## OP-SF WEB

### Extract from OP-SF NET

Topic #8     ------------   OP-SF NET 5.6  -----------  November 15, 1998
~~~~~~~~~~~~~
From: OP-SF NET Editors
Subject: Article on Weierstrass by U. Skornik

[The following item appeared in our Activity Group's Newsletter,
October 1998; it was reprinted, with permission, from the
Russian Newsletter "Integral Transforms and Special Functions" ]

Karl Theodor W. Weierstrass - Life and Work
On the 100th anniversary of his death
by U. Skornik
Institute of Mathematics,
Staromiejska 8/6, 40-013
Katowice, Poland
E-mail: skornik@impan.gov.pl

19 February 1997 marked the hundredth anniversary of the death of the
great German mathematician, the father of classical mathematical analysis
and theory of Abelian and special functions, Karl Theodor Wilhelm
Weierstrass. He was born on 31 October 1815, the first child of Wilhelm
Weierstrass, secretary of the mayor of Ostenfelde and Theodora
Vonderforst.

Nothing in his early life indicated that he would become a famous
mathematician. Karl attended the Catholic Gymnasium in Paderborn from 1829
to 1834 when he entered the University of Bonn in order to follow a course
in public finance, economics and administration. This choice, far from his
own interests, was dictated by his father with the result that, after four
years spent on fencing, drinking and mathematics, Karl returned home
without having taken any examinations.  The years in Bonn, however, were
not entirely wasted. It was during the stay there that Weierstrass
attended lectures of the famous geometer Plucker, studied "Mecanique
Celeste"  by Laplace, "Fundamenta nova" by Jacobi and extended his
knowledge by an accidentally found transcript of lectures on elliptic
functions by Gudermann. As a last resort Karl was sent in 1839 to the
Theological and Philosophical Academy at Munster where he was to prepare
for a career as a secondary school teacher. He attended lectures on
elliptic functions given by Cristof Gudermann. The theory of these
functions was initiated by Gauss and developed by Abel and Jacobi. In the
early XIXth century Abel considered the elliptic integral of the form

\alpha = int(0 to x)  {dx}\over \sqrt{(1-c^2x^2)(1+e^2x^2)}

and the function x=\varphi(\alpha), inverse to the integral. The function
\varphi(\alpha) is called an elliptic function and, when extended to the
whole complex plane, gives a doubly periodic function. Jacobi based his
theory on the integral

u =int( 0 to \varphi {d\varphi}\over \sqrt{1-k^2\sin^2\varphi}},

with a parameter k, between 0 and 1, called the modulus of the elliptic
integral. The elliptic functions obtained from that integral were called
modular functions by Gudermann. It was a new theory and Gudermann was the
first after Jacobi to give lectures on the subject.

On 2 May 1840 Weierstrass was given problems for his final examinations;
one of them was posed by Gudermann in response to a special wish of his
student and concerned elliptic functions. In autumn 1840 Weierstrass
presented the results of his research on the decomposition of modular
functions. He expressed Jacobi modular functions as ratios of entire
functions whose power series coefficients are polynomials in k^2.  In
memory of Abel, he called them Al functions.  Next, Weierstrass introduced
his famous \sigma-functions, which differ from Al functions by a
multiplier.

His dissertation contained significant new material and could ensure an
academic position for him in Germany or elsewhere.  It is not known why
this work, highly prized by Gudermann, was not published until 54 years
later in the first volume of Weierstrass's Collected Papers. Instead of
gaining mathematical fame, Weierstrass, after passing the second oral part
of his examinations in spring 1841, worked for 14 years as a secondary
school teacher.

After a probationary year in Munster, Weierstrass worked at the Catholic
gymnasium (a high level high school) in Deutsch-Krone (West Prussia) from
1842 to 1848 and then in a similar school in Braunsberg (East Prussia)
from 1848 to 1855. He taught not only mathematics and physics but also
German, botany, geography, history, gymnastics and calligraphy. In 1844
Weierstrass took part in a course for gymnastics teachers in Berlin.
