Računanje decimalk števila pi : magistrsko delo

Rus, Filip (author), Jezernik, Urban (mentor)

V delu predstavimo zgodbo števila pi. Prvi preboj pri računanju njegovih decimalk je delo Arhimeda. Po utemeljitvi infinitezimalnega računa v 17. stoletju zgodba dobi nove razsežnosti in doseže svoj vrh z iznajdbo formule bratov Chudnovsky. Ta nas popelje v svet eliptičnih in hipergeometrijskih funkcij ter kompleksnega množenja. Točnost implementacije formule lahko preverimo z BBP formulo. Za konec hipergeometrijske vrste uporabimo pri generiranju vrst za druge konstante, kar nam omogoči Zeilbergerjev algoritem.

• Language: Slovenian
• Keywords: pi, formula bratov Chudnovsky, eliptične funkcije, hipergeometrijske funkcije, kompleksno množenje, BBP formula, Zeilbergerjev algoritem
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-162405
• UDC: 517.5
• COBISS.SI-ID: 208482051
• Publication date in RUL: 22.09.2024

Secondary language

• Language: English
• Title: Calculating digits of pi

Abstract

In this thesis, we present the story of the number pi. The first breakthrough in the calculation of its digits was made by Archimedes. The story gains new dimensions with the invention of infinitesimal calculus in the 17th century and reaches its climax with the discovery of Chudnovsky formula. This formula presents us with the world of elliptic and hypergeometric functions, and of complex multiplication. The correctness of the implementation can be verified with the BBP formula. We finish by using hypergeometric series to generate series for other constants, which is made possible by Zeilberger's algorithm.

• Keywords: pi, Chudnovsky formula, elliptic functions, hypergeometric functions, complex multiplication, BBP formula, Zeilberger's algorithm