Mittag-Leffler's theorem

The Mittag-Leffler's theorem is named after the mathematician Gösta Mittag-Leffler named set of function theory . In its application-oriented formulation, it guarantees the existence of certain meromorphic functions .

sentence

Let be a discrete sequence of pairwise different complex numbers without an accumulation point in . Then there is a holomorphic function that has poles exactly at the points and has a given main part there . That is, each of these can be a polynomial choose without constant term, according to the set of lunch Leffler exists a meromorphic function whose Laurent development on a perforated disc to just the main body has. In particular, the degrees of the polynomials and thus the orders of the poles can be freely selected. ${\displaystyle (p_{n})}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} \backslash \{p_{n}\mid n\in \mathbb {N} \}}$ ${\displaystyle p_{n}}$ ${\displaystyle p_{n}}$ ${\displaystyle P_{n}(z)}$ ${\displaystyle p_{n}}$ ${\displaystyle P_{n}({\tfrac {1}{z-p_{n}}})}$

Instead of polynomials, it is also possible, more generally, to choose whole functions (i.e. power series that converge to whole ) without a constant term. In the case of non-terminating power series, however, the resulting function has significant singularities and is therefore only meromorphic for polynomials. ${\displaystyle \mathbb {C} }$

Method of the convergence-generating summands

The case of finitely many poles is trivial, because then one can simply take the finite sum of as the solution. ${\displaystyle P_{n}({\tfrac {1}{z-p_{n}}})}$

We therefore assume for the following that the number of poles is infinite, choose (if there is no pole in 0, we set ) and order the poles in such a way that applies to all . Since the set of poles is discrete, it follows . ${\displaystyle p_{0}=0}$ ${\displaystyle P_{0}=0}$ ${\displaystyle |p_{n}|\leq |p_{n+1}|}$ ${\displaystyle n}$ ${\displaystyle p_{n}\to \infty }$

The case of finitely many poles suggests the approach of simply adding the main parts, that is, to form them. The question then arises as to the convergence of the series with respect to the compact convergence . First of all, this is a suitable concept of convergence, because for every compact set in there is an index so that all with lie outside of this compact set and therefore the uniform convergence of the remaining sum on this compact set can be considered. It now turns out that the above approach does not generally converge. ${\displaystyle \textstyle f(z)=\sum _{n=0}^{\infty }P_{n}({\tfrac {1}{z-p_{n}}})=P_{0}({\tfrac {1}{z}})+\sum _{n=1}^{\infty }P_{n}({\tfrac {1}{z-p_{n}}})}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle p_{n}\to \infty }$ ${\displaystyle n_{0}}$ ${\displaystyle p_{n}}$ ${\displaystyle n\geq n_{0}}$ ${\displaystyle \textstyle \sum _{n=n_{0}}^{\infty }P_{n}({\tfrac {1}{z-p_{n}}})}$

Therefore one tries next to adapt the summands appropriately. For are the functions holomorphic around 0 and therefore have a Taylor series in 0. Let the Taylor polynomial be of degree , that is, the beginning of the Taylor series up to the -th power. The idea is now to the addends by replacing, with the be selected such that this convergence is generated. Since they are holomorphic as polynomials, nothing changes in the main parts. This actually leads to success and is obviously called the method of convergence-generating summands. With the terms introduced here, the following applies: ${\displaystyle n>0}$ ${\displaystyle \textstyle z\mapsto P_{n}({\tfrac {1}{z-p_{n}}})}$ ${\displaystyle z\mapsto T_{n}(z)}$ ${\displaystyle T_{n,m}}$ ${\displaystyle m}$ ${\displaystyle m}$ ${\displaystyle P_{n}({\tfrac {1}{z-p_{n}}})}$ ${\displaystyle P_{n}({\tfrac {1}{z-p_{n}}})-T_{n,m_{n}}(z)}$ ${\displaystyle m_{n}}$ ${\displaystyle T_{n,m_{n}}(z)}$

• There are numbers , so ${\displaystyle m_{n}}$

${\displaystyle f(z):=P_{0}({\tfrac {1}{z}})+\sum _{n=1}^{\infty }(P_{n}({\tfrac {1}{z-p_{n}}})-T_{n,m_{n}}(z))}$

compactly converged. The function is then meromorphic with poles exactly in the given points and has the main parts there . ${\displaystyle f}$ ${\displaystyle p_{n}}$ ${\displaystyle P_{n}({\tfrac {1}{z-p_{n}}})}$

It is also allowed, namely when it is not necessary to adapt the summand using a Taylor polynomial. ${\displaystyle m_{n}=-1}$

Examples

• The following simple example shows what is known as the partial fraction decomposition of a function. Consider . has poles of the second order exactly in the integers. The approach of simply choosing the term as polynomials and thus for the main parts in straight leads to . It can be shown that this sum is already converging. In particular, no convergence-generating summands are required. It turns out that the sum actually converges to, that is, it holds: ${\displaystyle \textstyle f(z)=\pi ^{2}/(\sin(\pi z))^{2}}$ ${\displaystyle f}$ ${\displaystyle z^{2}}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle 1/(z-n)^{2}}$ ${\displaystyle \textstyle \sum _{n=-\infty }^{\infty }1/(z-n)^{2}}$ ${\displaystyle f}$

${\displaystyle \pi ^{2}/(\sin(\pi z))^{2}=\sum _{n=-\infty }^{\infty }1/(z-n)^{2}}$

• If one specifies only simple polynomials with a residual 1, then one has the main parts whose sum does not converge. For is the 0th Taylor polynomial to and you can show that the series actually converges. One can then even show: ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle {\tfrac {1}{z-n}}}$ ${\displaystyle n\neq 0}$ ${\displaystyle -{\tfrac {1}{n}}}$ ${\displaystyle z\mapsto {\tfrac {1}{z-n}}}$ ${\displaystyle {\tfrac {1}{z}}+\sum _{n\in \mathbb {Z} ,n\neq 0}({\tfrac {1}{z-n}}+{\tfrac {1}{n}})}$

${\displaystyle \pi \cot(\pi z)={\frac {1}{z}}+\sum _{n\in \mathbb {Z} ,n\neq 0}({\frac {1}{z-n}}+{\frac {1}{n}})}$

Generalization to Riemannian surfaces

In order to generalize to Riemannian surfaces , we have to find a formulation that can be generalized. To this end, let's take a fresh look at the situation of the sentence.

