Thomas Joannes Stieltjes

How to use this course

Most students meet Stieltjes through one integral, \(\int f\,dg\), and never learn why he needed it. He needed it to solve a problem about continued fractions: which sequences of numbers are the moments of a mass distribution, and when do the moments fix the distribution? That question, posed and answered in his last memoir, links orthogonal polynomials, Gaussian quadrature, complex analysis, probability and, later, the spectral theory of operators. This course follows the route he took, alongside the story of a career built without a university degree.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 to 06 cover the work of the 1880s; Lessons 07 to 10 unpack the 1894 memoir Recherches sur les fractions continues. Notation is modern. Stieltjes wrote his measures as non-decreasing functions \(\varphi(u)\), and we follow him where it helps.

Timeline

Zwolle and Delft, 1856–1877

Thomas Joannes Stieltjes was born in Zwolle, in the eastern Dutch province of Overijssel, on 29 December 1856. He had the same name as his father, a distinguished civil engineer who held a Leiden doctorate, built harbour works around Rotterdam and sat in the Dutch parliament. The family had seven children; Thomas had two brothers and four sisters. Both Thomas Joannes and Thomas Jan appear in the literature.

In 1873 he entered the Polytechnic School at Delft, the obvious path for an engineer’s son. He did not follow it. Instead of attending lectures he read Gauss and Jacobi in the library, and he failed the propaedeutic examination. He failed it again in 1875 and in 1876. He never earned a university degree of any kind, a fact that would block his career in the Netherlands for years.

The reading was not wasted. Gauss’s numerical integration, Jacobi’s elliptic functions and the continued fractions that both men used run through everything Stieltjes later did. His first paper appeared in 1876, while he was still failing examinations.

The observatory and the letters to Hermite, 1877–1884

His father was a friend of H. G. van de Sande Bakhuyzen, director of the Leiden Observatory, and through him the young man found work. In April 1877 Stieltjes became an assistant for astronomical calculations at the observatory. His father died in Rotterdam on 23 June 1878, and the job became a necessity.

Astronomy gave him daily practice in numerical analysis: interpolation, quadrature, the reduction of observations. It also gave him a reason to write to Paris. On 8 November 1882 he sent his first letter to Charles Hermite, about a question in celestial mechanics. Hermite replied with interest, and the letters soon turned to pure mathematics. The correspondence lasted until Stieltjes’s death and runs to 432 letters. Hermite, one of the most generous mathematicians of his century, became his mentor, sponsor and friend.

Stieltjes married Elizabeth Intveld in May 1883. She encouraged him to leave astronomy for mathematics. From 1 January 1883 he was released from observing, and on 1 December 1883 he resigned from the observatory. That autumn he substituted at Delft for F. J. van den Berg, teaching analytical and descriptive geometry, the first time he had taught at a university.

Groningen, Leiden, Paris, 1884–1886

In 1883 a chair of analysis fell vacant at the University of Groningen. The faculty first ranked Diederik Korteweg ahead of Stieltjes; Korteweg declined, and the faculty then placed Stieltjes first, ahead of Floris de Boer. The ministry overruled them. A Royal Decree of 12 March 1884 appointed de Boer, and the reason given was the one Stieltjes could not answer: he had no degree.

His supporters moved quickly. On the proposal of David Bierens de Haan and van de Sande Bakhuyzen, with Hermite’s encouragement, the Senate of Leiden University considered an honorary doctorate in mathematics and astronomy on 27 May 1884, and the degree was conferred in June. Stieltjes missed the public ceremony on 17 June through a misunderstanding and wrote to thank the Senate two days later.

The honorary degree did not open a Dutch chair. In April 1885 Stieltjes moved with his family to Paris, closer to Hermite and to the most active school of analysis in Europe. The same year he was elected to the Royal Academy of Sciences in Amsterdam. In 1886 he took a French doctorate of science with the thesis Recherches sur quelques séries semi-convergentes, published in the Annales scientifiques de l’École normale supérieure. Hermite was his advisor; some sources also name Gaston Darboux. The degree qualified him for a French university post, and that year he was appointed to the Faculty of Sciences at Toulouse.

