In 1829, Weierstrass Was Accepted to the Catholic Gymnasium (High School) in Paderborn

The developments and discoveries of Karl Theodor Wilhelm Weierstrass had great impact on the math community, so much so, that it’s still being taught to students today. However, his life did not start out in a way that would suggest this outcome. Weierstrass was born October 31, 1815 in Ostenfelde, Westphalia (which is now part of Germany) to Wilhelm Weierstrass and Theodora Vonderforst, who were married five months before his birth. The eldest of four children, two sisters and a brother, none ever married, including Karl, himself. Little is known about Weierstrass’ mother’s family, except that they were moderate Catholics, like the Weierstrass family.11 (p887) Weierstrass’s father was an intelligent and well-educated man with extensive knowledge in the arts and sciences and had the potential to hold higher positions than he actually did.6 Wilhelm had a career as a bureaucrat in the Westphalian and what was known then, as the Prussian government. When Weierstrass was just eight years old, his father took a position as for the taxation service for the Prussian government. Consequently, the family moved around quite a bit and Weierstrass was forced to attend many different primary schools. In 1827, Weierstrass’s mother died suddenly and his father soon remarried, the following year.6 In 1829, Weierstrass was accepted to the Catholic Gymnasium (high school) in Paderborn. This was where his father was forced to move to become assistant tax commissioner of Paderborn. Because of financial difficulties at home, Weiestrass, at the age of fourteen, went to work as a bookkeeper for a local merchant’s wife.11 (p 887) This allowed Weiestrass’s talent of mathematical skill to surface, in addition to the numerous awards and prizes he won at the Gymnasium.6 In his spare time, Weiestrass would often read copies of the Journal fur die reine und angewandte Mathematik (Journal of Pure and Applied Mathematics), which helped to pique his interest in mathematics. Additionally, he would often “coach” his brother in mathematics. However, his brother did not find these coaching session to be particularly helpful and stated this “proofs were generally ‘knocking.’” 6 After completely his studies at the Gymnasium, Weierstrass enrolled to the University of Bonn to study public finance and administration, at the request of his father. This was the course of study for those interested in seeking positions in higher administration in Prussia.6 However, Weierstrass really wanted to study mathematics, and because of the stress of the internal conflict of wanted to study mathematics and following his father’s wishes, Weierstrass suffered from both mental and physical strain. This led to Weierstrass indulging in bouts of drinking and fencing as a means of rebellion. 11 (p. 888) Eventually, Weierstrass found Dietrich von Munchow, and understanding advisor in astronomy, mathematics, and physics. However, Munchow was of the old school, and since he only gave elementary lectures, was removed from the advances of modern mathematics.6 Weierstrass begain reading Jacobi’s Fundamenta nova theoriae functionum ellopticarum. However, he found this work particular difficult. Once reading a transcript by Christof Gudermann, a lecture on modular functions that showed the theory of elliptic variables to be understandable to him and inspired him to do his own research.6 After eight semesters at the University of Bonn, Weierstrass left, without taking the examinations for a degree. His father was extremely disappointed. However, a family friend, who was president of the court of justice of Paderborn, suggested that Weierstrass enroll at the nearby Theological and Philosophical Academy in Munster and take the

