The mathematical papers of Fuchs are very numerous, but
excepting a few of his earliest attempts, they are all connected
directly or indirectly with the theory of linear differential
equations. This was the province which, to quote the words of
Auwers when he introduced Fuchs to the Berlin Academy of
Sciences, he had added to the mathematical kingdom. Of course the
conquest of this new territory had been prepared by others. The
theory of functions, which was an essential
prerequisite had been built up by Cauchy and Riemann. The work
of Briot and Bouquet on differential equations of the first order was
an illustration of the application of this new theory to the treatment
of differential equations. But to Fuchs
belongs the credit of first applying the theory of functions to the
linear differential equations of any order, with rational
coefficients. His paper in the sixty-sixth volume of *Crelles
Journal* is a classic, and to this day I know of no clearer
exposition of the fundamental principles involved. It is true that
Riemaim was acquainted with these principles, as his posthumous
paper on this subject proves. It is true also that Fuchs took his
immediate inspiration from Riemann's famous paper on the
hypergeometric series. But all of this does not lessen the credit due
to Fuchs. To generalize is one of the functions of the
mathematician. This Fuchs did. Riemann also did this, but his paper
was never published until the theory had long been developed by
Fuchs. It is interesting to notice in this
connection the difference between the points of view adopted by
Riemann and Fuchs. Fuchs starts out with the linear differential
equation of the *n*th order whose coefficients are rational
functions of *x*. By a rigorous examination of the
convergence of the series, which formally satisfies the differential
equation he finds that these equations have a very important and
characteristic property. The singular points of the solutions are
fixed, *i. e.*, they are independent of the constants of
integration and can be found without first integrating the
differential equation. They are in fact included among the poles of
its coefficients. He then shows that a fundamental system of
solutions
undergoes a linear substitution with constant coefficients when the
variable *x* describes a circuit enclosing such a singular
point, and from this behavior of the solutions derives expressions
for them, valid in a circular region surrounding that point and
reaching as far as the next singular point. He thus establishes the
existence of systems of *n* functions, uniform, finite and
continuous, except in the vicinity of certain points, and undergoing
linear substitutions with constant coefficients when the variable
*x* describes circuits around these points.

Riemann's point of view was exactly opposite to this. With
him it was a matter of principle not to define a function by an
expression, but by a characteristic property. He, therefore, starts
out with the assumption of a system of n functions
uniform, finite and continuous, except in the vicinity of certain
arbitrarily assigned points, and undergoing an arbitrarily
assigned linear substitution when the variable *x* describes
a closed path around such a point. He then shows that such a
system of functions will satisfy a linear differential equation of
the *n*th order. But the theorem that the branch points and
the
substitutions may be arbitrarily assigned, ought to be proved at
the outset if this point of view is to be adopted. Much has since
been done on this question, which is really a fundamental one in
the theory of linear differential equations.

This is not the place to go into details. Suffice it to say that the theory of linear differential equations was placed by Fuchs upon a solid and adequate foundation. He and his followers have reared upon this a noble structure. He himself characterized a class of such equations, called after him Fuchsian, whose solutions are everywhere regular. He studied the question of algebraic integrability which has so many points of contact with other questions of importance. He studied by his own methods the periods of an elliptic integral as function of the modulus, for Legendre had shown that these verify a linear differential equation of the second order. By considering the modulus inversely as a function of the quotient of the periods, a uniform automorphic function, now known as the modular function, was obtained. The theory of modular functions, and more generally of automorphic functions owes much to Fuchs, as well as to Klein and to Poincaré, who as an indication of this fact has named a large class of such functions Fuchsian. Fuchs has introduced other transcendental functions into analysis, connected with a linear differential equation in much the same way as the abelian functions are connected with an algebraic equation. Little has yet been done with these beyond the proof of their existence.

We have already mentioned that one of the first results
obtained by Fuchs in his classical memoir was the fact that the
branch points of a linear differential equation are fixed. Fuchs
himself was the first to ask the question; are there other
equations of this kind ? In a beautiful paper in the
*Sitzungsberichte der Berliner Akademie*, he started
the investigation of this important question, confining himself,
however, to differential equations of the first order.
Poincaré completed the investigation
in a remarkable manner, the result being that no essentially new
functions could be defined by differential equations of the first
order with fixed singular points. Painlevé has since then
found that transcendental functions essentially new can be defined
by such equations of order higher than the first.

We will close this brief sketch by translating a sentence, which
is as characteristic of the modern theory of differential equations,
as the famous definition of Kirchhoff is of modern mechanics.
Fuchs says in his famous paper in *Crelle's Journal*:
" In the present condition of science it is not so much the
problem of the theory of differential equations to reduce a given
differential equation to quadratures, as to deduce from the
equation itself the behavior of its integrals at all points of the
plane, *i. e.*, for all values of the complex variable."

This is the present point of view in the theory of differential equations. The first chapter of this theory, that of linear differential equations, has been far advanced, although not completed, by Fuchs and his pupils. Something has been done on later chapters, but not much. The theory of non-linear differential equations is one of the central problems of modern mathematics, but it has not yet found its Fuchs.

E. J. WlLCZYNSKI.

University of California

Berkeley, July 8, 1902.

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Letzte Änderung: Mai 2014 *Gabriele Dörflinger*
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