During this time he visited the famous geometer Steiner. This, however,
did not change his situation. In Deutsch-Krone Weierstrass had neither
ideas with any other mathematician. He felt isolated and filled his life
with work. During those years he developed the theory of Abelian functions
which form a larger class than the elliptic functions. Abelian integrals
are defined like elliptic integrals by

u = \int(0 to v) R(t,\sqrt{f(t)})dt = I(v)

where R(x,y) is a rational function of x and y, except that the function f
is of a very general type which includes all polynomials.  Inversion then
yields Abelian functions, just as elliptic functions arise from the
inversion of elliptic integrals, i.e. the integrals where f(t)  is a
polynomial.

During Weierstrass' probationary year at the gymnasium in Munster he
worked on three papers. The first "Darstellung einer analytischen
Funktion, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt"
contained a proof of Cauchy's integral theorem without the use of double
or surface integrals.  He also introduced Laurent series and arrived at
the Laurent theorem two years before it was officially published in
Comptes Rendus by Cauchy. In that work Weierstrass expressed complex
numbers in the form

r{1+\lambda i}\over {1-\lambda i},

where r is the absolute value and \lambda is a real number given by
\lambda = \tan(\theta/2), where \thet is an amplitude of the complex
number. The paper "Zur Theorie der Potenzreihen" dates from autumn 1841.
In that work Weierstrass introduced the notion of uniform convergence and
examined series in several variables. In the next paper "Definition
analytischer Funktionen einer Veranderlichen vermittelst algebraischer
Differentialgleichungen", dating from 1842, he proved the theorem of
Cauchy concerning the solution of systems of differential equations

dx/dt = G_i(x_1,...,x_n)

with the initial conditions x_i(0) =a_i, where the G_i are polynomials. In
that paper Weierstrass described the process of analytic continuation of
power series. Those papers were published only in the first volume of his
Collected Papers in 1894 but they make it clear that, already in 1842, he
was in full possession of all the methods and ideas which allowed him to
construct his theory of functions. Unfortunately, his first published
paper "Bemerkung uber die analytischen Fakultaten" appeared in a
supplement to the school report of the year 1842 in Deutsch-Krone and
received little attention. The same happened to another paper on Abelian
functions which was published in 1848 in the Braunsberg school prospectus.
Weierstrass would have remained unnoticed but for the fact that in summer
1853 during his stay in Munster he read Gudermann's opinion of his
dissertation for the first time. Already in 1840 Cristof Gudermann not
only recognised Karl's rare talent but also placed him among famous
discoverers and suggested that his student should work at a university
rather than as a secondary school teacher. This note encouraged
Weierstrass to publish his paper Zur Theorie der Abelschen Funktionen".
Its appearance in 1854 in Crelle's Journal caused a sensation in the
mathematical world. The consequences were amazing. The first recognition
was the award of an honorary doctorate by the University of Konigsberg.
Then the Prussian ministry of education gave him a year's paid leave from
the Braunsberg gymnasium to enable him to concentrate on his research.

Weierstrass gained enough confidence to apply for the post at the
University in Breslau which was vacated by Kummer's appointment as
professor in Berlin. It may sound strange that Weierstrass's application
was rejected because of Kummer.  The reason was that Kummer, who spent 13
years teaching in Breslau, intended to take Weierstrass to Berlin. He
applied on 12 June 1856 to the university in Berlin with a request for a
post for Weierstrass. He was not successful on that occasion but his
eventual success gave to mathematics in Berlin three great names:
Weierstrass, Kummer and Kronecker.

Weierstrass's "Theorie der Abelschen Funktionen" published in Crelle's
Journal in 1856 contained results of his dissertation. D. Hilbert
considered these results concerning the solution of the Jacobi inversion
problem for the hyperelliptic integral the greatest achievement of
analysis. This publication was a turning point in the life of Karl
inquiries about him through Alexander von Humboldt. This pushed von
Humboldt to action and on 1 July 1856 Weierstrass was appointed professor
at the Industry Institute in Berlin.