1 teacher’s examination. Weierstrass enrolled on May 22, 1839 and a year later, sat for an examination for a teaching degree. 11 (p888) He asked for a particularly challenging problem on the mathematics exam, one that involved the representation of elliptical functions. Weierstrass provided an important advance in the new theory of elliptic functions and this contained important starting points for his subsequent investigations. This follows Abel, who Weierstrass had always held in high regard.6 Weierstrass had no knowledge of just how well he answered this problem. He was just told that he had passed the mathematics exam, along with the other sections of the exam. Another fact they chose to leave out, was that one of the external examiners for the mathematics exam was Gudermann. 11 (p888) Gudermann spoke highly of Weierstrass’s work and said it was “of equal rank with the discoverers who were crowned with glory.” 6 Gudermann recognized how Weierstrass’s answer was about to make important advances in elliptical function theory. 11 (p. 888) However, his work did not receive such high appraisal from the school superintendent. Despite this fact, Weierstrass was pleased with the fact that Gudermann spoke so highly of him, although his work contained criticism of Gudermann’s methodology. 6 After passing his oral examination in April 1841, Weierstrass taught for a year probationary period at the Gymnasium in Munster, then transferred to the Catholic secondary school in Deutsch-Krone, West Prussia, from 1842-1848.11 (p889) After that, Weierstrass transferred to the Catholic Gymnasium in Braunsberg, East Prussia from 1848-1855. In addition to teaching mathematics and physics, Weierstrass taught German, botany, geography, history, gymnastics, and calligraphy. This was where the peculiar P of the Weiesrstrass p-function found it’s origination.6 However, these fifteen years of teaching, proved themselves unsatisfactory to Weierstrass, finding himself surrounded by disinterested students, lack of colleagues for mathematical discussions, and no access to mathematical libraries. As a result, he spent all his free time, devoted to studying any contemporary mathematics that he could find.11 (p 889) Weierstrass’s first publication to the Braunsberg’s prospectus virtually went unnoticed, but he made his first publication, in 1854,on Abelian functions entitled “Zur Theorie der Abelschen Functionen” 6 “On the Theory of Abelian Functions”11 (p889) to the Journal of Pure and Applied Mathematics, which proved to be an important turning point in Weierstrass’s life. In this article, he “demonstrated the solution to the problem of inversion of the hyperelliptic integrals, which he accomplished by representing Abelian functions as the quotients of constantly converging power series.” 6 After suffering many years of stress, eventually the stress took a toll on Weierstrass’s health and by 1850, he had begun to suffer persistent attacks of vertigo that could last as long as an hour and would only stop once he experienced a violent bout of vomiting.11 (p889) These attacks, then, commonly referred to as “brain spasms”, recurred for twelve years and made it impossible for him to work.6 Despite this, his first publication, which was considered to be a “preliminary” statement, Liouville called it “one of those works that marks an epoch in science”6 Then, on March 31, 1854, Weierstrass received an honorary doctorate from the University of Konigsberg (now knows as Danzig). In order to retain him, Braunsberg promoted Weierstrass to senior lecturer and, in the fall of 1855, offered a year long sabbatical to continue his studies. Weierstrass did not want to return to the school, and applied for the post of Kummer’s successor (a professor of mathematics) at the University of Breslau in East Prussia (now

2 Wroclaw in Poland)11 (p.889) which was considered “an unusual mode of procedure” 6 However, Weierstrass did not receive the position.6 In his second published article on Abellian functions, in 1856 “Theorie der Abelschen Functionen” Weierstrass proved something he had only touched on previously. “He had realized one of the greatest achievements of analysis, the solution of the Jacobian inversion problem for hyperelliptic integrals”6 On June 14, 1856, Weierstrass, had accepted a position as a professor at the Industry Institute in Berlin, a forerunner of the Technische Hochschule. However, what Weierstrass was really seeking for was a position at the University of Berlin, and his aspiration of a position was finally realized while attending a conference of natural scientists in Vienna, in September 1856, when he was offered a special professorship with any Austrian university of his choosing. Unable to choose, he was invited to the University of Berlin as an associate professor. He accepted and on November 19, 1856, he became a member of the Berlin Academy. He was not able to leave the Industrial Institute until July 1864, but after that, he assumed his position as a chair at the university.6 Because he spent so much of his teaching life teaching elementary classes, something that is far removed from the centers of scientific activity, Weierstrass did find time for his own research, but only at the expense of his own health. At Berlin, the heavy demands took its toll on Weierstrass’s health, as well, and Weierstrass suffered a complete collapse on December 16, 1861. He did not return to work until the winter semester of 1862-1863. Consequently, he taught his lectures while seated and turned over the duty of work on the blackboard to an advanced student. Soon, the “brain spasms” were replaced by frequent attacks of bronchitis and phlebitis, which affected him until his death. Despite this, Weierstrass became a recognized master, mainly through his lectures. His publications were delayed, not because of “a basic aversion to printer’s ink”6 as was often rumored, but “because his critical sense invariably compelled him to base any analysis on a firm foundation, starting from a fresh approach and continually revising and expanding.” 6 At first, his lectures were often cited as “seldom clear, orderly, or understandable.”6 However, because he was known for lecturing on new theories, this attracted students from all around the world and eventually, he had over 250 students. Since no one else offered this same subjects, graduate students and lectures from universities were drawn to Berlin. To add to his popularity, Weierstrass was also generous in suggesting topics for dissertations and continuing investigations. 6 Among Weierstrass’s lectures was on the application of the Fourier series and integrals to problems of mathematical physics. However, Weierstrass had decided to discontinue this course due to lack of available works and his own unproductive efforts, which caused frustration on his part. Then in 1885, Weierstrass took up the representation of single-valued functions of a real variable by means of trigonometric series. On July 9, 1857, Weierstrass clarified his research applications and stated “mathematics occupies an especially high place because only through its aid can a truly satisfying understanding of natural phenomena be obtained.”6 His outlook was similar to that of Gauss, who believed “mathematics should be a friend of practice, but never its slave.”6 Weierstrass developed a great lecture cycle that included: “Introduction to the Theory of Analytic Functions”,“Theory of Elliptical Functions” sometimes beginning

