# Karl Theodor Wilhelm Weierstrass

# Karl Theodor Wilhelm Weierstrass

**1815-1897**

**German Mathematician**

An exceptional teacher and researcher, Karl Weierstrass was one of the most influential mathematicians of the nineteenth century. He taught and inspired some of Europe's most gifted mathematicians and made important contributions in elliptic functions, the calculus of variations, and other areas of mathematics.

Weierstrass was born in Prussia, now part of Germany, to a mid-level bureaucrat. His father, a well-educated and intelligent man, was seemingly content working at a lower level than his abilities permitted. His father's position with the main tax office convinced him that his son should study accounting and law, though Weierstrass's interests lay elsewhere, in mathematics. This dilemma played itself out internally, with Weierstrass paying little attention to his studies at all for a time.

Eventually he decided to pursue mathematics, showing an immediate aptitude for the field. He changed universities before taking his graduation exams, leaving the University of Bonn to attend the Theological and Philosophical Academy of Münster. There he studied under the mathematician Gudermann, whose reputation lured him to Münster.

Graduating from Münster in 1842, Weierstrass took a teaching position at the Gymnasium at Münster where, in addition to mathematics, he taught many other subjects. This position, like those his father held, was far below his abilities and he soon tired of it. At the same time, he continued working on his research, publishing some papers on elliptic and complex functions in relatively obscure publications. Unfortunately, the stress of his unloved teaching job and his research began to take a toll on Weierstrass's health, leaving him with frequent attacks of dizziness and nausea. These spells were to recur frequently for much of the rest of his life.

Weierstrass finally gained recognition with the publication of a paper on the theory of Abelian functions in August Crelle's journal in 1854. In fact, based solely on this paper, he was given an honorary doctorate from the University of Königsberg that same year.

Over the next few years a number of European universities fought to attract Weierstrass to join their faculty. He finally accepted an offer from the University of Berlin, his original university of choice.

As a lecturer, Weierstrass excelled, attracting students from all over the world. Among those influenced by him were many of the best mathematicians of his time, including Georg Cantor (1845-1918), Ferdinand Frobenius (1849-1917), Felix Klein (1849-1915), Hermann Schwarz (1843-1921), Gösta Mittag-Leffler (1846-1927), Sophus Lie (1842-1899), and Sonya Kovalevskaya (1850-1891). Weierstrass was especially taken with Kovalevskaya, helping her to receive an honorary doctorate from Göttingen and to receive a position in Stockholm later. Her early death was a severe blow to him.

Although Weierstrass published very little during his career, many of his findings were announced in his lectures, which were collected and published during his later years and after his death. A colleague, who noted Weierstrass's reputation as "the father of modern analysis," summarized his achievements: "Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations. He also advanced the theory of bilinear and quadratic forms." All in all, an impressive professional legacy.

Weierstrass died at the age of 81, spending the last three years of his life confined to a wheelchair. The first two volumes of his collected works were published before his death, and another five were published posthumously.

**P. ANDREW KARAM**

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