In September 1856 Weierstrass and Kummer went to Vienna. There, Graf Thun
offered Weierstrass 2000 Gulden and a professorship at an Austrian
university of his choice. Weierstrass declined that offer but it became
clear to Kummer that if they wanted to keep the great mathematician in
Germany he had to take action again.

Shortly afterwards as a result of Kummer's efforts, Weierstrass was
appointed Associate Professor at the University in Berlin. In November
1856 he became a member of the Berlin Academy. From then until 1890 he
lectured on a great variety of topics, including periodic lectures on
elliptic functions, lectures on geometry, and on mechanics. The famous
mathematical seminar which he initiated together with Kummer in 1861
attracted international interest. From 1862 it was customary to make
awards to the best participants.  It is worth mentioning that the first
mathematical-physical seminar was founded in Konigsberg in 1834 by
Jacobi, Neumann and Sohnke, the second in Halle in 1838 and the third in
Gottingen in 1850.

Weierstrass' heavy work load resulted in a breakdown in his health in
December 1861. He returned to teaching after a year but he lectured from a
sitting position with a student writing the necessary text on the
blackboard. For the rest of his life he suffered recurring bouts of
bronchitis and phlebitis, but his determination kept him teaching and
pursuing his research.  In his lectures Weierstrass initiated the logical
and rigorous development of analysis starting with his own construction of
the real number system. He established the epsilon-delta notions in the
concept of continuity and convergence, uniform convergence, absolute
value, neighbourhood (a-\delta, a+\delta) of a point $a$, and many others.
He rejected intuitive arguments which were still prevalent among many
contemporary mathematicians. It was Weierstrass who in his lectures of
1862 gave the famous example of a continuous nowhere differentiable
function. In that year he first developed in his lectures the theory of
the \gamma(u) and \sigma(u) functions.  His famous approximation theorem
appeared in connection with the heat equation and was published in July
1885 in the Proceedings of the Meetings of the Berlin Academy of Sciences.
He also contributed to the development of the Calculus of Variations on
which he lectured in 1879. His main lectures however concentrated on
Abelian functions.  His periodically presented lectures included
"Introduction to the Theory of Analytic Functions", "The Theory of
Elliptic Functions", once approached from the point of view of
differential equations, another time from the point of view of the theory
of functions, "Application of Elliptic Functions to Geometry and
Mechanics", "Application of Abelian Functions to Geometric Problems and
the Calculus of Variations". His lectures drew audience of up to 250,
among them over 100 future professors, including S. Kovalevsky, Schwarz,
Fuchs, G. Mittag-Leffler, L. Koenigsberger, H.  Minkowski, and Cantor.

Weierstrass became the leading influence on the mathematical world. He
obtained high recognition throughout Europe. The Swedish mathematician
Mittag-Leffler mentions a nice anecdote.  When he got a scholarship to
study abroad in 1873 he went to Paris where Hermite greeted him with the
words "Vous avez fait erreur, Monsieur, vous auriez du suivre les cours
de Weierstrass a Berlin". And of course Mittag-Leffler followed Hermite's

The most remarkable among Weierstrass's students was Sonia Kovalevsky, the
daughter of a Russian artillery general.  Women were not allowed to study
in Russia so in 1868 she contracted a marriage of convenience to a young
They first enrolled at the University of Heidelberg but after two years
they separated. Vladimir went to Jena and Sonia travelled to Berlin hoping
to attend Weierstrass's lectures.  Unfortunately, there was a ban on women
students at the University in Berlin and her application was rejected. So
she went straight to Weierstrass with her Heidelberg references. He gave
her some problems to solve and her solutions and enthusiasm impressed him
so he decided to teach her privately. During the four years that she spent
in Berlin, she became his close friend and an irreplaceable partner for
scientific discussions.  We do not know much about Weierstrass' private
life. Sonia, however, the second woman (after Maria Sklodowska Curie) to
hold a university post, a novelist and a revolutionary, became a wonderful
topic for many biographies and novels. Weierstrass regarded her as his
most talented student. In fact during her stay in Berlin she produced
three outstanding papers; on differential equations, on Abelian integrals
and on Saturn's rings, and managed to obtain a doctorate from the
University of Gottingen. However, Weierstrass, much to his regret, could

Their friendship, although based on purely scientific interaction, became
the subject of rumour. Weierstrass felt deeply hurt by persistent
insinuations surrounding his student and her mathematical achievements. On
return to Russia her interest in mathematics ceased. She couldn't work at
the University in Petersburg because her qualifications were not
recognised. She returned to family life, gave birth to a daughter in 1878
and together with her husband, Vladimir, tried to make money as an estate
entrepreneur, but this venture ended in bankruptcy.

