臺灣博碩士論文加值系統

本文研究高斯超幾何方程的施圖姆-劉維爾形式及其Zeta 行列式.基於Lesch 給出以wronskian 表示正則奇異施圖姆-劉維爾算子之Zeta 行列式的公式,以及拉馬努金的某項公式,便得出本文的主要結果: 超幾何方程之施圖姆-劉維爾形式的Zeta 行列式之closed form.

In the thesis, the eignevalue probelm of the Hypergeometric equation(for short: HGE or E(a,b,c)) is discussed. There are three parts in this thesis.First of all, I introduce some heuristic backgrounds and motivations of the eigenvalue problem of HGE. The second part is a survey about the theory of the HGE. Finally, based on Lesch’s formula of zeta determinant of Regular Singular Sturm-Liouville Operators, I calculate the zeta determinant with repect to the Sturm-Liouville form of HGE operator on a closed interval, by using Ramanujan''s identities.

口試委員會審定書……………………………………………………………… #
誌謝………………………………………………………………………………. i
中文摘要………………………………………………………………………… ii
Abstract……………………………..…………………………………………… iii
第一章 Introduction…………………………………………………………………..1
1.1 Motivation and main results…...…………………………………………… 1
1.2 Heat Kernel and Mellin Transform in the Case of Riemann Zeta………….. 4
1.3 Some Backgrounds………………………………………..…………………6
第二章A Survey of the Hypergeometric Equation……………..…………………. 8
2.1 Hypergeomteric Series.…………………………………..……………….…8
2.2 Hypergeomteric Equation..……………..………………..……………….…8
2.3 Integral representation of Hypergeomteric series.…….....……………….…9
2.4 Projective Equivalence implies Sturm-Liouville form of Hypergeomteric
equation..……………..………………..……………….…..……………..……..9
2.5 Schwarz derivative of Schwarz map..……. ....………....………………….10
第三章 Calculation of the Zeta Determinant….. ……………………….…………12
第四章 Appendix…………………………………………..………………………15
4.1 Laplacian in polar coordinates of half plane and Disk.…………………….15
4.2 Eigenproblem of Laplacian implies hypergeometric..…………………..….15
參考文獻…………………………………………………………………….…… 18

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