In the years 1865, 1866, and 1868, Lazarus Fuchs published three papers, each entitled “Zur Theorie der Linearen Differentialgleichungen mit veränderlichen Coefficienten” (“On the theory of linear differential equations with variable coefficients”). These will be surveyed in this chapter. In them he characterised the class of linear differential equations in a complex variable x, all of whose solutions have only finite poles and possibly logarithmic branch points. So, near any point x0 in the domain of the coefficients, the solutions become finite and singled-valued upon multiplication by a suitable power of (x − x0) unless it involves a logarithmic term. This class came to be called the Fuchsian class, and equations in it, equations of the Fuchsian type. As will be seen, it contains many interesting equations, including the hypergeometric. In the course of this work, Fuchs created much of the elementary theory of linear differential equations in the complex domain: the analysis of singular points; the nature of a basis of n linearly independent solutions to an equation of degree n when there are repeated roots of the indicial equation; explicit forms for the solution according to the method of undetermined coefficients, He investigated the behaviour of the solutions in the neighbourhood of a singular point, much as Riemann had done, by considering their monodromy relations — the effect of analytically continuing the solutions around the point — and, like Riemann, he did not explicitly regard the transformations so obtained as forming a group. One problem which he raised but did not solve was that of characterising those differential equations all of whose solutions are algebraic.
KeywordsSingular Point Linear Differential Equation Abelian Function Indicial Equation Hypergeometric Equation
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