ProofWiki:Mathematicians/Karl Theodor Wilhelm Weierstrass

German mathematician whose main work concerned the rigorous foundations of calculus.

Known as "the father of modern analysis".

Nationality

German

History

• Born: 31 Oct 1815, Ostenfelde, Westphalia (now Germany)
• Died: 19 Feb 1897, Berlin, Germany

Achievements

• Pioneered the $\epsilon-\delta$ definition of continuity.
• Was able thereby to formulate proofs of the Bolzano-Weierstrass Theorem (which had been proved earlier, independently of Weierstrass, by Bernhard Bolzano), the Intermediate Value Theorem and the Heine-Borel Theorem.
• Made significant advancements in the field of calculus of variations.

Theorems and Definitions

• Weierstrass Approximation Theorem
• Weierstrass-Casorati Theorem (with Felice Casorati)
• Weierstrass's Elliptic Functions
• Weierstrass-Erdmann Corner Conditions
• Weierstrass Function
• Weierstrass M-Test
• Weierstrass Preparation Theorem
• Weierstrass Factorization Theorem
• Bolzano-Weierstrass Theorem (independently of Bernhard Bolzano)
• Enneper-Weierstrass Parameterization (with Alfred Enneper)
• Hermite-Lindemann-Weierstrass Theorem (with Charles Hermite and Ferdinand von Lindemann)
• Sokhotski-Weierstrass Theorem (with Yulian Vasilievich Sokhotski)
• Stone-Weierstrass Theorem (with Marshall H. Stone) (a generalization of the Weierstrass Approximation Theorem)

Books and Papers

• 1854: Zur Theorie der Abelschen Funktionen
• 1856: Theorie der Abelschen Funktionen
• 1894: Abhandlungen-1
• 1897: Abhandlungen-2
• 1902: Vorl. ueber die Theorie der Abelschen Transcendenten
• 1915: Abhandlungen-3
• 1927: Vorl. ueber Variationsrechnung

Notable Quotes

"A mathematician who is not also something of a poet will never be a complete mathematician."

See also

• John J. O'Connor and Edmund F. Robertson: "Karl Theodor Wilhelm Weierstrass": MacTutor History of Mathematics archive

• Eric Temple Bell: Men of Mathematics (1937): Chapter $\text{XXII}$
• George F. Simmons: Calculus Gems (1992), Chapter $\text {A}.33$