# Lazarus Fuchs

## Abstract

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 characterized 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 x_{0} in the domain of the coefficients, the solutions become finite and singled-valued upon multiplication by a suitable power of (x−x_{0}) 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 characterizing those differential equations all of whose solutions are algebraic. It became very important, and is discussed in the next chapter.

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## Notes

- 1.These biographical details are taken from Biermann, [1973b, 68, 94, 103].Google Scholar
- 2.Weierstrass [1841]. These series are called Laurent series after Laurent’s work, reported on in Cauchy, [1843]. Laurent’s paper is reprinted in Peiffer [1978].Google Scholar
- 3.Singuläre Punkte = singular points, in this case poles of finite order.Google Scholar
- 5.See Hawkins [1977].Google Scholar
- 6.Fuchs did not call the w Eigenwerthe (eigenvalues), and had no special term for them other than ‘roots of the fundamental equation’.Google Scholar
- 7.I introduce the vector notation purely to abbreviate what Fuchs wrote in coordinate form.Google Scholar
- 8.Fuchs did not specify the sense in which a circuit of a point is to be taken, but it must be taken in an agreed way each time.Google Scholar
- 9.See Biermann [1973b, 69-70].Google Scholar
- 10.Strictly, Fuchs has made a mistake: (2.1.9) admits solutions n=1, ρ arbitrary or n=2, ρ=2.Google Scholar
- 11.See Biermann [1973b, 77].Google Scholar
- 15. New readers can start with Thomé’s own summary [1884].Google Scholar