic functions of several complex variables a generalization of the Mittag-Leffler problem on the construction of a function with given singularities is the first (additive) Cousin problem (cf. Cousin problems). In this connection the following equivalent statement of the Mittag-Leffler theorem is often useful. Let , where the
are open sets in
, and let there be given meromorphic functions
, respectively, on the sets
, where the differences
are regular functions on the intersections
for all
and
. Then there is on
a meromorphic function
such that the differences
are regular on
for all
(see , ).
For the Mittag-Leffler theorem on the expansion of single-valued branches of an analytic function in a star see Star of a function element.
References
[1] | G. Mittag-Leffler, "En metod att analytisk framställa en funktion at rationel karacte..." Öfversigt Kongl. Vetenskap-Akad. Förhandlinger , 33 (1876) pp. 3–16 |
[2] | G. Mittag-Leffler, "Sur la répresentation analytique des fonctions monogènes uniformes d'une variable indépendante" Acta Math. , 4 (1884) pp. 1–79 |
[3] | E. Goursat, "Cours d'analyse mathématique" , Gauthier-Villars (1927) |
[4] | A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian) |
[5] | B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian) |
[6] | L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) |
[7] | H. Behnke, F. Sommer, "Theorie der analytischen Funktionen einer komplexen Veränderlichen" , Springer (1972) |
[8] | L. Schwartz, "Analyse mathématique" , 2 , Hermann (1967) |
Comments
References
[a1] | J.B. Conway, "Functions of one complex variable" , Springer (1978) |
[a2] | M. Heins, "Complex function theory" , Acad. Press (1968) |
Mittag-Leffler theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mittag-Leffler_theorem&oldid=19271