Weierstrass, Karl Theodor Wilhelm 529

(

π

–

2

1

–

2

and was one of the first scholars to permit

negative and complex solutions to equations. He

argued that the number i, representing the square

root of –1, does have a place in real-world applica-

tions of mathematics. Wallis also introduced the sym-

bol ∞for infinity.

In addition to working as a research mathemati-

cian, Wallis also made significant contributions as a

mathematical historian. He restored several ancient

Greek texts and presented a comprehensive survey of

the history of algebra in his 1685 work Treatise on

Algebra. He died in Oxford, England, on November

8, 1703.

Wallis helped shape the entire course of mathemati-

cal work in England for the latter part of the 17th cen-

tury and paved the way for Newton to develop his

ideas. As a result, Britain became the center of mathe-

matical research in the late 1600s and remained so

until the influence of L

EONHARD

E

ULER

(1707–83) and

members of the B

ERNOULLI FAMILY

moved the focus

back to continental Europe in the mid 1700s.

See also W

ALLIS

’

S PRODUCT

.

Wallis’s product In 1656 English mathematician

J

OHN

W

ALLIS

discovered the following remarkable

expression for the number as an

INFINITE PRODUCT

:

He discovered this result while attempting to compute

INTEGRAL

s of the form

with nnot necessarily an integer. (For example, when

n= , four times this integral equals the area of a

circle.) The work of L

EONHARD

E

ULER

(1707–83) on

the

ZETA FUNCTION

also leads to a proof of Wallis’s

product, one that is relatively straightforward to follow.

Wallis’s formula led English colleague L

ORD

W

ILLIAM

B

ROUNCKER

(1620–84) to discover the follow-

ing astonishing

CONTINUED FRACTION

formula for π:

See also

PI

.

Weierstrass, Karl Theodor Wilhelm (1815–1897)

German Analysis Born on October 31, 1815, Ger-

man scholar Karl Weierstrass is remembered as a lead-

ing figure in the field of mathematical

ANALYSIS

.

Throughout his career he emphasized the need for

absolute rigor, and, following the efforts of A

UGUSTIN

-

L

OUIS

C

AUCHY

(1789–1857), worked to introduce very

precise definitions of fundamental notions in the study

of

CALCULUS

. He developed the famous ε– δdefinition

of a

LIMIT

that we use today, as well as precise clarifica-

tion of the meaning of continuity, convergence, and of

DIFFERENTIAL

s. To illustrate that intuitive understand-

ing alone never suffices, Weierstrass presented exam-

ples of pathological functions that are continuous but

have no well-defined tangent lines. Weierstrass also

solved the famous

ISOPERIMETRIC PROBLEM

with his

“calculus of variations.”

Weierstrass entered the University of Bonn in 1834

to pursue a degree in law and finance, but he never

completed the program, choosing to follow instead a

study of mathematics at the Theological and Philosoph-

ical Academy of Münster in 1839. He began his aca-

demic career as a provincial mathematics schoolteacher.

While teaching, Weierstass published a number of

papers on the study of real and complex functions. The

bulk of his early work went unnoticed by the mathe-

matics community, and it was not until the publication

of his 1854 piece Zur Theorie der Abelschen Functio-

nen (On the theory of Abelian functions) that Weier-

strass’s genius as a mathematician was recognized. He

was immediately awarded an honorary doctoral degree

from the University of Königsberg and was granted a

year’s leave from the school to pursue advanced mathe-

matical study. He never returned to school teaching,

however. In 1856, at the age of 40, Weierstrass

accepted a professorship at the University of Berlin,

which he retained for the remainder of his life.