Karl Weierstrass

How to use this course

Karl Weierstrass is the reason a first course in analysis looks the way it does. The definition of continuity with \(\varepsilon\) and \(\delta\), the notion of uniform convergence, the theorem that a bounded sequence has a convergent subsequence, and the construction of the real numbers all reached the world through his Berlin lectures. In complex analysis he built the whole theory on power series, and his \(\wp\)-function is still the standard way to write elliptic functions.

His life is unusual. He left university without a degree, spent fourteen years teaching in small Prussian secondary schools, and published his first major paper at 38. Two years later he was in Berlin, and within a decade his lecture hall was the centre of European analysis.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Weierstrass published little; most of his results became known through lecture notes taken by his students. The dates given for results are those of first publication or of the lecture course where they are documented.

Timeline

Westphalia, Paderborn, Bonn, 1815–1838

Karl Theodor Wilhelm Weierstrass was born on 31 October 1815 in Ostenfelde, a village in Westphalia now part of the town of Ennigerloh. The region had just passed to Prussia after the Napoleonic Wars. His father Wilhelm was secretary to the local mayor and later a customs and tax official; the family was Catholic, and the father was a strict, educated man with interests in the humanities. Karl was the eldest of four children. His mother Theodora died in 1827 and his father remarried the following year. None of the four children ever married.

The family moved with the father’s postings. In 1829 Karl entered the Catholic Gymnasium Theodorianum in Paderborn, where he won prizes regularly. He also worked part-time as a bookkeeper to help the family finances.

The wrong degree

His father wanted a career in the Prussian civil service. In 1834 Weierstrass enrolled at the University of Bonn in law, finance and economics. He found this work dull and the mathematics lectures at Bonn little better, and he spent much of his energy on student life. Alone, he worked through Pierre-Simon Laplace’s celestial mechanics and Carl Gustav Jacob Jacobi’s work on elliptic functions. He left Bonn in 1838 after four years without taking any examination, to his family’s dismay.

The way forward was to train as a secondary school teacher at the Academy in Münster, a shorter route that required no university degree.

Münster and Gudermann, 1839–1842

Weierstrass enrolled at the Münster Academy on 22 May 1839. The decisive figure there was Christoph Gudermann, who was lecturing on elliptic functions and had developed them through power series expansions. Gudermann had also used, in an 1838 paper, the idea that a series converges “with the same speed” throughout a region, the germ of uniform convergence.

In 1840 Weierstrass sat the state teacher’s examination. One of the written topics was the representation of elliptic functions by power series. He answered with original research. Gudermann’s report placed the candidate on a level with the celebrated discoverers of the subject, a rare tribute for a teaching certificate. The oral examinations followed in 1841.

He spent 1841–42 as a probationary teacher at the Gymnasium in Münster. In these years he wrote several mathematical papers that he did not publish: on the representation of analytic functions by power series, on functions defined by differential equations, and on series in an annulus. They contain uniform convergence used as a working tool, and a Laurent expansion two years before Pierre Laurent. They appeared only in his collected works in 1894.

Fourteen years in the schoolroom, 1842–1856

After his probation he took a post at the Pro-Gymnasium in Deutsch Krone, a small town in West Prussia (today Wałcz in Poland). In 1848 he moved to the Lyceum Hosianum in Braunsberg in East Prussia (today Braniewo). Besides mathematics and physics he taught botany and gymnastics, among other subjects. He had no access to a research library and almost no contact with other mathematicians. From about 1850 he suffered attacks of dizziness that recurred for over a decade.

He worked at night on the problem that had defeated many: the inversion of hyperelliptic integrals, the natural generalization of the elliptic case that Niels Henrik Abel and Jacobi had opened. His first results appeared in school prospectuses, where no mathematician would see them.

The 1854 paper

In 1854 he published Zur Theorie der Abelschen Functionen in Crelle’s Journal. It solved the Jacobi inversion problem for hyperelliptic integrals of arbitrary genus. The effect was immediate. On 31 March 1854 the University of Königsberg gave him an honorary doctorate. When Kummer left Breslau for Berlin in 1855, Weierstrass applied for the Breslau chair and was passed over. Peter Gustav Lejeune Dirichlet wrote to the minister in his support.