Weierstrass and Sonia Kovalevsky corresponded fairly regularly from the
time they met till her death in 1891. His letters and encouragement to
take up mathematics again inspired her and helped her through financial
difficulties and her husband's suicide in 1883. In 1884 Mittag-Leffler
succeeded in getting her a post as lecturer at the University in
Stockholm. In 1888 she achieved great success, when her famous paper {\it
"On the rotation of a solid body about a fixed point"} won the French

Sadly, Weierstrass burned all Sonia's letters after her early death. His
letters, however, survived and are preserved at the Mittag-Leffler
Institute in Djursholm, Sweden. The vast correspondence to Mittag-Leffler,
H. Schwarz, Paul du Bois-Reymond, L. Koenigsberger, Riemann, L. Fuchs and
Sonia, which contains mostly mathematical problems, also illuminates
Weierstrass's life. He was considered successful but, in fact, his life
was full of sufferings and personal problems. He did not marry after an
engagement in Deutsch-Krone was broken due to the unfaithfulness of the
fiancee, according to his brother, Peter. His career in Berlin had hardly
started when his health collapsed.  Moreover, towards the end of his life,
rather than enjoying fame and appreciation, he felt isolated.  This was
because of a conflict with the mathematician and philosopher Leopold
Kronecker.

Weierstrass and Kronecker were friends for more than twenty years, sharing
many fruitful mathematical discussions and ideas.  Unfortunately, at the
end of 1870's their views on mathematics, especially the foundations of
mathematics, diverged gradually. Weierstrass's work on limits and
convergence led him to develop a theory of irrational numbers based on
convergent sequences of rationals which he called aggregates". His
student, Georg Cantor, founded the theory of transfinite numbers which
caused a revolution in mathematical thought. Before Cantor, mathematicians
had accepted the notion of the infinite in the situation of a sequence
"tending to infinity", but were not prepared to accept an actual infinity
per se. Cantor's great achievement was to introduce this precise concept,
in fact a whole class of different infinities -- that corresponding to the
countable sets such as the natural numbers, that of the continuum of real
numbers, and so on. For Kronecker, the type of non-constructive reasoning
used by Weierstrass and Cantor was deeply suspect. His famous dictum, "God
made the integers, all the rest is the work of man", pronounced at the
Berlin Congress in 1886, encapsulated his philosophy, and he envisaged the
inclusion of essentially the whole of mathematics within the terms of
arithmetic.

Kronecker did not hide his objections to the work of Weierstrass and
Cantor and criticized them openly in front of students. No wonder that
this mathematical conflict turned into personal quarrels. Towards the end
of the 1880's Weierstrass evidently admitted that his long friendship with
Kronecker was over though Kronecker himself seemed unaware of it.
Weierstrass even considered the possibility of leaving Berlin for
Switzerland to avoid the continuing conflict; but, since he did not wish
his successor at the university to be chosen by Kronecker, he decided to
stay. It became clear to him, however, that if he did not publish his
lectures and works his achievements might fall into oblivion.

His worry was probably exaggerated because his students were spread all
over Europe and continued his research. His greatest successor was H.
Poincare in Paris.  Nevertheless the situation in Berlin was tense. In
1885 a special commission responsible for editing his works was set up.
