Joseph Liouville

How to use this course

Few mathematicians have their name on as many different theorems as Joseph Liouville. Physicists meet him in Hamiltonian mechanics and in Sturm–Liouville eigenvalue problems. Analysts meet him in the theorem on bounded entire functions. Number theorists meet him as the first person to write down a transcendental number. Computer algebra systems use his theorem every time they decide that an integral has no closed form.

He was also an institution builder. For almost forty years he edited the Journal de Mathématiques Pures et Appliquées, and in 1846 he published the collected work of Évariste Galois, which had been ignored for fourteen years. This course follows his life and then his mathematics, one lesson per major theme.

Each technical lesson begins with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons run roughly in the order of his work: fractional calculus (1832–1837), eigenvalue problems (1836–1838), elementary integration (1833–1841), phase space (1838), transcendence (1844), elliptic functions (1847) and geometry (1847–1850).

Timeline

Saint-Omer to the Polytechnique, 1809–1827

Joseph Liouville was born on 24 March 1809 in Saint-Omer, in the far north of France. His father, Claude-Joseph Liouville, was a captain in Napoleon’s army, and both parents came from Lorraine. While his father was on campaign, the young Joseph lived with an uncle. After the wars the family settled in Toul, in Lorraine, and Liouville kept a family house there for the rest of his life, returning to it every summer.

He went to school in Commercy and Toul, then moved to Paris to the Collège Saint-Louis, where he studied advanced mathematics. In November 1825 he entered the École Polytechnique. His examiners included Gaspard de Prony and Siméon Denis Poisson. He attended André-Marie Ampère’s course in analysis and mechanics and lectures by François Arago, who later became a close friend and political patron.

The Polytechnique of the 1820s trained engineers for the state, and its graduates were assigned to the technical corps by rank. In 1827 Liouville passed into the École des Ponts et Chaussées, the school of the civil engineering corps. Augustin-Louis Cauchy was then the dominant figure in Paris analysis. Liouville did not attend his courses, but read his work closely, and Cauchy’s methods shaped his own.

From engineer to analyst, 1827–1836

Liouville’s first research, done while he was still an engineering student, was in mathematical physics: electrodynamics in the style of Ampère and the theory of heat in the style of Joseph Fourier and Claude-Louis Navier. He presented memoirs to the Academy from 1828. Engineering field work did not suit his ambitions, and in 1830 he resigned from the corps to make a career in mathematics.

In 1831 he married his maternal cousin Marie-Louise Balland. They had three daughters and a son. The same year he became an assistant (répétiteur) to Claude-Louis Mathieu in the analysis course at the École Polytechnique. Like most young Paris mathematicians, he assembled an income from several posts, teaching in private schools and at the École Centrale.

The years 1832–1836 were extraordinarily productive. He developed a calculus of derivatives of arbitrary order (Lesson 04), began the theory of integration in finite terms (Lesson 06), and started work with his friend Charles Sturm on eigenvalue problems (Lesson 05). He did not obtain a doctorate until 1836, with a thesis on the expansion of functions in series of the kind Fourier had used for heat conduction. The Mathematics Genealogy Project lists Poisson and the chemist Louis Jacques Thénard as his advisors.

The Journal

In 1836 Liouville founded the Journal de Mathématiques Pures et Appliquées. It was modelled on August Leopold Crelle’s Journal für die reine und angewandte Mathematik, founded in Berlin ten years before. Paris had no specialized mathematical journal; the Comptes rendus of the Academy only began the year before and printed short notes. Liouville edited the Journal, almost single-handed, for 39 volumes, and it came to be called simply the Journal de Liouville. It still appears today.

The Journal, the Academy and Galois, 1836–1846

Liouville’s career advanced quickly after 1836. In 1836 he lost the Polytechnique chair left vacant by Navier’s death to Jean-Marie Duhamel. In 1837 he substituted for Jean-Baptiste Biot at the Collège de France, and in 1838 he became professor of analysis and mechanics at the Polytechnique. In 1839 he was elected to the Académie des Sciences, in the astronomy section, against strong opposition from Guglielmo Libri. In 1840, after Poisson died, he joined the Bureau des Longitudes.

The quarrel with Libri became one of the bitterest in French science. Libri, an Italian count and a historian of mathematics, was a skilled academic politician. In 1843 he won the chair at the Collège de France left by Sylvestre François Lacroix, for which Liouville had also applied. Liouville resigned his substitute post there in protest in March 1843. Libri later fled France in 1848, accused of stealing rare books and manuscripts from public libraries, and his chair was declared vacant.