Semi-convergent series

For the general reader. Some infinite series diverge, yet their first few terms give excellent approximations. Astronomers had used such series for a century. In 1886 Stieltjes, in his thesis, and Henri Poincaré, in a paper in Acta Mathematica, independently made the idea precise. The modern theory of asymptotic expansions starts here.

Definition

The coefficients are unique when they exist, since \(a_N=\lim x^N\big(f(x)-\sum_{n<N}a_nx^{-n}\big)\). The series need not converge for any \(x\). Stieltjes, following older usage, called such series semi-convergent.

The model example

For \(x>0\) let \(F(x)=\displaystyle\int_0^\infty\frac{e^{-t}}{x+t}\,dt\). Expanding \(1/(x+t)\) with an exact remainder,

and integrating against \(e^{-t}\) using \(\int_0^\infty t^ne^{-t}dt=n!\), we get

The bound follows from \(1/(x+t)\le1/x\). Since the terms \(n!/x^{n+1}\) decrease while \(n<x\) and then grow, the best accuracy comes from stopping near \(N\approx x\), where Stirling’s formula gives an error of order \(\sqrt{2\pi/x}\,e^{-x}\). Exponentially small errors from a divergent series are the reason asymptotic methods dominate applied analysis.

Functions of the form \(\int_0^\infty d\varphi(t)/(x+t)\) with \(\varphi\) non-decreasing are now called Stieltjes functions. The error behaviour above, alternating and bounded by the first omitted term, holds for every Stieltjes function whose moments exist. In 1889 Stieltjes used integral representations of this kind to analyse the remainder in Stirling’s series for \(\log\Gamma\), including complex arguments. Lesson 07 shows how continued fractions turn the divergent series back into a convergent object.

Zeros as charges

For the general reader. Place a few electric charges on a wire between two fixed charges and let them settle. Stieltjes discovered in 1885 that the resting positions are exactly the roots of the classical polynomials of mathematical physics. The physical picture explains why those roots are real, distinct and interlaced, and it still guides research on random matrices.

Jacobi polynomials

For \(\alpha,\beta>-1\), the Jacobi polynomial \(P_n^{(\alpha,\beta)}\) is the degree-\(n\) polynomial solution of

They are orthogonal for the weight \((1-x)^\alpha(1+x)^\beta\) on \([-1,1]\); Legendre polynomials are the case \(\alpha=\beta=0\).

Proof

The energy tends to \(+\infty\) when points collide or approach \(\pm1\), so a minimum exists, and at a minimum \(\partial E/\partial x_i=0\):

Let \(y(x)=\prod_j(x-x_j)\). Differentiating \(\log y\) twice shows \(\sum_{j\ne i}\frac{1}{x_i-x_j}=\frac{y''(x_i)}{2y'(x_i)}\). Multiply the equilibrium condition by \(2(1-x_i^2)\,y'(x_i)\):

The left side, as a function of \(x\), is a polynomial of degree \(n\) that vanishes at all \(n\) zeros of \(y\), so it equals \(-\lambda\,y\). Comparing leading coefficients gives \(\lambda=n(n-1)+(\alpha+\beta+2)n=n(n+\alpha+\beta+1)\). So \(y\) solves the Jacobi equation and is a multiple of \(P_n^{(\alpha,\beta)}\). Since the polynomial solution is unique up to scale, the critical point is unique, and it is the minimum. As a corollary, the zeros of \(P_n^{(\alpha,\beta)}\) are real, simple and inside \((-1,1)\), and Stieltjes used the extremal property to compute the discriminant of these polynomials.

Lamé equations

The same idea works with several fixed charges. Let \(a_0<a_1<\dots<a_p\) be real, \(\rho_0,\dots,\rho_p>0\), \(A(x)=\prod_k(x-a_k)\), and consider

where \(V\) is a polynomial of degree at most \(p-1\) to be chosen. This generalizes the equation of Gabriel Lamé, studied for polynomial solutions by Eduard Heine.