The fuller version, Theorie der Abelschen Functionen, appeared in 1856. On 14 June 1856 he accepted a chair at the Gewerbeinstitut in Berlin, the industrial institute that later became the Technische Hochschule. In October 1856 he became an extraordinary professor at the University of Berlin, and in November a full member of the Berlin Academy. He was 41.

Epsilon, delta and the real line

For the general reader. Calculus worked for two centuries without anyone being able to say exactly what “approaching a limit” meant. Weierstrass replaced talk of motion and infinitely small quantities with inequalities between ordinary numbers. Every precise statement about limits that students now learn has his form.

Static definitions

Augustin-Louis Cauchy in the 1820s defined continuity by saying that an infinitely small increment of the variable produces an infinitely small increment of the function. Bernard Bolzano had a similar definition in 1817. Weierstrass, in his lectures from the late 1850s, removed the dynamic language entirely.

The order of the quantifiers carries the content. Swapping “for every \(x\)” and “there is \(\delta\)” changes continuity into uniform continuity, a distinction made explicit in the Berlin circle and published by Eduard Heine in 1872.

Completeness

Such definitions say what a limit is, but they cannot prove that a limit exists unless the real numbers have no gaps. From the winter semester of 1863–64 Weierstrass gave a construction of the real numbers in his introductory course, building them from rational numbers through convergent sums of “aggregates”. Richard Dedekind (cuts, 1872) and Cantor (Cauchy sequences, 1872) published other constructions. All three give the same complete ordered field.

Proof sketch. Let \(x_n\in[a,b]\). Halve the interval; one half contains \(x_n\) for infinitely many \(n\). Repeat, obtaining nested intervals \(I_k\) of length \((b-a)/2^k\), each containing infinitely many terms. Choose \(n_1<n_2<\cdots\) with \(x_{n_k}\in I_k\). The intervals shrink to a single point \(L\) by completeness, and \(|x_{n_k}-L|\le(b-a)/2^k\). Bolzano used the bisection idea in 1817; Weierstrass made it the central tool of his lectures.

If \(f\) were unbounded, pick \(x_n\) with \(|f(x_n)|>n\); a subsequence converges to some \(c\in[a,b]\) and continuity at \(c\) gives a contradiction. Let \(M=\sup f\) and pick \(f(x_n)\to M\); a subsequence converges to \(c\), and \(f(c)=M\). Weierstrass’s insistence that a supremum need not be a maximum without proof became decisive in Lesson 10.

Uniform convergence

For the general reader. Add up infinitely many smooth curves and you may get a curve with a jump. Cauchy believed in 1821 that a convergent series of continuous functions is always continuous, and Fourier series showed he was wrong. The repair is to ask that the series converge at the same rate everywhere. Weierstrass made this “uniform” convergence the workhorse of analysis.

Proof. Given \(\varepsilon\) and \(a\), pick \(n\) with \(\sup|f_n-f|<\varepsilon/3\), then \(\delta\) with \(|f_n(x)-f_n(a)|<\varepsilon/3\) for \(|x-a|<\delta\). The triangle inequality gives \(|f(x)-f(a)|<\varepsilon\). This “\(\varepsilon/3\) argument” is Weierstrass’s signature. For integrals, \(\big|\int(f_n-f)\big|\le(b-a)\sup|f_n-f|\).

Gudermann had used the idea in 1838, Weierstrass in his unpublished 1841 work on power series, and Philipp Ludwig von Seidel and George Gabriel Stokes independently in 1847–48. It was through Weierstrass’s teaching that the concept spread.

The partial sums are uniformly Cauchy: \(\big|\sum_{n=m}^{k}u_n(x)\big|\le\sum_{n=m}^{k}M_n\), which is small independently of \(x\).