The rescue of Galois

Évariste Galois died in a duel in 1832, aged twenty. His manuscripts had been rejected or lost by the Academy, and his brother Alfred and his friend Auguste Chevalier sent copies to leading mathematicians, without effect. Around 1842 the papers reached Liouville. He read them carefully, filled the gaps in the arguments, and on 4 July 1843 told the Academy that he had found in them a method, as exact as it is deep, for deciding when an equation is solvable by radicals. He published the papers in the October and November issues of his Journal for 1846.

Liouville promised a commentary and never published it. He did, however, lecture privately on Galois theory to younger mathematicians. Joseph Bertrand later said that it was Liouville who taught him Galois’s discoveries, and the audience is thought to have included Joseph-Alfred Serret and possibly Charles Hermite. Through Serret’s algebra textbook of 1866 and Camille Jordan’s Traité of 1870, Galois theory entered the mainstream.

Derivatives of arbitrary order

For the general reader. We know what the first and second derivatives are. Is there a sensible “half derivative”, an operation that gives the ordinary derivative when applied twice? Gottfried Leibniz asked the question in 1695. Liouville gave the first systematic answer in a series of memoirs from 1832, and his ideas, refined by Bernhard Riemann, are now the foundation of fractional calculus.

Liouville’s starting point

Since \(D^n e^{ax}=a^ne^{ax}\) for integers \(n\), Liouville defined, for a function expanded as a sum of exponentials \(f(x)=\sum_k c_k e^{a_kx}\),

for any real or complex \(\alpha\). Writing \(x^{-\beta}\) as an integral of exponentials with Euler’s Gamma integral, \(x^{-\beta}=\frac{1}{\Gamma(\beta)}\int_0^\infty t^{\beta-1}e^{-xt}\,dt\) for \(x,\beta>0\), and differentiating under the integral sign, he obtained

The integral form

The cleanest modern formulation starts from Cauchy’s formula for repeated integration, \(\int_a^x\!\int_a^{t_{n-1}}\!\!\cdots\int_a^{t_1}f\,dt\cdots=\frac{1}{(n-1)!}\int_a^x(x-t)^{n-1}f(t)\,dt\), and replaces \((n-1)!\) by \(\Gamma(\alpha)\).

Proof of the semigroup law. By Fubini,

Substituting \(t=s+u(x-s)\), the inner integral is \((x-s)^{\alpha+\beta-1}B(\alpha,\beta)\), and \(B(\alpha,\beta)=\Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)\). The limit \(a=-\infty\) is Liouville’s own convention, natural for his exponentials; the base point \(a=0\) comes from Riemann’s student paper of 1847, published after his death.

With \(a=0\), a direct computation gives \(D^{\alpha}x^{\mu}=\frac{\Gamma(\mu+1)}{\Gamma(\mu+1-\alpha)}x^{\mu-\alpha}\) for \(\mu>-1\).

Sturm–Liouville theory

For the general reader. A vibrating string of uneven thickness, or a rod cooling unevenly, has a set of natural modes, each with its own frequency or decay rate. Charles Sturm and Liouville showed in 1836–1837 that every problem of this kind behaves like the sine series of a uniform string: the modes are ordered, the n-th mode crosses zero n−1 times, and any reasonable initial shape is a sum of modes.

The problem

On a bounded interval \([a,b]\) let \(p,p',q,w\) be continuous and real, with \(p>0\) and \(w>0\). Consider

with real \((\alpha_1,\alpha_2)\ne0\) and \((\beta_1,\beta_2)\ne0\). A number \(\lambda\) is an eigenvalue if a non-zero solution exists.

Sturm’s contribution was the qualitative theory: comparison and oscillation of zeros. Liouville’s was the expansion problem. In his memoirs of 1836–1837 he gave the first arguments that the eigenfunction series of a function converges to it, and in 1837–1838 he obtained asymptotic formulas for \(\lambda_n\) and \(y_n\). Full completeness proofs came only with the theory of integral equations around 1900.

Reality and orthogonality

Write \(Ly=-(py')'+qy\). Integrating by parts twice gives Green’s identity

and the boundary terms vanish when \(u,v\) satisfy the same separated conditions. If \(Ly_m=\lambda_mwy_m\) and \(Ly_n=\lambda_nwy_n\), then \((\lambda_n-\lambda_m)\int y_my_nw=0\). Applying the same identity to \(y\) and \(\bar y\) shows \(\lambda=\bar\lambda\). Simplicity follows because two solutions satisfying the same condition at \(a\) have zero Wronskian there, hence are proportional.