3 with differential calculus, or starting with the theory of functions, “Applications of Elliptic Functions to Problems in Geometry and Mechanics”, “Theory of Abelian Functions”, “Application of Abelian Functions to the Solution of Selected Geometric Problems”, and “Calculus of Variations.”6 Weierstrass did lecture for seven semesters (1864-1873) on synthetic geometry as a promise to Jakob Steiner, before his death in 1863. However, Weierstrass had no interest in this subject matter, and only did this to uphold his promise to Steiner.6 Among Weierstrass’ students are: Heinrich Bruns, Georg Frobenius, Georg Hettner, Ludwig Kiepert, Wilhelm Killling, Johannes Knoblauch, Ernst Kotter, Reinhold von Lilienthal, Hans von Mangoldt, Felix Muller, Eugen Netto, Friedrich Schottky, Ludwig Stickelberger, and Wilhelm Ludgwig Thome. Participants in his seminar included: Paul Bachmann, Oskar Bolza, Friedrich Engel, Leopold Gegenbauer, August Gutzmer, Lothar Heffter, Kurt Hensel, Otto Holder, Adolf Hurwitz, Felix Klein, Adolf Kneser, Leo Koenigsberger, Frtiz Kotter, Mathias Lerch, Sophus Lie, Jacob Luroth, Franz Mertens, Hermann Minkowski, Gosta Mittag-Leffler, Hermann Amandus Schwarz, and Otto Stolz.6 Many of Weierstrass’s students accepted his theories as undeniable truths, even though many of Weierstrass’s work was very difficult to check because he often referenced himself. Despite this, opposition was sparse, but there was some, including Felix Klein, who had openly cited such problems. 6 Weierstrass possessed an unusual teaching style. His classes contained new and important ideas, but he allowed his students to distribute his information, rather than publishing it himself. Weierstrass felt it was his job to “create” significant mathematics. And, in this, he was successful. Because of his great work, proving theorems, and repairing subtle misconceptions, he became known as the “father of modern analysis.”7 (p 129) He also experienced some criticism from his colleges. This hurt him deeply. In the late 1870’s, Weierstrass had a conflict with a close friend, Leopold Kronecker, which diminished their friendship. Weierstrass realized and worked on Cantor’s accomplishments and began some of his work on the concept of countability. However, Kronecker disagreed and made it his goal “of investigating the error of every conclusion used in the so called present method of analysis.”6 As a reaction to this, Weierstrass decided to leave Germany and go to Switzerland, under the impression that everything he had worked so hard far was about to fail. However, to prevent such destruction, Weierstrass decided to stay in Berlin to fight for his belief. The problem was still that his work still had to be endorsed by Kronecker. But, this was no longer a problem when, in 1891, Kronecker’s death allowed for the appointment of Hermann Amandus Schwartz, as his successor. 6 In 1870, at the age of fifty-five, Weierstrass met Sonya Kovalevsky, a twenty year old Russian who had come to Berlin from Heidelberg. She had taken her first semester under Leo Koenigsberger. Weierstrass taught Kovalevsky privately, since her admission to the university could not be secured. He claimed that she was a “refreshingly enthusiastic participant” in regards to all of his thoughts and things that he had troubles with became clearer after talking them over with her. In an August 20, 1873 letter, Weierstrass writes of their having, “dreamed and been enraptured of so many riddles that remain for us to solve, on finite and infinite spaces, on the stability of the world system, and on all major problems of the mathematics and the physics of the future. [She had]