The zeta function and the Mertens claim

For the general reader. In July 1885 Stieltjes told Hermite he could prove the Riemann hypothesis, the most famous open problem in mathematics. Hermite presented a short note to the Académie. The proof never appeared, and the stronger statement it rested on is now believed to be false. The episode left behind a set of constants that bear his name and a lesson about numerical evidence.

The Stieltjes constants

The zeta function \(\zeta(s)=\sum_{n\ge1}n^{-s}\) has a simple pole at \(s=1\) with residue 1. In his letters to Hermite, Stieltjes described the full Laurent expansion there.

Sketch. For \(\operatorname{Re}s>1\), compare \(\sum_{k\le m}k^{-s}\) with \(\int_1^m x^{-s}dx\) and expand \(k^{-s}=k^{-1}e^{-(s-1)\log k}\) in powers of \(s-1\). The integral contributes \(1/(s-1)\) plus the subtracted terms \((\log m)^{n+1}/(n+1)\); the limits converge, and analytic continuation extends the identity.

Mertens function and the Riemann hypothesis

Let \(\mu\) be the Möbius function: \(\mu(n)=(-1)^k\) if \(n\) is a product of \(k\) distinct primes, and 0 if \(n\) has a square factor. Let \(M(x)=\sum_{n\le x}\mu(n)\). For \(\operatorname{Re}s>1\), summation by parts gives

Stieltjes’s note Sur une fonction uniforme (Comptes Rendus, July 1885) asserted that the series for \(1/\zeta(s)\) converges for \(\operatorname{Re}s>\tfrac12\), which would give the Riemann hypothesis. In letters to Hermite he explained that the partial sums of \(\mu\) compensated each other remarkably well and seemed to stay between \(\pm\sqrt n\). He admitted the key bound was not fully proved, and in 1887, when Gösta Mittag-Leffler asked for details, he conceded that his confidence may have been exaggerated. Franz Mertens conjectured \(|M(x)|<\sqrt x\) for \(x>1\) in 1897. Andrew Odlyzko and Herman te Riele disproved that inequality in 1985. The weaker bound \(M(x)=O(\sqrt x)\) that Stieltjes claimed is still open, though most experts expect it to be false as well.

The analytic theory of continued fractions

For the general reader. A continued fraction is a fraction whose denominator contains another fraction, and so on forever. Before Stieltjes, mathematicians treated them as clever formulas. He studied them as analytic functions of a complex variable and found a simple rule that decides when they converge. He is often called the father of the analytic theory of continued fractions.

Stieltjes fractions

Given positive numbers \(a_1,a_2,\dots\), Stieltjes studied

now called an S-fraction. Its \(n\)-th convergent \(P_n(z)/Q_n(z)\) is a rational function. Stieltjes proved that the zeros of the \(Q_n\) are real, simple and non-positive, and that those of successive even denominators interlace. Expanding formally in powers of \(1/z\), the fraction corresponds to a series

and each convergent agrees with this series to one more term than the last. Conversely, a series with suitable positivity (Lesson 08) has a unique S-fraction with all \(a_k>0\).

Ideas of the proof. Each convergent can be written \(\sum_k w_k/(z+u_k)\) with \(w_k>0\) and \(u_k\ge0\), a Stieltjes function of a discrete measure. Such functions are bounded on compact sets away from \((-\infty,0]\), so the family of convergents is normal. Stieltjes showed directly that the even and odd convergents are monotone for \(z>0\), with a gap controlled by \(\sum a_k\). When the sum diverges the gap closes on the positive axis, and a uniqueness argument for analytic functions (later formalized as the Stieltjes–Vitali theorem) spreads convergence to the whole cut plane. The limit is again a Stieltjes function, which led him to the moment problem and to the integral of Lesson 10.

The moment problem

For the general reader. Imagine a rod with mass spread along it. Its total mass, centre of mass, moment of inertia and higher moments are numbers you can measure. Can you recover the mass distribution from them? Stieltjes posed this question in 1894, named it the problem of moments, and solved it for mass on a half-line. The answer is sometimes no: different distributions can share every moment.

