Higher Special Functions: A Theory of the Central Two-Point Connection Problem Based on a Singularity Approach

Higher special functions emerge from boundary eigenvalue problems of Fuchsian differential equations with more than three singularities. This detailed reference provides solutions for singular boundary eigenvalue problems of linear ordinary differential equations of second order, exploring previously unknown methods for finding higher special functions. Starting from the fact that it is the singularities of a differential equation that determine the local, as well as the global, behaviour of its solutions, the author develops methods that are both new and efficient and lead to functional relationships that were previously unknown. All the developments discussed are placed within their historical context, allowing the reader to trace the roots of the theory back through the work of many generations of great mathematicians. Particular attention is given to the work of George Cecil Jaffé, who laid the foundation with the calculation of the quantum mechanical energy levels of the ionised hydrogen molecule ion.

高階特殊函數源於具有三個以上奇異點的Fuchsian微分方程的邊界特徵值問題。這本詳細的參考書提供了二階線性常微分方程奇異邊界特徵值問題的解法，探索了尋找高階特殊函數的先前未知方法。從微分方程的奇異點決定其解的局部和全局行為的事實出發，作者開發了既新穎又高效的方法，並得到了先前未知的函數關係。所有討論的發展都放置在其歷史背景中，讓讀者能夠追溯到許多偉大數學家的工作。特別關注喬治·塞西爾·賈菲的工作，他通過計算電離氫分子離子的量子力學能級奠定了基礎。