Karl Theodor Wilhelm
Weierstrass (WeierstraÃŸ) (October 31,
1815 â€“ February 19, 1897) was a German mathematician who is often cited as the "father
of modern analysis".
Biography
Weierstrass was born in Ostenfelde, part of
Ennigerloh, Province of Westphalia.
Weierstrass was the son of Wilhelm Weierstrass, a government
official, and Theodora Vonderforst. His interest in mathematics
began while he was a
Gymnasium student.
He was sent to the
University of
Bonn upon graduation to prepare for a government
position. Because his studies were to be in the fields of
law, economics, and finance, he was immediately
in conflict with his hopes to study mathematics. He resolved the
conflict by paying little heed to his planned course of study, but
continued private study in mathematics. The outcome was to leave
the university without a degree.
After that he studied mathematics at the
University of
MÃ¼nster (which was even at this time very famous for
mathematics) and his father was able to obtain a place for him in a
teacher training school in MÃ¼nster.
Later he was certified as a teacher in that city. During this
period of study, Weierstrass attended the lectures of
Christoph Gudermann and became
interested in
elliptic
functions.
After 1850 Weierstrass suffered from a long period of illness, but
was able to publish papers that brought him fame and distinction.
He took a
chair at the Technical University of Berlin, then known as the Gewerbeinstitut. He was
immobile for the last three years of his life, and died in Berlin
from
pneumonia.
Mathematical contributions
Soundness of calculus
Weierstrass was interested in the
soundness of calculus. At the time, there
were somewhat ambiguous definitions regarding the foundations of
calculus, and hence important theorems could not be proven with
sufficient rigour. While
Bolzano had
developed a reasonably rigorous definition of a
limit as early as 1817 (and possibly
even earlier) his work remained unknown to most of the mathematical
community until years later,and many had only vague definitions of
limits and
continuity of functions.
Cauchy gave a form of the
definition of
limit, in the context of formally defining the derivative, in
the 1820s,{{citation
first=A.L. 
last=Cauchy 
authorlink=Augustin Louis Cauchy 
title=RÃ©sumÃ© des leÃ§ons donnÃ©es Ã lâ€™Ã©cole royale polytechnique
sur le calcul infinitÃ©simal 
place=Paris 
year=1823 
url=http://mathdoc.ujfgrenoble.fr/cgibin/oeitem?id=OE_CAUCHY_2_4_9_0 
chapter=SeptiÃ¨me LeÃ§on  Valeurs de quelques expressions qui se
prÃ©sentent sous les formes indÃ©terminÃ©es \frac{\infty}{\infty},
\infty^0, \ldots Relation qui existe entre le rapport aux
diffÃ©rences finies et la fonction dÃ©rivÃ©e 
chapterurl=http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f45n5.capture 
postscript=, p. 44.}}but did not correctly distinguish between
continuity at a point versus uniform continuity on an interval, due
to insufficient rigor. Notably, in his 1821 Cours
d'analyse, Cauchy gave a famously incorrect proof that the
(pointwise) limit of (pointwise) continuous functions was itself
(pointwise) continuous. The correct statement is rather that the
uniform limit of uniformly
continuous functions is uniformly continuous.This required the
concept of uniform convergence,
which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper,
where Gudermann noted the phenomenon but did not define it or
elaborate on it. Weierstrass saw the importance of the concept, and
both formalized it and applied it widely throughout the foundations
of calculus.
The definition of
limit, as formulated by Weierstrass, is as follows:
\displaystyle f(x) is continuous at \displaystyle x = x_0 if for
every \displaystyle \varepsilon > 0\ \exists\ \delta > 0 such
that
 \displaystyle xx_0 \delta \Rightarrow f(x)  f(x_0)
\varepsilon.
Using this definition and the concept of uniform
convergence,Weierstrass was able to write proofs of
several thenunproven theorems such as the intermediate value theorem (for
which Bolzano had given an
insufficiently rigorous proof), the BolzanoWeierstrass theorem, and
HeineBorel
theorem.
Calculus of variations
Weierstrass also made significant advancements in the field of
calculus of variations. Using
the apparatus of analysis that he helped to develop, Weierstrass
was able to give a complete reformulation of the theory which gave
way for the modern study of calculus of variations. Among several
significant results, Weierstrass established a necessary condition
for the existence of strong extrema
of variational problems. He also helped devise the WeierstrassErdmann corner
conditions which give sufficient conditions for an extremal to
have a corner.
Other analytical theorems
Selected works
Students of Karl Weierstrass
Honours and awards
The lunar
crater Weierstrass is named after him.
Notes
External links