Weierstrass himself was no longer able to supervise the whole process so
his students undertook the task of gathering and polishing up his lectures
based on their own notes or transcripts. Weierstrass intended to give to
mathematics an extensive treatment of analysis, clear and complete. So
Knoblauch and Hettner were to prepare the theory of Abelian functions, his
greatest aim in life. His dream was not fulfilled, however, and the draft
was highly unsatisfying in his opinion and not up to the standard he
expected. The lack of precision and printing errors worried him too. The
first volume appeared in 1894 and contained his collected papers. The
second was printed in 1895 but the next five volumes were published
between 1902 and 1927, after his death. It is worth mentioning that it is
to Weierstrass that we owe the publication of the collected papers of
Jacobi, Dirichlet, Steiner and the letters of Gauss. In his old age he
worked as an editor to supplement his salary which was not sufficient to
maintain his family. He lived in Berlin with his sisters Elise and Klara
who kept house, his father who died in 1869 and his uncle's grandson whom
he took custody of as a two-year-old in 1884. He also took care of
Borchardt's six children after his closest friend's death in 1880.

In 1892 Weierstrass received the Helmholtz medal and in 1895 he was
awarded the Copley Medal, the highest honour of the Royal Society of
London. In the same year he celebrated his 80th birthday among his
students. He spent his last three years in a wheelchair and died on 19
February 1897, after an inflammation of the lungs.

References

1.  Ahrens, W.:  Skizzen aus dem Leben Weierstrass'. Math.-naturwiss.
Blatter 4 (1907), 41-47.

2.  ---:  Gudermanns Urteil uber die Staatsexamensarbeit von Weierstrass.
Math.-naturwiss Blatter 13 (1916), 44-46.

3.  Baker, A. C.:  Karl Theodor Wilhelm Weierstrass, the father of modern
analysis. Math. Spectrum 1996/7, vol.29, No.2, 25-29.

4.  Biermann, K. R.:  Karl Weierstrass, Ausgewahlte Aspekte seiner
Biographie. Journal fur die reine und angewandte Mathematik, 223 (1966),
191-220.

5.  Bolling, R.:  Karl Weierstrass - Stationen eines Lebens.  Jahresber.
Dt. Math.-Vereinigung, 96 (1994), 56-75.

6.  Dorofeeva, A. & Chernova, M.:  Karl Weierstrass. Mathematics and
Cybernetics 7 (1985), Moscow (in Russian).

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Weierstrass. Forsch. u. Fortschr. 35 (1961), 236-239.

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1-37.

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Kgl. Ges. d. Wissensch. zu Gottingen 1897, 60-69.

10.  ---:  Uber das Unendliche. Mathematische Annalen 95 (1926), 161-185.

11.  Kiepert, L.:  Personliche Erinnerungen an Karl Weierstrass.
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12.  Kochina, P.:  Karl Weierstrass. Moscow "Nauka" 1985 (in Russian).

13.  Lampe, E.:  Karl Weierstrass. Jahresber. d. Dt. Math.-Vereinigung 6
(1899), 27-44.

14.  ---:  Zur hundertsten Wiederkehr des Geburtstages von Karl
Weierstrass. Jahresber. d. Dt. Math.-Vereinigung 34 (1915), 416-438.

15.  von Lilienthal, R.:  Karl Weierstrass. Westfalische Lebensbilder 2
(1931), 164-179.

16.  Lorey, W.:  Die padagogischen Ansichten des Mathematikers Karl
Weierstrass. Blatter fur hoheres Schulwesen 32 (1915), 626-629.

17.  ---:  Karl Weierstrass zum Gedachtnis. Zeitschr. f. math. u.
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19.  Mittag-Leffler, G.L:  Une page de la vie de Weierstrass. Compte Rendu
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20.  ---:  Zur Biographie von Weierstrass. Acta Mathematica 35 (1912),
29-65.

21.  ---:  Die ersten 40 Jahre des Lebens von Weierstrass. Acta
Mathematica 39 (1923), 1-57.

22.  ---:  Sophie Kovalevsky: Notice biographique. Acta Mathematica 16
(1892/93), 385-395.

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133-198.

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26.  ---:  Bericht uber die Herausgabe des 7. Bands der Mathematischen
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