Since the sequence in the above sentence is discrete, one can find an open neighborhood around each point that does not contain any other of these points. By possibly enlarging the or by adding further points (with suitable open neighborhoods), for which one chooses the main part polynomials 0, one can assume that there is an open cover of and that each of the given sequence contains only the point . If one sets , the main parts are meromorphic and the differences are holomorphic. The above theorem from Mittag-Leffler says that there is a (global) meromorphic function , so that all differences are holomorphic, more precisely: holomorphic can be added (see Riemann's theorem ). denotes the restriction of the function to the specified amount. This motivates the following concept formation. ${\displaystyle (p_{n})_{n}}$ ${\displaystyle p_{n}}$ ${\displaystyle U_{n}}$ ${\displaystyle U_{n}}$ ${\displaystyle {\mathcal {U}}=(U_{n})_{n}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle U_{n}}$ ${\displaystyle p_{n}}$ ${\displaystyle f_{n}=P_{n}({\tfrac {1}{z-p_{n}}})}$ ${\displaystyle f_{n}}$ ${\displaystyle f_{n}-f_{m}\colon U_{n}\cap U_{m}\to \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle f|_{U_{n}}-f_{n}}$ ${\displaystyle U_{n}}$ ${\displaystyle f|_{U_{n}}}$

For a Riemann surface, let and be the sheaves of holomorphic or meromorphic functions. A Mittag-Leffler distribution is a family of meromorphic functions on open sets such that an open cover of is and holds for all . One solution to such a Mittag-Leffler distribution is a globally defined meromorphic function , so that all can be continued holomorphically . With these terms: ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle f_{n}\in {\mathcal {M}}(U_{n})}$ ${\displaystyle U_{n}\subset X}$ ${\displaystyle (U_{n})_{n}}$ ${\displaystyle X}$ ${\displaystyle f_{n}-f_{m}\in {\mathcal {O}}(U_{n}\cap U_{m})}$ ${\displaystyle n\neq m}$ ${\displaystyle f\in {\mathcal {M}}}$ ${\displaystyle f|_{U_{n}}-f_{n}\colon U_{n}\to \mathbb {C} }$ ${\displaystyle U_{n}}$

• Every Mittag-Leffler distribution can be solved on a non-compact Riemann surface.

On compact Riemann surfaces the situation is more complicated, as will now be explained. In continuation of the above conceptualizations is clear that for a Mittag-Leffler distribution family a cocycle from hence and with designated item in the Garbenkohomologiegruppe defined. The criterion ${\displaystyle \mu =(f_{n})_{n}}$ ${\displaystyle \delta \mu :=((f_{n}-f_{m})|_{U_{n}\cap U_{m}})_{n,m}}$ ${\displaystyle Z^{1}({\mathcal {U}},{\mathcal {O}})}$ ${\displaystyle [\delta \mu ]}$ ${\displaystyle H^{1}(X,{\mathcal {O}})}$

• A Mittag-Leffler distribution of a Riemann surface can be solved if and only if the zero element is. ${\displaystyle \mu }$ ${\displaystyle X}$ ${\displaystyle [\delta \mu ]\in H^{1}(X,{\mathcal {O}})}$

is not very profound against the background of these concepts, but shows the difference between compact and non-compact Riemann surfaces. For non-compact Riemann surfaces always applies , which is why the above theorem applies to non-compact Riemann surfaces. This is not the case for compact Riemannian surfaces with gender . Indeed, one of the possible equivalent definitions of gender is for Riemann surfaces, and therefore one can always construct Mittag-Leffler distributions for compact Riemann surfaces of sex , which are not solvable. ${\displaystyle H^{1}(X,{\mathcal {O}})=0}$ ${\displaystyle g\geq 1}$ ${\displaystyle g=\mathrm {dim} H^{1}(X,{\mathcal {O}})}$ ${\displaystyle g\geq 1}$

literature

• Eberhard Freitag , Rolf Busam: Function Theory 1 . Springer , 2006, ISBN 3-540-31764-3 .
• Klaus Jänich : Function Theory . 6th edition. Springer, 2004, ISBN 3-540-20392-3 .

Individual evidence

1. ↑ Wolfgang Fischer, Ingo Lieb : Function theory. Vieweg, 1980, ISBN 3-528-07247-4 , chap. VII, Theorem 1.3 (Theorem of Mittag-Leffler).
2. ↑ Wolfgang Fischer, Ingo Lieb: Function theory. Vieweg, 1980, ISBN 3-528-07247-4 , chap. VII, sentence 3.1.
3. ↑ Wolfgang Fischer, Ingo Lieb: Function theory. Vieweg, 1980, ISBN 3-528-07247-4 , chap. VII, sentence 3.2.
4. ↑ Otto Forster : Riemann surfaces. Springer, 1977, ISBN 3-540-08034-1 , 26.3.
5. ↑ Otto Forster: Riemann surfaces. Springer, 1977, ISBN 3-540-08034-1 , January 18.
6. ↑ Otto Forster: Riemann surfaces. Springer, 1977, ISBN 3-540-08034-1 , January 26.