Liouville’s transformation

Liouville introduced the change of variables

which turns the equation into the normal form \(-u''+Q(t)u=\lambda u\) on \([0,\ell]\), \(\ell=\int_a^b\sqrt{w/p}\,dx\), for a continuous \(Q\) when \(p\) and \(w\) are twice differentiable. Comparing with \(-u''=\lambda u\) gives Liouville’s asymptotic law for Dirichlet conditions,

The same substitution, used independently by George Green in 1837, is the origin of the Liouville–Green (WKB) approximation.

Integration in finite terms

For the general reader. Calculus students learn that some integrals, such as \(\int e^{-x^2}dx\), have “no formula”. That is a theorem, and Liouville proved it. Between 1833 and 1841 he showed exactly when an integral can be written with the usual functions: polynomials, exponentials, logarithms and their combinations. His criterion is the basis of the algorithms that computer algebra systems use today.

The algebraic setting

A differential field is a field \(F\) with a derivation \(f\mapsto f'\) satisfying \((f+g)'=f'+g'\) and \((fg)'=f'g+fg'\). Its constants are the elements with \(c'=0\). An extension \(E\supseteq F\) is elementary if it is obtained by adjoining finitely many elements \(t\), each of which is algebraic over the previous field, or a logarithm (\(t'=u'/u\)), or an exponential (\(t'=u't\)) of an element \(u\) of the previous field. Over \(\mathbb C(x)\) with \(d/dx\) this captures every formula built from rational functions, roots, \(\exp\), \(\log\), and therefore also trigonometric functions and their inverses.

The meaning is that an elementary antiderivative can only use new functions in one way: as constant multiples of logarithms. Exponentials, roots and nested constructions cannot appear unless they were already in \(F\).

Sketch. Apply the theorem with \(F=\mathbb C(x,e^g)\). Comparing the coefficients of \(e^g\) on both sides, viewed as a Laurent polynomial in the transcendental \(t=e^g\), shows that the logarithmic terms contribute nothing at degree 1 and that \(v\) has the form \(R\,t+\cdots\). Since \((Rt)'=(R'+g'R)t\), the condition follows.

Liouville also proved that some differential equations cannot be solved “by quadratures”. His 1841 result on the Riccati equation \(y'+y^2=x^m\) states that it is solvable in finite terms exactly when \(m=-2\) or \(m=-4k/(2k\pm1)\) for a non-negative integer \(k\), Daniel Bernoulli had found the integrable exponents; Liouville proved there are no others. Robert Risch turned these ideas into a decision procedure in 1969.

Volume in phase space

For the general reader. Imagine a cloud of possible starting states for a mechanical system, such as a swinging pendulum with slightly uncertain position and speed. As time passes the cloud moves, stretches and folds, but if there is no friction its total volume in the space of positions and momenta never changes. This fact, which goes back to a note Liouville published in 1838, became the foundation of statistical mechanics.

The Jacobian of a flow

Let \(\dot x=X(x,t)\) on \(\mathbb R^m\) and let \(\varphi_t\) be its flow. The matrix \(J(t)=D\varphi_t(x_0)\) satisfies the variational equation \(\dot J=A(t)J\) with \(A=DX(\varphi_t(x_0),t)\).

Proof: \(\det(I+hA)=1+h\operatorname{tr}A+O(h^2)\), so \(\det J(t+h)=\det J(t)\,(1+h\operatorname{tr}A+O(h^2))\). Here \(\operatorname{tr}A=\operatorname{div}X\). For a second-order linear equation \(y''+P(x)y'+Q(x)y=0\), the same formula gives Abel’s identity for the Wronskian, \(W(x)=W(x_0)\exp\big(-\int_{x_0}^xP\big)\).

Hamiltonian systems

For Hamilton’s equations \(\dot q_i=\partial H/\partial p_i\), \(\dot p_i=-\partial H/\partial q_i\) on \(\mathbb R^{2n}\),

Liouville’s 1838 note, Sur la théorie de la variation des constantes arbitraires, proved the determinant identity for systems of differential equations and applied it to the constants of integration in dynamics. The statistical reading, that \(\rho\) is constant along trajectories, was made by Ludwig Boltzmann and above all by Josiah Willard Gibbs, who made it the fundamental equation of his Elementary Principles in Statistical Mechanics (1902). The density form also implies Poincaré recurrence: a bounded Hamiltonian flow returns arbitrarily close to almost every initial state.