The link with Lesson 07 is formal: \(\int_0^\infty\frac{d\varphi(u)}{z+u}=\sum_k(-1)^k c_k z^{-k-1}\) as an asymptotic series. So solving the moment problem means finding a Stieltjes function with a prescribed asymptotic expansion.

Stieltjes stated the conditions through the \(a_k\), which are ratios of products of Hankel determinants. Positivity of \(\Delta_n\) says the quadratic form \(\sum c_{i+j}\xi_i\xi_j=\int\big(\sum\xi_iu^i\big)^2d\varphi\) is positive; positivity of \(\Delta^{(1)}_n\) says the same with an extra factor \(u\ge0\), which is where the half-line enters. Hans Hamburger (1920) treated mass on the whole line and Felix Hausdorff (1921) mass on \([0,1]\). In 1926 Torsten Carleman gave a convenient sufficient condition: the Stieltjes problem is determinate if \(\sum_n c_n^{-1/(2n)}=\infty\).

Indeterminate examples

In the 1894 memoir Stieltjes gave explicit densities on \((0,\infty)\) with all moments equal to zero, so that adding them changes a distribution without changing a single moment. One is \(u^{-\log u}\sin(2\pi\log u)\); Exercise 8.1 checks it. The log-normal distribution of probability theory is indeterminate for the same reason.

Moment inequalities

For the general reader. If you know only a few moments of a distribution, you cannot know the distribution, but you can bound it. Pafnuty Chebyshev stated the sharp bounds in 1874 without proof. In 1884 Stieltjes and Andrei Markov proved them independently, and a courteous dispute about priority followed. The bounds use the nodes and weights of Gaussian quadrature, which Stieltjes had studied since his student days.

Gauss quadrature

Let \(\mu\) be a positive measure on \(\mathbb R\) with infinite support and finite moments. Its monic orthogonal polynomials \(p_n\) satisfy a three-term recurrence and \(p_n\) has \(n\) simple real zeros \(x_1<\dots<x_n\). There are weights \(w_k>0\), the Christoffel numbers, such that

The rational function \(\sum_k w_k/(z-x_k)\) is the \(n\)-th convergent of the continued fraction of \(\int d\mu(t)/(z-t)\), with denominator \(p_n\). Stieltjes proved in 1884 that Gauss quadrature converges for every continuous function on a bounded interval, a theorem he obtained from the positivity of the weights.

Proof of the upper bound. By Hermite interpolation there is a polynomial \(q\) of degree \(2n-2\) with \(q(x_j)=1\) for \(j\le k\), \(q(x_j)=0\) for \(j>k\), and \(q'(x_j)=0\) for \(j\ne k\). One checks by counting sign changes of \(q'\) that \(q\ge\mathbf 1_{(-\infty,x_k]}\) on \(\mathbb R\). Then

The lower bound follows by the same argument applied to \(1-\mathbf 1_{(-\infty,x_k)}\). In the hero illustration, the straight line is the distribution function of the uniform measure and the staircase jumps by \(w_k\) at each node: the line crosses every riser. As \(n\to\infty\) the gaps \(w_k\) shrink, and when the moment problem is determinate any measure with the same moments has the same distribution function. Markov used exactly this to prove the central limit theorem by the method of moments in 1898.

The Stieltjes integral and transform

For the general reader. To describe mass that may be spread smoothly, concentrated at points, or both, Stieltjes integrated with respect to a non-decreasing function instead of with respect to length. He introduced the tool in passing in his 1894 memoir. Fifteen years later it became the way to describe every continuous linear functional, and through spectral theory it entered quantum mechanics.

Definition

Stieltjes took \(g=\varphi\) non-decreasing, the cumulative mass. If \(\varphi\) is continuously differentiable, \(\int f\,d\varphi=\int f\varphi'\,dx\); if \(\varphi\) jumps by \(m_k\) at \(u_k\), the integral picks up \(\sum_k m_kf(u_k)\). Both kinds of mass are handled at once.