A continuous, nowhere differentiable function

For the general reader. Most nineteenth-century mathematicians assumed that a continuous curve must be smooth except at isolated corners. In 1872 Weierstrass showed the Berlin Academy a curve that is continuous everywhere and has a tangent nowhere. It ended reliance on geometric intuition in analysis, and its jagged self-similar graph is an ancestor of fractals.

Weierstrass presented the example to the Berlin Academy on 18 July 1872. It was first printed by Paul du Bois-Reymond in 1875. Bolzano had constructed such a function in the 1830s, but his manuscript remained unpublished until the twentieth century. Bernhard Riemann was reported to have proposed \(\sum\sin(n^2x)/n^2\) as an example; much later it was shown to be differentiable at certain rational multiples of \(\pi\). In 1916 G. H. Hardy proved that \(W\) is nowhere differentiable whenever \(0<a<1\) and \(ab\ge1\), with \(b>1\) real.

Proof sketch

Continuity follows from the M-test with \(M_n=a^n\). For non-differentiability, fix \(x_0\) and \(m\ge1\). Write \(b^mx_0=\alpha_m+\xi_m\) with \(\alpha_m\) an integer and \(-\tfrac12\le\xi_m<\tfrac12\). Set \(x_m=(\alpha_m-1)/b^m\) and \(x_m'=(\alpha_m+1)/b^m\), points on either side of \(x_0\) at distance at most \(\tfrac32 b^{-m}\). Split each difference quotient into terms \(n<m\) and \(n\ge m\).

• For \(n<m\), the mean value theorem bounds each term by \(\pi a^nb^n\), so the low terms together are at most \(\pi\,(ab)^m/(ab-1)\).
• For \(n\ge m\), since \(b\) is odd, \(\cos(b^n\pi x_m)=\cos(b^{n-m}\pi(\alpha_m-1))=-(-1)^{\alpha_m}\), and similarly at \(x_m'\). All these terms have the same sign, fixed by \((-1)^{\alpha_m}\), and the \(n=m\) term alone contributes at least \(\tfrac23(ab)^m\) in absolute value. The sign is opposite for the two sides.

When \(\tfrac23>\pi/(ab-1)\), which is exactly \(ab>1+\tfrac{3\pi}2\), the high terms dominate. The two difference quotients then have opposite signs and absolute value at least a constant times \((ab)^m\to\infty\). No finite derivative can exist.

The approximation theorem

For the general reader. Monster functions like the one in Lesson 06 might suggest that continuous functions are hopelessly wild. In 1885, at the age of 69, Weierstrass proved the reassuring counterpart: any continuous function on a closed interval can be imitated as closely as you like by a polynomial. The theorem is a foundation of numerical analysis and of modern functional analysis.

The paper, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, appeared in the Berlin Academy’s proceedings in 1885.

Weierstrass’s proof

Extend \(f\) to a bounded, uniformly continuous function on \(\mathbb R\). For \(k>0\) define the Gaussian smoothing

The kernel has integral 1 and concentrates near \(u=x\) as \(k\to0\). Splitting the integral into \(|u-x|<\delta\), where uniform continuity makes \(|f(u)-f(x)|\) small, and the tails, where the Gaussian is tiny, gives \(F_k\to f\) uniformly. For fixed \(k\), \(F_k\) is an entire function of \(x\); its Taylor series converges uniformly on \([a,b]\), so a partial sum approximates \(F_k\) within \(\varepsilon/2\). In modern terms \(F_k\) is the solution of the heat equation at time \(k^2/4\) with initial data \(f\).

Later proofs and generalizations

Sergei Bernstein gave a constructive probabilistic proof in 1912 using

Marshall Stone generalized the result in 1937 and 1948: a subalgebra of \(C(K)\), for \(K\) compact Hausdorff, that contains the constants and separates points is dense in the uniform norm.

Analytic functions from power series

For the general reader. Cauchy built complex analysis on integrals, Riemann on geometry and surfaces. Weierstrass built it on power series alone: a function is a collection of power series, each one continuing the next. His approach is algebraic and needs no geometric intuition, and it gave the theory of entire functions its central theorem.