Integrability

In a note of 1855 Liouville showed that if a Hamiltonian system with \(n\) degrees of freedom has \(n\) independent integrals \(F_1=H,\dots,F_n\) in involution, \(\{F_i,F_j\}=0\), then it can be solved by quadratures. Vladimir Arnold added the geometry in 1963: compact connected level sets are tori carrying quasi-periodic motion. This is the Liouville–Arnold theorem.

The first transcendental numbers

For the general reader. A number is algebraic if it solves a polynomial equation with integer coefficients, like \(\sqrt2\). Otherwise it is transcendental. For centuries mathematicians suspected that numbers like \(\pi\) are transcendental, but nobody could show that any transcendental number existed. In 1844 Liouville proved that algebraic numbers cannot be approximated too well by fractions, and then wrote down numbers that are approximated far too well.

Proof. Let \(P\in\mathbb Z[x]\) be the minimal polynomial of \(\alpha\), of degree \(d\). It is irreducible of degree at least 2, so it has no rational root, and \(q^dP(p/q)\) is a non-zero integer: \(|P(p/q)|\ge q^{-d}\). If \(|\alpha-p/q|\ge1\) there is nothing to prove with \(c\le1\). Otherwise, by the mean value theorem, \(|P(p/q)|=|P(p/q)-P(\alpha)|=|P'(\xi)|\,|\alpha-p/q|\le M|\alpha-p/q|\), where \(M=\max_{|x-\alpha|\le1}|P'(x)|\). Hence \(|\alpha-p/q|\ge 1/(Mq^d)\), and \(c=\tfrac12\min(1,1/M)\) works with strict inequality.

Indeed, a Liouville number is irrational (a rational \(a/b\) satisfies \(|a/b-p/q|\ge 1/(bq)\) when the two differ), and if it were algebraic of degree \(d\) the approximations with \(n>d\) and \(q\) large would contradict the theorem.

Liouville’s first examples, announced in the Comptes rendus in 1844, were defined by continued fractions whose partial quotients grow very fast. In a memoir in his Journal in 1851 he gave the decimal example now called Liouville’s constant,

Five years earlier, in 1840, he had proved that \(e\) is not a quadratic irrational, and that the same holds for \(e^2\). Hermite proved \(e\) transcendental in 1873 and Ferdinand von Lindemann did the same for \(\pi\) in 1882. Georg Cantor showed in 1874 that algebraic numbers are countable, so almost every real number is transcendental, though his argument names none.

After Liouville

The exponent \(d\) was improved by Axel Thue (1909) to \(d/2+1+\varepsilon\), by Carl Ludwig Siegel (1921) to about \(2\sqrt d\), and by Klaus Roth (1955) to \(2+\varepsilon\), independent of \(d\). Roth received the Fields Medal in 1958. By Dirichlet’s approximation theorem the exponent 2 cannot be lowered.

Doubly periodic functions and bounded entire functions

For the general reader. Functions such as sine repeat along one direction. Elliptic functions, studied by Abel and Carl Gustav Jacob Jacobi, repeat in two independent directions of the complex plane. In 1847 Liouville rebuilt their whole theory from one principle: a doubly periodic function with no poles must be constant. Its general form, that a bounded function analytic on the whole plane is constant, now bears his name.

Proof. By Cauchy’s integral formula on the circle of radius \(R\) about \(z_0\), \(|f'(z_0)|=\big|\frac{1}{2\pi i}\oint\frac{f(z)}{(z-z_0)^2}dz\big|\le\frac{M}{R}\). Let \(R\to\infty\): \(f'\equiv0\).

History

The attribution is a historical accident. Cauchy published the general statement in 1844. Liouville proved the case he needed, that a doubly periodic entire function is constant, and built his 1847 lectures on elliptic functions on it. He presented the theory privately that year to Ferdinand Joachimsthal and Carl Wilhelm Borchardt and lectured on it at the Collège de France in 1850–51. Borchardt’s notes were published only in 1879, and Liouville’s name became attached to the general theorem.