The proof is Niels Henrik Abel’s summation by parts applied to the approximating sums: \(\sum_i f(\xi_i)\big(g(t_i)-g(t_{i-1})\big)\) rearranges into the boundary term minus a Stieltjes sum for \(\int g\,df\) on the partition formed by the \(\xi_i\).

The Stieltjes transform and its inversion

For non-decreasing \(\varphi\) on \([0,\infty)\) with \(\int d\varphi/(1+u)<\infty\), set \(F(z)=\int_0^\infty\frac{d\varphi(u)}{z+u}\), analytic for \(z\notin(-\infty,0]\). The mass is recovered from the jump of \(F\) across the cut.

Proof. \(\operatorname{Im}\dfrac{1}{u-x+i\varepsilon}=-\dfrac{\varepsilon}{(u-x)^2+\varepsilon^2}\), and \(\frac{1}{\pi}\frac{\varepsilon}{(u-x)^2+\varepsilon^2}\) is the Poisson kernel, an approximate identity in \(x\). By Fubini, \(-\frac1\pi\int_a^b\operatorname{Im}F(-x+i\varepsilon)\,dx=\int\Big(\frac1\pi\int_a^b\frac{\varepsilon\,dx}{(u-x)^2+\varepsilon^2}\Big)d\varphi(u)\). The inner integral tends to 1 for \(a<u<b\), to 0 outside \([a,b]\), and to \(\tfrac12\) at the endpoints, which carry no mass.

Afterlife

In 1909 Frigyes Riesz proved that every continuous linear functional on \(C[a,b]\) has the form \(f\mapsto\int_a^b f\,dg\) with \(g\) of bounded variation. David Hilbert’s spectral theory of bounded quadratic forms (1906) was recast with Stieltjes integrals, and the spectral theorem \(A=\int\lambda\,dE(\lambda)\) is a Stieltjes integral with projection-valued \(E\). Henri Lebesgue and Johann Radon (1913) extended the idea to Lebesgue–Stieltjes measures. In probability, every distribution is a Stieltjes measure \(dF\), and \(E[X]=\int x\,dF(x)\). The Stieltjes transform of a spectral measure is the resolvent trace that random-matrix theory uses to derive the Wigner semicircle law.

Toulouse, 1886–1894

Stieltjes arrived at the Faculty of Sciences in Toulouse in 1886 and stayed there for the rest of his life. In 1889 he was given the chair of differential and integral calculus. He took a large part in the faculty’s new journal, the Annales de la Faculté des Sciences de Toulouse, whose first volume appeared in 1887. He sat on its editorial committee, published much of his own work in it from 1888 onward, and worked to give it standing beyond France. The journal is still published.

The Toulouse years were productive. Besides continued fractions he wrote on interpolation, on the gamma function and elliptic functions, on differential equations, and on number theory. He kept up his letters to Hermite, who read every result as it was found, and to other mathematicians across Europe. In 1892 he was a candidate for the Paris Académie des Sciences but was not elected. In 1893 the Académie awarded him the Petit d’Ormoy Prize for his work on continued fractions.

He sent Hermite a note on continued fractions in October 1892, and the full theory grew over the next eighteen months. On 25 May 1894 he delivered the manuscript of Recherches sur les fractions continues. A summary appeared in the Comptes Rendus on 18 June 1894, and Poincaré’s report to the Académie praised the work. The memoir, more than a hundred pages long, appeared in the Toulouse Annales in two parts, in 1894 and 1895. It introduced the moment problem, the convergence theory of S-fractions and the Stieltjes integral, and it is regarded as his masterpiece.

Death and legacy

By the time the memoir appeared Stieltjes was seriously ill. He spent a period of rest in Algeria, and Eugène Cosserat took over his teaching during his illness. Sources differ on the illness itself. A history of his Toulouse years in the faculty’s Annales calls it incurable at the time; another account names the influenza epidemic of those years. Hermite wrote to him for the last time on 15 December 1894. Stieltjes died in Toulouse on 31 December 1894, two days after his 38th birthday, and was buried in the Terre Cabade cemetery on 2 January 1895. Cosserat succeeded him in the chair and wrote a notice on his work. Shortly before his death he had been named a corresponding member of the Academy of Sciences in St Petersburg.