4 been close …throughout [his] entire life…and never have I found anyone who could bring me such understanding of the highest aims of science and such joyful accord with my intentions and basic principles as you!”6 She received her doctorate in absentia at Gottingen in 1874.6 However, their friendship did have its troubles. Kovalevsky was associated with socialist circles, maintained a literary career as an author of novels, and had a strong advocacy of the emancipation of women, and these factors deteriorated her relationship to Weierstrass. However, many of his letters to her were unanswered. He was influential in her gaining the appointment as a lecturer in mathematics in Stockholm in 1883 and a life professorship in mathematics in 1889. This misinterpretation of their relationship and her early death in 1891, caused Weierstrass much physical pain. During the last three years of his life, he was confined to a wheelchair, immobile and dependent. He died of pneumonia, on February 19, 1897, in Berlin Germany.6 As for the publication of his works, Weierstrass was not satisfied with the transcripts of his lectures nor with textbooks that followed his concepts, and his major ideas and methodologies went unpublished. In 1887, Weierstrass had decided to publish his work, with the assistance of younger mathematicians. He lived to see the first two volumes appear in print, in 1894 and 1895. In compliance with his wishes, volume four was treated with preferential treatment and was published in 1902, entitled “Lectures on the Theory of Abelian Transcendentals.” Volume three was published the following year, and twelve years later, volume five (“Lectures on Elliptical Functions”) and six (“Selected Problems on Geometry and Mechanics to be Solved With the Aid of the Theory of Elliptic Functions”) made its appearance. And, twelve years after that, volume seven (“Lectures on the Calculus of Variations”) made its appearance. Weierstrass tried to ensure that his publication would be published soon after this death, but this did not happen. Volumes VIII-X, which contained works on hyperelliptic functions, a second edition of elliptic functions, and the theory of functions never made it to publication. Weierstrass was best known for his development of the “epsilon method.” This was first developed in Weierstrass’s class “Introduction to Analysis” during the 1859- 1860 school year. This method provided a rigorous way for mathematicians to work with the notion of an infinite sequence or series reaching a limit. Here is an example to illustrate his explanation: Consider the geometric series that begins with 1 and continues with successive power of ½.

1 + ½ + ¼ + 1/8 + 1/16 +1/32

This is continued to infinity. And, the formula for the sum of a finite number of terms of a geometric progression is :

n-1 a  rk = a r n -1 k=0 r – 1

This formula can be applied to any finite number n of terms in the terms in the series a =1 and r= ½, resulting in:

5 1 - ½ n 1 – ½

It was argued that the numerator of this expression reach 1 in the limit as n reached infinity since ½ n reached 0 as n reached infinity. But, the definition of n to “reach infinity” was unclear. Weierstrass’s epsilon method provided a solution: that n no longer had to reach infinity, rather the limit was defined for n being in the process of reaching infinity. Weierstrass defined an infinite sequence as having a limit if any epsilon, , then it would be able to find an integer n so that all integers m  n, with mth term of the sequence was always within  of the limit.11 (p891) The epsilon method also had an important impact on the theory of functions. Mathematicians had been primarily concerned with functions, like the sine wave, that can be at least partially drawn, without lifting a pencil from the paper. This was a sign on continuity within geometry. Therefore, the epsilon method freed mathematicians from thinking about continuity in such a manner. This is because Weierstrass defined continuity of a function f at a point xo : If for every , a  be found so that for all values of x such that:

xo -  < x < xo -  then,

11 |f(x) – f(xo)| <  (p891)

The definition above fit the idea that if you draw a continuous function near a point, then you are getting closer to the value of the function at that point. However, this was not the only thing this proved. Gustav Dirichlet had defined two very unusual functions. The first X (x) is called the characteristic function of the rationals in the reals. The values are:

X (x) = 1 if x if rational = 0 if x if irrational11 (p 891)

This second function, called the Dirichlet function, D(x), is even more unusual. It takes the following values:

D(x) = 1/b if x is rational and can be expressed in lowest terms as the fraction a/b. = 0 if x is irrational11 (p 891)

