Nultočke holomorfnih funkcija

master's thesis

• Author: Marija Blaslov
• Mentor: Marcela Hanzer (mentor)
• Committee member: Marcela Hanzer (predsjednik povjerenstva)
• Committee member: Boris Guljaš (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Ivica Nakić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-07-13, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu proučavaju se nultočke holomorfnih funkcija. Bitan rezultat o kojem govorimo u radu je da ukoliko neki skup točaka \(Z\) nema gomilište, tada postoji holomorfna funkcija \(f\) čiji je skup nultočaka \(Z(f)\) upravo jednak skupu \(Z\), i obratno, ukoliko je \(f\) holomorfna funkcija koja nije identički jednaka nuli, tada skup svih nultočaka od \(f\) nema gomilište. O tome nam govori Weierstrassov teorem. Također smo iskazali i dokazali Mittag-Lefflerov teorem koji za meromorfne funkcije, točnije za unaprijed zadane glavne dijelove meromorfnih funkcija, pronalazi meromorfnu funkciju s upravo tim, unaprijed zadanim, glavnim dijelovima. Kako bi to bilo moguće prije svega trebali smo uvesti niz pojmova posebice iz kompleksne analize i teorije mjere. Zatim smo definirali beskonačne produkte i proučavali neka njihova opća svojstva. Na putu dokazivanja Weierstrassovog teorema o faktorizaciji upoznali smo pojam elementarnih faktora, kao i pojam kanonskog produkta. Jensenova formula nam je dala mogućnost određivanja uvjeta koje moraju zadovoljavati nultočke nekonstantnih holomorfnih funkcija, uz neke dodatne uvjete. Upoznali smo se s pojmom Baschkeovih produkta kao i sa nekim njihovim svojstvima. Također, naveli smo i dokazali Müntz-Szaszov teorem. U konačnici smo riješili neke zadatke koji koriste uvedene i dokazane rezultate.

Abstract (english)

In this paper we examined the zeros of holomorphic functions. An important result we deal with in this work is that if the set Z has no limit point, then there exists holomorphic function f such that its zero set Z(f) is precisely the set Z, and conversely, if the function f is holomorphic and it is not identically equal to 0, then the zero set Z(f) has no limit point. The Weierstrass theorem asserts that. We also demonstrated and proven Mittag-Leffler theorem which, for given main parts of Laurent expansions around given set of points, gives a meromorphic function with exactly that Laurent expansions near its poles. In order to accomplish that, we needed to introduce numerous concepts and ideas especially from complex analysis and measure theory. Then we defined infinite products and studied some of their general properties. On the way to prove the Weierstrass factorization theorem we introduced the concept of elementary factors, as well as the concept of canonical product. Jensen’s formula gave us the possibility to determine the precise conditions which the zeros of nonconstant holomorphic functions, with some additional properties, must satisfy. We introduced the concept of Baschke products as well as with some of their properties. Also, we demonstrated and proven Müntz-Szasz theorem. At the end, we resolve some exercises using the results we previously obtained.

• Keywords: holomorfne funkcije; nultočke holomorfnih funkcija; gomilište; Weierstrassov teorem; Mittag-Lefflerov teorem; beskonačni produkt; elementarni faktor; kanonski produkt; Jensenova formula; Baschkeov produkt; Müntz-Szaszov teorem
• Keywords (english): holomorphic function; zeros of holomorphic functions; limit point; Weierstrass theorem; Mittag-Leffler theorem; infinite product; elementary factor; canonical product; Jensen's formula; Baschke product; Müntz-Szasz theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:190941
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-05-31 10:25:18