His correspondence with Hermite was published in two volumes in 1905, and his Œuvres complètes in Groningen in 1914–18, the city whose chair he had been refused. In 1966 Toulouse named a lecture theatre after him; his family restored his grave in 1990. The Dutch mathematical community awards an annual Stieltjes Prize for the best doctoral thesis in mathematics.

Legacy

• Integration: the Riemann–Stieltjes integral, its Lebesgue–Stieltjes extension, Riesz representation and the spectral theorem.
• Moments and probability: the moment problem, the Chebyshev–Markov–Stieltjes inequalities, and the method of moments for limit theorems.
• Continued fractions and approximation: the analytic theory of continued fractions, Padé approximation, and the line of ideas that runs through Jacobi matrices to Hilbert-space operator theory.
• Special functions: electrostatic interpretation of zeros, Heine–Stieltjes polynomials, Stieltjes–Wigert polynomials, and the Stieltjes constants of the zeta function.
• Asymptotics: with Poincaré, the founding definitions of asymptotic expansions.

Connections

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

Teachers and advisors

• Charles Hermite 1882-1894
• Gaston Darboux

Influenced by

• Carl Friedrich Gauss
• Carl Gustav Jacob Jacobi
• Pafnuty Chebyshev

Influenced

• David Hilbert
• Felix Hausdorff
• Frigyes Riesz
• Henri Lebesgue

Colleagues and collaborators

• Gösta Mittag-Leffler 1885-1887
• Henri Poincaré 1886-1894
• Diederik Korteweg 1883-1884
• Eugène Cosserat

Rivals

• Andrey Markov 1884

Also mentioned in this life

• Niels Henrik Abel
• Charles-Augustin de Coulomb
• Leonhard Euler
• Fibonacci
• Adrien-Marie Legendre
• August Ferdinand Möbius
• Siméon Denis Poisson
• Bernhard Riemann

Fields

• analysis 20 lives
• number theory 45 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Thomas Jan Stieltjes, MacTutor History of Mathematics.
2. Thomas Joannes Stieltjes, Wikipedia.
3. Mathematical Institute, Leiden University, Thomas Joannes Stieltjes: life.
4. Stieltjes, Dictionary of Scientific Biography (MacTutor copy).
5. H. Strick, Thomas Jan Stieltjes (MacTutor).
6. J. Cassinet, Stieltjes à Toulouse, Annales de la Faculté des Sciences de Toulouse (1995).
7. B. Baillaud and H. Bourget (eds.), Correspondance d’Hermite et de Stieltjes, 2 vols., Gauthier-Villars, 1905.
8. Mathematics Genealogy Project entry.

Mathematics

1. T. J. Stieltjes, Recherches sur quelques séries semi-convergentes, Ann. Sci. École Norm. Sup. (3) 3 (1886).
2. T. J. Stieltjes, Sur une fonction uniforme, C. R. Acad. Sci. Paris 101 (1885).
3. T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse 8 (1894) and 9 (1895).
4. T. J. Stieltjes, Œuvres complètes, 2 vols., Noordhoff, Groningen, 1914–18; reprinted with commentary by G. van Dijk (ed.), Springer, 1993.
5. “Marvelous cancellations”: T. J. Stieltjes’ letters concerning the zeta function, arXiv:1410.8592.
6. A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. reine angew. Math. 357 (1985).
7. G. Szegő, Orthogonal Polynomials, AMS Colloquium Publications 23 (Chapter 6, on zeros, electrostatics and Heine–Stieltjes polynomials).
8. N. I. Akhiezer, The Classical Moment Problem, Oliver and Boyd, 1965.
9. B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998).
10. K. Schmüdgen, The Moment Problem, Springer, 2017.