This chapter does three things. It gives a short account of the work of Euler, Gauss, Kummer, and Riemann on the hypergeometric equation, with some indication of its immediate antecedents and consequences. It therefore looks very briefly at some of the work of Gauss, Legendre, Abel and Jacobi on elliptic functions, in particular at their work on modular functions and modular transformations. It concludes with a description of the general theory of linear differential equations supplied by Cauchy and Weierstrass. There are many omissions, some of which are rectified elsewhere in the literature.1 The sole aim of this chapter is to provide a setting for the work of Fuchs on linear ordinary differential equations, to be discussed in Chapter II, and for later work on modular functions, discussed in Chapters IV and V.
KeywordsBranch Point Analytic Continuation Elliptic Function Power Series Expansion Algebraic Function
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