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 $\{a_n\}_{n=1}^\infty$ be a sequence of distinct complex numbers,

$$ |a_1|\leq|a_2|\leq\dots,\qquad\lim_{n\rightarrow\infty}a_n=\infty, $$

and let $\{g_n(z)\}$ be a sequence of rational functions of the form

\begin{equation} g_n(z)=\sum_{k=1}^{l_n}\frac{c_{nk}}{(z-a_n)^k}, \end{equation}

so that $a_n$ is the unique pole of the corresponding function $g_n(z)$. Then there are meromorphic functions $f(z)$ in the complex $z$-plane $\mathbb{C}$ having poles at $a_n$, and only there, with given principal parts (1) of the Laurent series corresponding to the points $a_n$. All these functions $f(z)$ are representable in the form of a Mittag-Leffler expansion

\begin{equation} f(z)=h(z)+\sum_{n=1}^\infty[g_n(z)+p_n(z)], \end{equation}

where $p_n(z)$ is a polynomial chosen in dependence of $a_n$ and $g_n(z)$ so that the series (2) is uniformly convergent (after the removal of a finite number of terms) on any compact set $K\subset\mathbb{C}$ and $h(z)$ is an arbitrary entire function.

The Mittag-Leffler theorem implies that any given meromorphic function $f(z)$ in $\mathbb{C}$ with poles $a_n$ and corresponding principal parts $g_n(z)$ of the Laurent expansion of $f(z)$ in a neighbourhood of $a_n$ can be expanded in a series (2) where the entire function $h(z)$ is determined by $f(z)$. G. Mittag-Leffler gave a general construction of the polynomials $p_n(z)$; finding the entire function $h(z)$ relative to a given $f(z)$ 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 [3]–[5]).

A generalization of the quoted theorem, also due to Mittag-Leffler, states that for any domain $D$ of the extended complex plane $\bar{\mathbb{C}}$, any sequence $\{a_n\}$ of points $a_n\in D$ all limit points of which are in the boundary $\partial D$, and corresponding principal parts (1), there is a function $f(z)$, meromorphic in $D$, having poles at $\{a_n\}$, and only there, with the given principal parts (1). In this form the Mittag-Leffler theorem generalizes to open Riemann surfaces $D$ (see [7]); 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 $g_n$, $f:D\rightarrow F$, $D\subset\bar{\mathbb{C}}$, with values in a Banach space $F$ (see [8]).

Another generalization of the Mittag-Leffler theorem states that for any sequence $\{a_n\}\subset\mathbb{C}$, $|a_1|\leq|a_2|\leq\dots$, $\lim a_n=\infty$, and corresponding functions

$$ g_n(z)=\sum_{k=1}^\infty\frac{c_{nk}}{(z-a_n)^k} $$

that are entire functions of the variable $w_n=1/(z-a_n)$, there is a single-valued analytic function $f(z)$ having singular points at $a_n$, and only there, and with principal parts $g_n(z)$ (see [3]).

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 $\Omega=\cup_j\Omega_j$, where the $\Omega_j$ are open sets in $\mathbb{C}$, and let there be given meromorphic functions $g_j$, respectively, on the sets $\Omega_j$, where the differences $g_j-g_k$ are regular functions on the intersections $\Omega_j\cap\Omega_k$ for all $j$ and $k$. Then there is on $\Omega$ a meromorphic function $f$ such that the differences $f-g_j$ are regular on $\Omega_j$ for all $j$ (see [5], [6]).

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) MR1296666 MR1296665 MR1296664 MR1519291
[4]	A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[5]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001
[6]	L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) MR0344507 Zbl 0271.32001
[7]	H. Behnke, F. Sommer, "Theorie der analytischen Funktionen einer komplexen Veränderlichen" , Springer (1972) MR1532738 MR0147622 MR0073682 Zbl 0273.30001
[8]	L. Schwartz, "Analyse mathématique" , 2 , Hermann (1967) MR0226973 MR0226972 Zbl 0171.01301

References

[a1]	J.B. Conway, "Functions of one complex variable" , Springer (1978) MR0503901 Zbl 0887.30003 Zbl 0277.30001
[a2]	M. Heins, "Complex function theory" , Acad. Press (1968) MR0239054 Zbl 0155.11501