These two functions cannot be “drawn” in the same manner as a sine curve can be. The first one is discontinuous at every point. Within two years of formulating the epsilon method, Weierstrass proved the Dirichlet function was continuous at any irrational value of x. For example, suppose you -want to use the epsilon method for x= 2. with  = 0.1, or that  = 1/10. We know that

6 D(2) = 0. So, what is the value of , so that D(x) for all x within  of 2 that is not greater than 0.1, the choice for , not including 0, the value of D(x) for x = 2?11 (p 891) This example makes sense when once it’s realized that the only rational numbers for which D (x) is greater than 0.1 (the value of ) must have a denominator no greater than 10.11 (p 891) Because of the great work of Weierstrass, especially in the Dirichlet function, he was well respected by Henri Poincare, who “had complained of the surge of monstrous mathematical objects such as the Dirichlet function.”11 (p891) Poincare considered Weierstrass to be Germany’s third greatest mathematician of the nineteenth century, followed by Gauss and Reimann.11 (p 891) Cauchy based his calculus upon limits, which he states, “When the values successively attributed to a variable approach indefinitely to a fixed value, in a manner so as to end by differing from it by as little as one wishes, this last is called the limit of all others.”7 (p 129) Based upon these words by Cauchy, Weierstrass polished them and came up with the definition, as stated in the previous example:

lim f(x) = L if and only if, for every  > 0, there exists a  > 0 xa

so that, if 0 < |x-a| < , then |f(x) – L| < 

This statement shows that Weierstrass grasped an understanding that those previous to him, did not. This is an example of uniform continuity, an important property that Cauchy did not recognize. Cauchy defined continuity as a point-by-point basis, saying that: f is continuous at a if lim f(x)= f(a). xa

To Weierstrass, this means that for every  > 0, there corresponds a  > 0 so that, if 0 < |x – a| < , then |f(x) – f(a)| < .7 (p 130) For all students of Calculus, this “definition” is a basic one, learned early in the course, as the definition of a limit. This is commonly referred to as the “epsilon-delta definition of a two-sided limit.” 1 (p 135) This “breakthrough” makes the transition from an informal concept of a limit to a precise definition.1 (p 135) In addition to this, Weierstrass also was well known for proving important theorems. Here are four theorems that involve uniform convergence. 1. Theorem 1- If {fk} is a sequence of continuous functions converging uniformly to f on [a,b], then f itself is continuous.7 (p. 138) 2. Theorem 2- if {fk} is a sequence of bounded, Riemann-integrable functions converging uniformily to f on [a,b], then f is Reimann-integrable on [a,b] and

b b b 7 lim [a fk (x)dx] = a [lim fk (x)]dx = a fk (x)dx (p 138) k k

7 From this theorem, it shows the interchange of limits and integrals is allowed for uniform converging sequences of functions.7 (p 138) 3. Theorem 3- (Weierstrass approximation theorem) If f is a continuous function defined on a closed, bounded interval [a,b], then there exists a 7 Sequence of polynomials Pk converging uniformly to f on [a,b]. (p 138) These three theorems show uniform convergence. These allow for “the transfer and integrability from individual functions to their limit and provide a vehicle for approximating continuous functions by polynomials.” 7(p138) However, the fourth theorem, shows an easier way to establish uniform convergence initially. 4. Theorem 4- (Weierstrass M- test) If a sequence {fk} of functions defined on a common domain has the property that, for each k, there

exists a positive number Mk so that |fk(x)|  Mk for all x in the domain and if the infinite series  converges, then the series of functions

 Mk k=1

 7  fh (x) converges uniformly. (p 139) k=1

This leads to a comparison test between functions and numbers, where convergence of the series of functions.7 (p 139)

Let’s look at another example:

F(x) =   x k = x + x 2 + x 4 + …. 3 3 3 4 k=1 (k+1) 2 3 4 so, we have

k 7 |fk (x)| = | x |  __1___  __1___ for all x in [0,1] (p 139) |(k +1)3| (k +1)3 k2

