Lastnosti holomorfnih funkcij v okolici singularnih točk

magistrsko delo

Maja Rebernišek (Author), Marko Jakovac (Mentor)

Abstract

Vsako holomorfno funkcijo v okolici singularne točke lahko razvijemo v Laurentovo vrsto. Glede na število členov z negativno potenco v tej vrsti ločimo med odpravljivo singularnostjo, polom ▫$n$▫-tega reda, ▫$n\in\mathbb{N}$▫, in bistveno singularnostjo. Funkcija ▫$f$▫ ima v točki ▫$a\in\mathbb{C}$▫ odpravljivo singularnost, če vrsta ne vsebuje členov z negativno potenco. Za take funkcije bomo pokazali, da so v okolici singularne točke omejene in da jih lahko holomorfno razširimo v tej točki singularnosti. Funkcija ima pol ▫$n$▫-te stopnje, ko ima Laurentova vrsta ▫$n$▫ členov z negativno potenco. Za take funkcije bomo pokazali, da v okolici singularne točke postanejo funkcijske vrednosti zelo velike. Funkcije, ki so holomorfne povsod, razen v točkah singularnosti, kjer imajo pole, imenujemo meromorfne. Za te funkcije bomo dokazali Mittag-Lefflerjev izrek, ki pravi, da lahko konstruiramo meromorfno funkcijo, ki ima v točkah poljubnega zaporedja brez stekališč vnaprej predpisane končne glavne dele Laurentove vrste. Pri bistveni singularnosti ima Laurentova vrsta neskončno mnogo členov z negativno potenco. Za takšne funkcije bomo pokazali, da za točke v okolici singularnosti funkcija doseže vse kompleksne vrednosti, razen morda ene. To je t. i. veliki Picardov izrek.

Keywords

magistrska dela;singularne točke;holomorfne funkcije;Rungejev izrek;Mittag-Lefflerjev izrek;mali Picardov izrek;veliki Picardov izrek;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.09 - Master's Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[M. Rebernišek]
UDC:	517.5(043.2)
COBISS:	24218632
Views:	642
Downloads:	57
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Properties of holomorphic functions around singularities
Secondary abstract:	Each holomorphic function in the neighbourhood of a singular point can be expanded into the Laurent series. Regarding the number of elements with a negative exponent, we can differentiate between a removable singularity, a pole of order ▫$n$▫ and an essential singularity. The function ▫$f$▫ has a removable singularity in the point ▫$a\in\mathbb{C}$▫ if the series does not contain elements with a negative exponent. We will show that such functions are bounded in the neighbourhood of a singular point and that such functions can be holomorphically extended in this particular point of singularity. A function has a pole of order ▫$n$▫ when the Laurent series has ▫$n$▫ elements with a negative exponent. We will show for such functions that in the neighbourhood of a singular point the function values become particularly large. Functions that are holomorphic everywhere, except for the points of singularity, which are the poles of a function, are called meromorphic. For such functions, we will prove the Mittag-Leffler's theorem. This theorem suggests that we can construct a meromorphic function, which in certain points of an arbitrary sequence without accumulation points already contains finite principal parts of the Laurent series that are in this case determined in advance. Regarding essential singularity, the Laurent series contains an infinite number of elements with a negative exponent. We will show for such functions that for all points in the neighbourhood of the singularity the function assumes the whole complex plane or the whole complex plane minus one point. This is the so-called Picard's great theorem.
Secondary keywords:	master theses;singularities;holomorphic functions;Runge's theorem;Mittag-Leffler's theorem;Little Picard's theorem;Great Picard's theorem;
URN:	URN:SI:UM:
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	VIII, 61 f.
ID:	10980594