Wielandtova karakterizacija funkcije gama

delo diplomskega seminarja

Saša Trstenjak (Author), Miran Černe (Mentor)

Abstract

Glavna tema dela je Wielandtova karakterizacija funkcije $\Gamma$. Wielandtova karakterizacija holomorfno funkcijo $f$, definirano na desni kompleksni polravnini, za katero velja $f(z+1) = zf(z)$, prepozna kot večkratnik funkcije gama, če je le $f$ omejena na navpičnem pasu oblike $\{z \in \mathbb{C} |1 \le \rm {Re}(z) < 2\}$. S pomočjo karakterizacije navedemo alternativen dokaz znanih rezultatov, kot so povezava med Eulerjevima funkcijama gama in beta, Gaussov produkt in Eulerjeva formula, Gaussova produktna formula ter Stirlingova formula z oceno za napako aproksimacije. Dokažemo še, da se da pogoj omejenosti v Wielandtovi karakterizaciji nadomestiti s precej milejšim pogojem, ki zahteva le, da funkcija $f$ na navpičnih pasovih ne raste prehitro.

Keywords

matematika;funkcija gama;integrali s parametrom;enakomerna konvergenca;holomorfne funkcije;

Data

Language:	Slovenian
Year of publishing:	2019
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[S. Trstenjak]
UDC:	517.53
COBISS:	18722393
Views:	1129
Downloads:	165
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Wielandt’s Characterization of the Gamma Function
Secondary abstract:	The main theme of the work is Wielandt's characterization of the $\Gamma$ function. Wielandt's characterization claims, that a holomorphic function $f$, defined on the right complex half-plane, for which the recursion $f(z+1) = zf(z)$ applies, is a multiple of gamma, if only f is bounded on vertical band $\{z \in \mathbb{C} \|1 \le \rm {Re}(z) < 2\}$. Using this characterization, we provide alternative proofs of known results, such as the relation between Euler's functions gamma and beta, Gauss product and Euler's formula, multiplication formula of Gauss and Stirling's formula with an estimate of its error. We prove that the boundedness condition in Wielandt's characterization can be replaced by a weaker assumption, which demands, that the growth rate of function $f$ on vertical bands is not too great.
Secondary keywords:	mathematics;gamma function;integrals with parameter;uniform convergence;holomorphic functions;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	30 str.
ID:	11223554