A HISTORY OF MATHEMATICS.
416
variable pass from one limit to the other by a succession
of
imaginary values along arbitrary paths. Cauchy established
severalfundamentaltheorems, andgavethefirstgreatimpulse
to the study of the general theory of functions. His researches
were continued in France by Puiseux and Liouville. But more
profound investigations were made in Germany by Riemann.
Georg Friedrich Bernhard Riemann (1826-1866) was
born at Breselenz in Hanover.
His father wished him to
study theology, and he accordingly entered upon philological
and theological studies at Gottingen. He attended also some
lectures on mathematics.
Such was his predilection for this
science that he abandoned theology. After studying for a time
under Gauss and Stern, he was drawn, in 1847, to Berlin by a
galaxy of mathematicians, in which shone Dirichlet, Jacobi,
Steiner, and Eisenstein.
Returning to Gottingen in 1850,
he studied physics under Weber, and obtained the doctorate
the following year.
The thesis presented on that occasion,
Grundlagen fur eine allgemeine Theorie der Funktionen einer
veranderlichen complexen Grosse,
excited the admiration
of Gauss to a very unusual degree, as did also Riemann's
trial lecture, Ueber die Hypothesen welche der Geometrie zu
Grunde liegen.
Riemann's Habilitationsschrift was on the
Representation of a Function by means of a Trigonometric
Series, in which he advanced materially beyond the position of
Dirichlet. Our hearts are drawn to this extraordinarily gifted
but shy genius when we read of the timidity and nervousness
displayed when he began to lecture at Gottingen, and of
his jubilation over the unexpectedly large audience of eight
THEORY OF FUNCTIONS.
417
students at his first lecture on diferential equations.
Later he lectured on Abelian functions to a class of three
only,—Schering, Bjerknes, and Dedekind. Gauss died in 1855,
and was succeeded by Dirichlet.
On the death of the latter,
in 1859, Riemann was made ordinary professor.
In 1860
he visited Paris, where he made the acquaintance of French
mathematicians. The delicate state of his health induced him
to go to Italy three times. He died on his last trip at Selasca,
and was buried at Biganzolo.
Like all of Riemann's researches, those on functions were
profound and far-reaching.
He laid the foundation for a
general theory of functions of a complex variable.
The
theory of potential, which up to that time had been used
only in mathematical physics, was applied by him in pure
mathematics. He accordingly based his theory of functions on
the partial diferential equation,
 2u
 u
x2 + y22 = ∆u = 0, which
must hold for the analytical function w = u+iv of z = x+iy. It
had been proved by Dirichlet that (for a plane) there is always
one, and only one, function of x and y, which satisfies ∆u = 0,
and which, together with its diferential quotients of the first
two orders, is for all values of x and y within a given
area
one-valued and continuous, and which has for points on the
boundary of the area arbitrarily given values.[86]
called this "Dirichlet's principle,"
Riemann
but the same theorem
was stated by Green and proved analytically by Sir William
Thomson. It follows then that w is uniquely determined for
all points within a closed surface, if u is arbitrarily given for all