Uniform convergence follow immediately from the M-test. To be able to conclude the f is continuous and to evaluate its integral exactly is a pretty significant accomplishment that was made possible by the M-test.7 (p 139) Weierstrass additionally made an important contribution to the theory of functions. In 1885, Weierstrass published his “approximation theorem” which “stated that any continuous function can be approximated uniformly over a finite range of values by a sequence of polynomials of powers, or of sines and cosines.” 10 (p 498) This related a valuable link between “the major category of general functions and two of the principle ways of handling functions numerically.”10 (p 498). The information about Weierstrass comes from varied sources. While the Dictionary of Scientific Biography lists much detail and gives a lot of information about Weierstrass’ life and his great mathematical accomplishments, there is also information in general encyclopedias, such as the Encyclopedia Britannica. While the general

8 encyclopedia doesn’t give any different information than the Dictionary of Scientific Biography, the Dictionary of Scientific Biography does give a much more detail account of Weierstrass’s life and detail accounts of his many accomplishments. In determining which is more of a reliable source, it is necessary to look at the information that is given in both sources and examine the information to determine any discrepancies. However, there are few, if any discrepancies between these two sources. The only difference is the amount of information and detail given in each source. The audience of the two different sources is quite different, which may account for the level of explanation. For more general information, it would be more likely to use the Encyclopedia Britannica, and for those that are more interested in math or science, it is better to consult the Dictionary of Scientific Biography, as the level of difficulty in the explanation of the discoveries is directed to more of an audience with a background in math and/or science. The same goes for the websites. Much of the information on the web is “incomplete.” They often just give a more general description and don’t give too much information or go too far in depth about some of Weierstrass great mathematical accomplishments. Again, the audience on the web may not be looking for a very detailed description and would find the more generalized explanation to be more beneficial. The contributions that Karl Weierstrass has made to the mathematics had a tremendous impact in the mathematics world that still affects mathematics today. His theories are still taught in calculus classes and the foundings that he made were great advancements during his time period. Although his life didn’t start out as would be expected, as he had plans to attend school to go into public finance and administration, but soon learned that his hidden talents and passions lie in the field of mathematics. He didn’t even get a degree after attending a university! Weierstrass spend fifteen years teaching to disinterested students and it took quite a while for him to submit a journal article that finally got him the attention that he so well deserved. Although his communication style was difficult to understand, his great achievement and ideas attracted students from all around the world. He, himself, had little that he published and allowed his ideas to be disseminated by his students. One that is well renowned is Sonya Kovalevsky, who, not only was the first woman to receive a doctorate, but who made some great accomplishments on her own. His great work is still making an impression on mathematics today.

1. Anton, H., Bivens, I. Davis, S. Calculus, Eighth Edition, Wiley, New Jersey, 2005. 2. Berlinski, D. A Tour of the Calculus, Pantheon Books, New York, 1995. 3. Biographic Dictionary of Mathematicians, Vol. 4, 1999, pp 2549-2555. 4. Bottazzi, U., The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Springer-Verlag, New York, 1986. 5. Dauben, J. The History of Mathematics from Antiquity to the Present A Selective Biography, Garland Publishing, 1985. 6. Dictionary of Scientific Biography, Vol. XIV, 1976, pp 219-224. 7. Dunham, W., The Calculus Gallery: Masterpieces from Newton to Lebesgue. Princeton University Press, New Jersey, 2005. 8. Encyclopedia Americana, 2006, pp 574-575. 9. Encyclopedia Britannica, 2005, p 559.

9 10. Grattan-Guinness, I., The Norton History of the Mathematical Sciences. W.W. Norton & Company, New York, 1997. 11. Hawking, S., God Created the Integrals. Running Press, Pennsylvania, 2005. 12. Katz, V., A History of Mathematics. Addison Wesley Longman, Massachusetts. 1998. 13. http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Weierstrass.html, visited site on March 21, 2007. 14. http://en.wikipedia.org/wiki/Karl_Weierstrass , visited site on March 24, 2007. 15. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html, visited site on March 21, 2007. 16. http://pirate.shu.edu/~wachsmut/ira/history/weierstr.html, visited site on March 25, 2007. 17. http://scienceworld.wolfram.com/biography/Weierstrass.html, visited site on March 25, 2007. 18. http://scidiv.bcc.ctc.edu/Math/Weierstrass.html, visited site on March 25, 2007.