Function elements

A function element is a pair \((P,D)\) of a power series \(P(z)=\sum c_n(z-z_0)^n\) and its disc of convergence \(D\). Re-expanding \(P\) about a point \(z_1\in D\) gives a new element whose disc may extend outside \(D\). The analytic function in Weierstrass’s sense is the totality of elements reachable by chains of such re-expansions. This construction is the origin of the Riemann surface of a function as a space of germs, and it explains why \(\sqrt z\) and \(\log z\) are “many-valued”.

Entire functions with given zeros

A polynomial is determined up to a constant by its zeros. Weierstrass asked whether every sequence \(a_n\to\infty\) is the zero set of an entire function. Naive products \(\prod(1-z/a_n)\) need not converge. His solution uses the elementary factors

which satisfy \(|1-E_p(u)|\le|u|^{p+1}\) for \(|u|\le1\).

For fixed \(|z|\le R\), eventually \(|z/a_n|\le\tfrac12\), so \(|1-E_{n-1}(z/a_n)|\le2^{-n}\) and the product converges by the M-test applied to logarithms. A corollary is that every meromorphic function on \(\mathbb C\) is a quotient of two entire functions.

Felice Casorati and Yulian Sokhotski published it in 1868; Weierstrass included it in his 1876 paper. Picard’s great theorem strengthens it: \(f(U)\) omits at most one value.

It reduces the local study of analytic sets in several variables to polynomials in one variable, and it is a basic tool of several complex variables and singularity theory.

Elliptic and Abelian functions

For the general reader. The arc length of an ellipse cannot be written with ordinary functions. Inverting such integrals produces functions with two independent periods, the elliptic functions. Weierstrass gave them their most economical form, the \(\wp\)-function, and spent decades on their higher analogues, the Abelian functions, which brought him fame in 1854.

The \(\wp\)-function

Let \(\Lambda=\mathbb Z\omega_1+\mathbb Z\omega_2\) be a lattice in \(\mathbb C\) (with \(\omega_1/\omega_2\notin\mathbb R\)). Write \(\sum'\) for a sum over \(\omega\in\Lambda\setminus\{0\}\).

Derivation. Near \(0\), \(\wp(z)=z^{-2}+3G_4z^2+5G_6z^4+O(z^6)\). Then \(\wp'^2=4z^{-6}-24G_4z^{-2}-80G_6+O(z^2)\) and \(4\wp^3=4z^{-6}+36G_4z^{-2}+60G_6+O(z^2)\). The difference \(\wp'^2-4\wp^3+60G_4\wp+140G_6\) is elliptic, has no poles, and vanishes at \(0\), so it is identically zero by Liouville’s theorem.

The map \(z\mapsto(\wp(z),\wp'(z))\) identifies the torus \(\mathbb C/\Lambda\) with the cubic curve \(y^2=4x^3-g_2x-g_3\). Addition on the torus becomes the chord-and-tangent law on the curve, the starting point of the arithmetic of elliptic curves. Weierstrass also introduced the entire function \(\sigma(z)=z\prod'\big(1-\frac z\omega\big)\exp\big(\frac z\omega+\frac{z^2}{2\omega^2}\big)\), a product of the kind in Lesson 08, with \(\wp=-(\log\sigma)''\).

The Jacobi inversion problem

For a hyperelliptic curve \(y^2=f(x)\) with \(\deg f=2p+1\) or \(2p+2\), there are \(p\) independent holomorphic differentials \(x^{j-1}dx/y\), \(j=1,\dots,p\). A single integral cannot be inverted, because it has too many periods. Jacobi proposed instead to consider the system

and to show that symmetric functions of \(x_1,\dots,x_p\) are single-valued \(2p\)-fold periodic functions of \((u_1,\dots,u_p)\). Weierstrass solved this for every \(p\) in 1854–56, with explicit theta-type functions. Riemann published his solution for arbitrary algebraic curves in 1857, by quite different methods. Weierstrass withdrew a memoir on general Abelian integrals he had submitted to the Berlin Academy, and he spent years rebuilding the theory on his own foundations.