Liouville’s theorems on elliptic functions

Let \(\Lambda=\mathbb Z\omega_1+\mathbb Z\omega_2\) be a lattice and \(f\) a meromorphic function with \(f(z+\omega)=f(z)\) for \(\omega\in\Lambda\). Let \(P\) be a fundamental parallelogram with no poles on its boundary.

Part 1: \(f\) is continuous on the compact closure of \(P\), hence bounded there, hence bounded on \(\mathbb C\) by periodicity. Part 2: \(\oint_{\partial P}f\,dz=0\) because opposite sides cancel by periodicity. Part 3 applies part 2 to \(f'/(f-c)\). Part 4 integrates \(zf'(z)/f(z)\) around \(\partial P\).

Conformal rigidity in space

For the general reader. A conformal map preserves angles. In the plane there are vast numbers of them: every analytic function with non-zero derivative is one, which is why mapmakers have so many choices. Liouville proved in 1850 that in three or more dimensions the choice collapses. Apart from rescaling, rotating and moving, the only angle-preserving maps of space are inversions in spheres.

Inversion

Inversion in the unit sphere is \(\sigma(x)=x/|x|^2\) on \(\mathbb R^n\setminus\{0\}\). Its derivative is

a reflection matrix times the scalar \(|x|^{-2}\). So \(\sigma\) is conformal and reverses orientation. In the 1840s William Thomson, the future Lord Kelvin, used inversion to solve problems in electrostatics. A note of his prompted Liouville in 1847 to study the geometry of inversion, which he called transformation by reciprocal radii.

Liouville published the theorem in Note VI of his 1850 edition of Monge’s Application de l’analyse à la géométrie, where he extended the classical question of mapping the sphere to the plane to three dimensions. The regularity hypothesis was weakened step by step by Philip Hartman (1947), Harley Flanders (1966) and Yuri Reshetnyak (1967), who proved it for Sobolev maps in \(W^{1,n}_{\mathrm{loc}}\).

Proof idea

Write \(Df=e^{\phi}R\) with \(R\) orthogonal. The condition says the pull-back metric is \(e^{2\phi}\) times the Euclidean metric and still flat. For \(n\ge3\), flatness of \(e^{2\phi}\delta\) forces \(\partial_i\partial_j(e^{-\phi})=\lambda\,\delta_{ij}\) for a single function \(\lambda\), and then \(\lambda\) must be constant. So \(e^{-\phi}=a|x|^2+b\cdot x+c\), which is exactly the conformal factor of a Möbius map. In the plane the corresponding condition is just that \(\phi\) be harmonic, which leaves infinitely many solutions.

Parliament, Collège de France, Sorbonne, 1848–1874

Liouville was a moderate republican. When the revolution of February 1848 overthrew the July Monarchy, his friend Arago, now a minister in the provisional government, encouraged him to stand for election. On 23 April 1848 he was elected to the Constituent Assembly as a deputy for the Meurthe, the department of his family home in Toul. He sat with the moderate republicans and was defeated in the elections of May 1849. The defeat embittered him, and his political activity ended entirely after Louis-Napoleon’s coup of December 1851. His mathematical output fell sharply during his political years.

Libri’s flight in 1848 reopened the chair of mathematics at the Collège de France. Cauchy and Liouville were again rivals. Cauchy had refused to swear the oath of allegiance after 1830 and lost his posts as a result; Liouville had no such scruples. He was elected on 18 January 1851 after a contested vote that Cauchy protested vigorously, and he resigned his Polytechnique chair immediately. In 1857 he also took the chair of rational mechanics at the Faculté des Sciences (the Sorbonne).

From 1856 he turned to number theory. Between 1858 and 1865 he published eighteen papers under the general title Sur quelques formules générales qui peuvent être utiles dans la théorie des nombres, and nearly two hundred short notes in the Journal on representations of integers by quadratic forms and on arithmetic functions. Many of his identities were stated without proof and proved by others decades later. The arithmetic function \(\lambda(n)=(-1)^{\Omega(n)}\), where \(\Omega(n)\) counts prime factors with multiplicity, is named after him.

His friendships shaped his work. Peter Gustav Lejeune Dirichlet was his close friend and main mathematical correspondent until Dirichlet’s death in 1859. He corresponded with Jacobi and Kelvin, and he guided the early careers of Bertrand, Serret and Hermite. Eugène Catalan wrote his doctorate under him, and the Russian Nikolai Bugaev studied with him in Paris.

