Mittag-Leffler's theorem

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

Set

Be a discrete sequence of distinct complex numbers without accumulation point in. Then there exists a holomorphic function on which only has poles in there and each has a predetermined body. That is, for each of these you can choose a polynomial without constant term, by the theorem of Mittag-Leffler exists a meromorphic function whose Laurent expansion about just having on a perforated circular disk the main part. In particular, the degree of the polynomial, and thus the order of the pole can be freely selected.

Comments

• With the choice of a polynomial to a pole you also determines the order of the pole, it is equal to the degree of the polynomial.
• If the pole set is finite, then converges the sum of the principal parts trivially.
• If the pole set is infinite, one can not assume that the sum of the principal parts converge in general. In this case, the so-called convergence- enhancing summands are introduced (also convergence- generating p ) for each main section. In most cases, these are Taylor polynomials which do not alter the main part, but only the corresponding side of the Laurent expansion.

Examples

In a simple example we obtain the partial fraction expansion of a function. Look. has exactly the integers poles of second order. The approach to choose just as simple polynomials and thus for the main parts in the term that leads to. It can be shown that this sum converges and is already the same. In particular, no convergence improving summands are needed.

Generalization

Also entire functions can be selected, ie power series without constant term which converge to the whole instead of polynomials. But that the resulting function has essential singularities and is no longer meromorphic.