Asimptotski razvoji, integralne sredine i primjene na specijalne funkcije

doctoral thesis

• Title (english): Asymptotic expansions, integral means and applications to the special functions
• Author: Lenka Mihoković
• Mentor: Neven Elezović (mentor)
• Committee member: Sanja Varošanec (predsjednik povjerenstva)
• Committee member: Andrea Aglić-Aljinović (član povjerenstva)
• Committee member: Tomislav Burić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-06-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Kao uvod u glavni dio ovog rada izveden je asimptotski razvoj integralne sredine poligama funkcije. Dana je rekurzivna relacija koja određuje koeficijente u tom razvoju i dokazana su neka njihova svojstva. Taj je pristup potom generaliziran na integralne sredine funkcija koje imaju asimptotski razvoj u red potencija. U tu svrhu dobiven je algoritam za rješavanje jednadžbi oblika \(B(A(x)) = C(x)\), gdje su asimptotski razvoji funkcija B i C poznati, a zatim su rezultati ilustrirani na primjerima nekih bitnih integralnih sredina. Proučavanje asimptotskih razvoja funkcija u red potencija pokazalo se korisnim i kod usporedbe sredina. Tako su izvedeni nužni uvjeti za usporedbu parametarskih sredina. Rad je podijeljen u nekoliko cjelina. U uvodnom poglavlju definirani su asimptotski razvoji i opisana su osnovna svojstva vezana uz operacije s asimptotskim razvojima. Drugo poglavlje vezano je uz gama funkciju. Korištenjem poznatih rezultata i zapisa pomoću kvocijenta gama funkcija dobiven je asimptotski razvoj Wallisovog niza \(W_n\) na temelju čega su poboljšane neke postojeće nejednakosti vezane za ovaj niz. Na sličan način su dobiveni koeficijenti u asimptotskom razvoju Wallisove potencije i opisana su neka njihova svojstva. Izveden je asimptotski razvoj Wallisove funkcije preko poligama funkcija. Zatim su dobiveni i analizirani koeficijenti u razvoju integralne sredine poligama funkcije što dovodi do promatranja integralnih sredina općenito. U trećem poglavlju opisan je algoritam za računanje koeficijenata u asimptotskom razvoju integralne sredine funkcije koja ima asimptotski razvoj te su dobiveni rezultati objašnjeni na nekim konkretnim primjerima. Slične tehnike korištene su u četvrtom poglavlju kod analize klasičnih te parametarskih sredina. Poznavajući koeficijente u asimptotskim razvojima izvedeni su nužni uvjeti za usporedbu tih sredina. Također su promatrane konveksne kombinacije dviju sredina, posebno kada su uključene Seiffertove ili Neuman-Sándorova sredina. Dokazane su neke nejednakosti dok su druge ostavljene u obliku slutnji.

Abstract (english)

As an introduction to the main part of this work, the asymptotic expansion of the integral mean of polygamma function was derived. The recursive relation which determines coefficients in this expansion was given. This approach is then generalized to the integral mean of the functions which possess asymptotic power series expansion. To this end, algorithm for solving equations of the form \(B(A(x)) = C(x)\) is obtained, where asymptotic expansions of functions B and C are known, and the results are illustrated by the examples of some relevant integral means. Studying asymptotic expansions turned out to be useful when comparing means. Thus, the necessary conditions for the comparison of parametric means were derived. This thesis is divided in several parts. In the introduction, asymptotic expansions were defined and some basic properties about manipulations with asymptotic series were described. Second chapter deals with gamma and related functions. Using known results and expressions through quotient of gamma functions, asymptotic expansion of Wallis sequence \(W_n\) was derived, which provided improvements of some existing inequalities related to this sequence. In the same manner, coefficients in asymptotic expansion of the Wallis power function were found and some properties were described. Asymptotic expansion of the Wallis function through polygamma function was derived. Then the coefficients in asymptotic expansion of integral mean of polygamma function were found which motivated the studying of integral means in general. In the third and main chapter the algorithm for calculating coefficients in asymptotic expansion of integral mean of a function possessing asymptotic expansion was described and obtained results were explained on some specific examples. Similar techniques were used in the fourth chapter in the analysis of classical and parameter means. By knowing coefficients in asymptotic expansions, necessary conditions for comparison of those means were derived. Convex combinations of two means which include Seiffert or Neuman-Sándor mean were also observed. Some new inequalities were proved while others were stated in form of conjectures.

• Keywords: asimptotski razvoj; integralna sredina; poligama funkcije; usporedba parametarskih cjelina; gama funkcija; Wallisov niz; Wallisova potencija; Wallisova funkcija; Seiffertova sredina; Neuman-Sándorova sredina
• Keywords (english): asymptotic expansion; integral mean; polygamma function; comparison of parametric means; gamma function; Wallis sequence; Wallis power function; Wallis function; Seiffert mean; Neuman-Sándor mean
• Language: croatian
• URN:NBN: urn:nbn:hr:217:868731
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: 134 str.
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2019-03-19 09:23:10