The calculus of variations

For the general reader. Many problems ask for the curve that makes some quantity smallest. Euler and Lagrange found the equation such a curve must satisfy, but solving that equation does not prove that a minimum exists or that you have found it. Weierstrass gave the first sufficient conditions, and a famous counterexample showing that a minimum may simply fail to exist.

Consider \(J[y]=\int_{x_0}^{x_1}F(x,y,y')\,dx\) over curves with fixed endpoints. A curve is a strong minimum if \(J\) is smallest among nearby curves whose slopes may differ greatly, and a weak minimum if the slopes must also be close.

\(\mathcal E\) measures how far \(F\) lies above its tangent line in the slope variable. The proof inserts a short segment of slope \(q\) into the curve and compares. Weierstrass combined this with a field of extremals around \(y\), and showed that a strictly positive excess function on such a field gives a sufficient condition for a strong minimum. He presented this in his lectures of 1879.

The Dirichlet principle

Riemann had used the “Dirichlet principle”: the energy \(\int|\nabla u|^2\) attains its minimum among functions with given boundary values, and the minimizer is harmonic. In 1870 Weierstrass showed that a positive functional need not attain its infimum. The criticism stalled the principle until David Hilbert rescued it in 1900 with what became the direct method.

The Berlin school, 1856–1890

Weierstrass, Kummer and Kronecker made Berlin the leading mathematical centre of Germany for a generation. In 1861 Weierstrass and Kummer founded a seminar for pure mathematics, the first of its kind in Germany, where students presented their own work. In December 1861 Weierstrass collapsed from overwork and needed about a year to recover. Afterwards he lectured seated while a student wrote on the board. The dizzy spells ceased, but chest problems followed.

The lecture cycle

His lectures ran in a cycle over several semesters: introduction to analytic functions, elliptic functions, Abelian functions, and the calculus of variations or applications of elliptic functions. He revised them each time and rarely published. Students’ notes circulated widely and spread his methods. His audiences were large and international. Among those who attended, and later worked in their own directions, were Cantor, Felix Klein, Sophus Lie, Adolf Hurwitz, Hermann Minkowski and Gösta Mittag-Leffler.

His doctoral students included Hermann Amandus Schwarz, Lazarus Fuchs, Leo Königsberger, Georg Cantor, Georg Frobenius, Wilhelm Killing, Friedrich Schottky, Ludwig Stickelberger, Arthur Schoenflies, Hans von Mangoldt, Carl Runge and Eugen Netto. Edmund Husserl attended his lectures and later turned to philosophy.

Sofia Kovalevskaya

In 1870 the Russian mathematician Sofia Kovalevskaya came to Berlin. Women could not attend the university, and Weierstrass taught her privately for four years. He considered her his most gifted student. In 1874 Göttingen awarded her a doctorate on the strength of three papers, including the Cauchy–Kovalevskaya theorem on partial differential equations, without an oral examination. Weierstrass later helped secure her a post in Stockholm, through Mittag-Leffler. Their correspondence lasted until her death in 1891. He burned her letters to him; about 150 of his letters to her survive.

The break with Kronecker

Kronecker had been a close friend. From the late 1870s he attacked the use of irrational numbers and infinite processes in analysis and opposed Cantor’s set theory, which Weierstrass’s methods supported. Weierstrass took the criticism personally. In 1885 he considered leaving Berlin for Switzerland, but stayed.

Last years and legacy

Weierstrass served as rector of the University of Berlin in 1873–74. He received the order Pour le Mérite in 1875, became a Foreign Member of the Royal Society in 1881, and received its Copley Medal in 1895. His two unmarried sisters kept house for him in Berlin. He stopped lecturing regularly around 1890 and spent his last years preparing his collected works. Volumes 1 and 2 appeared in 1894 and 1895; later volumes, edited from lecture notes, appeared long after his death.

For his last three years he could not walk and used a wheelchair. He died of pneumonia in Berlin on 19 February 1897, aged 81, and was buried in the Catholic St. Hedwig cemetery.