Last years and legacy

From 1867 Liouville published less, and around 1874 he stopped altogether. In 1874–75 he gave up his courses at the Sorbonne and handed the Journal, after 39 volumes, to Henri Résal. He continued to attend the Academy and the Bureau des Longitudes, and he lectured at the Collège de France until 1879, when Ossian Bonnet took over his teaching. His wife Marie-Louise died in 1880. Liouville died in Paris on 8 September 1882, aged 73.

He left more than 400 papers and a large collection of unpublished notebooks, which Jesper Lützen used for his 1990 study of Liouville’s mathematics, the standard modern account. Much of his number theory was never fully published, and many results in the notebooks anticipate later work.

Legacy

• Analysis: Sturm–Liouville theory became the model for spectral theory, from Hilbert’s integral equations to quantum mechanics. The Liouville–Green approximation is the WKB method.
• Mechanics and physics: the Liouville equation is the starting point of statistical mechanics; Liouville integrability organizes the theory of integrable systems.
• Algebra and computation: his theorem on elementary integrals became differential algebra and the Risch algorithm.
• Number theory: Liouville’s inequality opened Diophantine approximation and transcendental number theory.
• Geometry: his conformal rigidity theorem is the reason higher-dimensional conformal geometry is governed by a finite-dimensional group.
• Mathematical culture: the Journal de Liouville gave France a research journal, and his edition of Galois launched group theory in France.

Connections

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

Teachers and advisors

• Siméon Denis Poisson 1836
• Louis Jacques Thénard 1836

Students

• Eugène Catalan
• Nikolai Bugaev

Influenced by

• André-Marie Ampère
• Gaspard Monge
• Joseph Fourier
• Évariste Galois

Influenced

• Camille Jordan
• Charles Hermite
• Josiah Willard Gibbs
• Joseph Bertrand
• Joseph-Alfred Serret

Colleagues and collaborators

• Carl Gustav Jacob Jacobi
• Ernst Kummer 1847
• Gaston Darboux 1873-1878
• Pafnuty Chebyshev
• Urbain Le Verrier
• William Thomson, Lord Kelvin
• Charles Sturm

Rivals

• Augustin-Louis Cauchy
• Guglielmo Libri 1839-1848

Friends

• François Arago
• Peter Gustav Lejeune Dirichlet

Also mentioned in this life

• Niels Henrik Abel
• Jean-Baptiste Biot
• Ludwig Boltzmann
• Georg Cantor
• Daniel Bernoulli
• Leonhard Euler
• George Green
• William Rowan Hamilton
• David Hilbert
• Henri Lebesgue
• Gottfried Wilhelm Leibniz
• August Ferdinand Möbius
• Claude-Louis Navier
• Henri Poincaré
• Bernhard Riemann
• Carl Ludwig Siegel
• Karl Weierstrass

Fields

• analysis 20 lives
• differential equations 13 lives
• mathematical physics 45 lives
• number theory 45 lives
• differential geometry 15 lives

On CoursesHub

• ODEs / PDEs / Green Functions · Physical Mathematics
• Differential Geometry

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Joseph Liouville, MacTutor History of Mathematics.
2. R. Taton, Liouville, Joseph, Dictionary of Scientific Biography.
3. Joseph Liouville, Wikipedia.
4. J. Lützen, Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics, Springer, 1990.
5. Mathematics Genealogy Project, entry for Joseph Liouville.

Mathematics

1. J. Liouville, Sur la théorie de la variation des constantes arbitraires, J. Math. Pures Appl. 3 (1838).
2. J. Liouville, Sur des classes très étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationnelles algébriques, C. R. Acad. Sci. Paris 18 (1844); J. Math. Pures Appl. 16 (1851).
3. É. Galois, Œuvres mathématiques, ed. J. Liouville, J. Math. Pures Appl. 11 (1846).
4. G. Monge, Application de l’analyse à la géométrie, 5th ed., with notes by J. Liouville, Bachelier, 1850.
5. M. Rosenlicht, Integration in finite terms, Amer. Math. Monthly 79 (1972).
6. Liouville’s theorem (Hamiltonian), Liouville’s theorem (complex analysis), Liouville’s theorem (conformal mappings), Liouville number, Wikipedia.
7. W. N. Everitt, A catalogue of Sturm–Liouville differential equations, in Sturm–Liouville Theory: Past and Present, Birkhäuser, 2005.
8. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1978.
9. Y. Bugeaud, Approximation by Algebraic Numbers, Cambridge University Press, 2004.