# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### The Existence of Meromorphic Functions with Prescribed Poles and Principal Parts

(2020) In Bachelor’s Theses in Mathematical Sciences MATK11 20201
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Abstract
The Swedish mathematician Gösta Mittag-Leffler (1846-1927) is well-known for founding Acta Mathematica, the famous international mathematical journal. His mathematical contributions are connected chiefly with complex analysis. The Mittag-Leffler theorem which built on Karl Weierstrass' work asserts the existence of meromorphic function with prescribed poles and principal parts.

The main purpose of this thesis is to make a well-organised note of the Mittag-Leffler theorem which plays a significant role in complex analysis. The Weierstrass theorem which prescribes the zeros of holomorphic functions will be proved by using the result of the Mittag-Leffler's theorem. Moreover, the concept of the Mittag-Leffler star, a starshaped open domain... (More)
The Swedish mathematician Gösta Mittag-Leffler (1846-1927) is well-known for founding Acta Mathematica, the famous international mathematical journal. His mathematical contributions are connected chiefly with complex analysis. The Mittag-Leffler theorem which built on Karl Weierstrass' work asserts the existence of meromorphic function with prescribed poles and principal parts.

The main purpose of this thesis is to make a well-organised note of the Mittag-Leffler theorem which plays a significant role in complex analysis. The Weierstrass theorem which prescribes the zeros of holomorphic functions will be proved by using the result of the Mittag-Leffler's theorem. Moreover, the concept of the Mittag-Leffler star, a starshaped open domain will also be defined and proved in detail. (Less)
Popular Abstract
We summarise some of the important concepts and results.

The Swedish mathematician Gösta Mittag-Leffler (1846-1927) was one of the most influential researchers in Swedish mathematics. His mathematical contributions are connected chiefly with complex analysis. He studied with Karl Weierstrass while he travelled to Berlin and some of his research were built on Weierstrass' work, such as the Mittag-Leffler theorem which is the main topic of this paper.

The Mittag-Leffler theorem proves the existence of meromorphic functions with prescribed poles and principal parts. Interestingly, this study began as an extension of the Weierstrass theorem. As a continuation of Weierstrass's work, it was undertaken with the tools acquired from... (More)
We summarise some of the important concepts and results.

The Swedish mathematician Gösta Mittag-Leffler (1846-1927) was one of the most influential researchers in Swedish mathematics. His mathematical contributions are connected chiefly with complex analysis. He studied with Karl Weierstrass while he travelled to Berlin and some of his research were built on Weierstrass' work, such as the Mittag-Leffler theorem which is the main topic of this paper.

The Mittag-Leffler theorem proves the existence of meromorphic functions with prescribed poles and principal parts. Interestingly, this study began as an extension of the Weierstrass theorem. As a continuation of Weierstrass's work, it was undertaken with the tools acquired from Weierstrass' lectures. These facts also show that Mittag-Leffler was indeed an important contributor to Weierstrass's research program concerning the foundations of analysis.

Mittag-Leffler published the final version of the theorem in 1884, in his newly-established journal Acta Mathematica which served as a comprehensive account of essentially all of his work on the subject. The theorem was seen as an important and fundamental element in complex analysis. It generated a number of research in the following generation of mathematicians, including the well-known figures Picard, Appell, and Poincaré, and it remained popular and widely-studied in different languages, such as French, Russian and German. (Less)
author
supervisor
organization
course
MATK11 20201
year
type
M2 - Bachelor Degree
subject
keywords
The Mittag-Leffler Theorem, Meromorphic Function, Complex Analysis, The Mittag-Leffler Star
publication/series
Bachelor’s Theses in Mathematical Sciences
report number
LUNFMA-4099-2020
ISSN
1654-6229
other publication id
2020:K14
language
English
id
9024200
2020-08-24 15:08:55
date last changed
2020-08-24 15:08:55
```@misc{9024200,
abstract     = {The Swedish mathematician Gösta Mittag-Leffler (1846-1927) is well-known for founding Acta Mathematica, the famous international mathematical journal. His mathematical contributions are connected chiefly with complex analysis. The Mittag-Leffler theorem which built on Karl Weierstrass' work asserts the existence of meromorphic function with prescribed poles and principal parts.

The main purpose of this thesis is to make a well-organised note of the Mittag-Leffler theorem which plays a significant role in complex analysis. The Weierstrass theorem which prescribes the zeros of holomorphic functions will be proved by using the result of the Mittag-Leffler's theorem. Moreover, the concept of the Mittag-Leffler star, a starshaped open domain will also be defined and proved in detail.},
author       = {Liu, Peicen},
issn         = {1654-6229},
keyword      = {The Mittag-Leffler Theorem,Meromorphic Function,Complex Analysis,The Mittag-Leffler Star},
language     = {eng},
note         = {Student Paper},
series       = {Bachelor’s Theses in Mathematical Sciences},
title        = {The Existence of Meromorphic Functions with Prescribed Poles and Principal Parts},
year         = {2020},
}

```