Legacy

• Rigour: the \(\varepsilon\)–\(\delta\) style, uniform convergence and the arithmetic construction of the reals define what a proof in analysis is. Later writers called this programme the arithmetization of analysis.
• Counterexamples: the nowhere differentiable function and the failure of the Dirichlet principle taught mathematicians to distrust intuition and to prove existence.
• Complex analysis: analytic continuation, the product theorem and the preparation theorem remain basic tools. The \(\wp\)-function is the standard model of elliptic curves.
• Approximation: the 1885 theorem led to Bernstein polynomials, the Stone–Weierstrass theorem and much of approximation theory.
• Honours: the Weierstrass Institute for Applied Analysis and Stochastics in Berlin, a lunar crater and asteroid 14100 Weierstrass bear his name.

Connections

People linked to Karl Weierstrass across this collection. A name without a link has no page here yet. See Karl Weierstrass in the network of all 337 lives →

Teachers and advisors

• Christoph Gudermann 1839-1841

Students

• Georg Cantor 1867
• Georg Frobenius 1870
• Gösta Mittag-Leffler
• Sofia Kovalevskaya 1870-1874
• Arthur Schoenflies 1877
• Carl Runge 1880
• Edmund Husserl
• Eugen Netto 1870
• Friedrich Schottky 1875
• Hans von Mangoldt 1878
• Hermann Amandus Schwarz 1864
• Lazarus Fuchs 1858
• Leo Königsberger 1860
• Ludwig Stickelberger 1874
• Wilhelm Killing 1872

Influenced by

• Augustin-Louis Cauchy
• Bernard Bolzano
• Carl Gustav Jacob Jacobi
• Niels Henrik Abel
• Pierre-Simon Laplace

Influenced

• Adolf Hurwitz 1877-1882
• David Hilbert
• G. H. Hardy
• Gösta Mittag-Leffler
• Hermann Minkowski
• Josiah Willard Gibbs 1867-1868
• Teiji Takagi

Colleagues and collaborators

• Bernhard Riemann
• Camille Jordan
• Charles Hermite
• Ernst Kummer 1856-1890
• Henri Poincaré 1885-1889
• Jakob Steiner
• Leopold Kronecker 1856-1891
• Peter Gustav Lejeune Dirichlet

Rivals

• Bernhard Riemann

Also mentioned in this life

• Richard Dedekind
• Leonhard Euler
• Joseph Fourier
• Felix Hausdorff
• Felix Klein
• Joseph-Louis Lagrange
• Sophus Lie
• Joseph Liouville
• George Gabriel Stokes

Fields

• real analysis 10 lives
• complex analysis 13 lives
• calculus of variations 9 lives

On CoursesHub

• Complex Analysis · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Karl Theodor Wilhelm Weierstrass, MacTutor History of Mathematics.
2. Karl Weierstrass, Wikipedia; Karl Weierstraß, German Wikipedia.
3. Weierstrass Institute, Karl Weierstrass: curriculum vitae.
4. Karl Weierstraß, Mathematics Genealogy Project.
5. K.-R. Biermann, Die Mathematik und ihre Dozenten an der Berliner Universität 1810–1933, Akademie-Verlag, 1988.
6. R. Bölling (ed.), Briefwechsel zwischen Karl Weierstraß und Sofja Kowalewskaja, Akademie Verlag, 1993.

Mathematics

1. K. Weierstrass, Mathematische Werke, 7 vols., Mayer & Müller, Berlin, 1894–1927.
2. K. Weierstrass, Zur Theorie der Abelschen Functionen, J. reine angew. Math. 47 (1854).
3. K. Weierstrass, Zur Theorie der eindeutigen analytischen Functionen, Abh. Akad. Wiss. Berlin (1876).
4. K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Sitzungsber. Akad. Wiss. Berlin (1885).
5. G. H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), 301–325.
6. U. Bottazzini, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer, 1986.
7. U. Bottazzini and J. Gray, Hidden Harmony, Geometric Fantasies: The Rise of Complex Function Theory, Springer, 2013.
8. A. Pinkus, Weierstrass and approximation theory, J. Approx. Theory 107 (2000), 1–66.