Jacques Hadamard

How to use this course

Jacques Hadamard lived from the Second Empire to the space age. In those 97 years he proved the prime number theorem, created the modern theory of entire functions, found the sharp bound for determinants, began the study of chaotic geodesic flows, and defined what it means for a problem of mathematical physics to be well posed. He also taught or supervised a remarkable line of students, including Maurice Fréchet, Paul Lévy and André Weil, and his seminar at the Collège de France set the pattern for mathematical seminars in France.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow the order of his work: complex analysis and number theory (1892–1896), determinants (1893), geometry (1898), and partial differential equations and functionals (1900–1932).

Timeline

Versailles and Paris, 1865–1884

Jacques Salomon Hadamard was born in Versailles on 8 December 1865. His father, Amédée Hadamard, came from a Jewish family and taught Latin, grammar and history in Parisian lycées. His mother, Claire Marie Jeanne Picard, gave private piano lessons.

He was educated at the Lycée Charlemagne, where his father taught, and from 1876 at the Lycée Louis-le-Grand. He took the baccalaureate in both letters and sciences. In 1883 he won first prize in mathematics in the Concours général.

In 1884 he came first in the entrance examinations of both the École normale supérieure and the École polytechnique. Like Émile Borel five years later, he chose the École normale, the school of research and teaching.

The École normale and the thesis, 1884–1892

His teachers at the École normale and the Sorbonne included Jules Tannery, Charles Hermite, Gaston Darboux, Paul Appell, Édouard Goursat and Émile Picard. He graduated in 1888 and, like most young normaliens, went to teach in secondary schools: briefly at Caen, then at the Lycée Saint-Louis, and from 1890 to 1893 at the Lycée Buffon in Paris. There he recognized the talent of a pupil, Maurice Fréchet, and encouraged him toward mathematics.

His doctoral thesis, Essai sur l’étude des fonctions données par leur développement de Taylor, defended in 1892 under Picard and Tannery, asked a question that Weierstrass’s school had left open: how can one read the singularities of a function from the coefficients of its Taylor series? It contains the formula for the radius of convergence now named after Cauchy and Hadamard, and the first gap theorem (Lesson 04).

The same year the Académie des sciences awarded him its Grand Prix des sciences mathématiques. The competition concerned the number of primes below a given bound, the question Bernhard Riemann had attacked in his 1859 memoir. Hadamard’s memoir developed a theory of entire functions of finite order and applied it to Riemann’s function \(\xi(s)\) (Lesson 05). In June 1892 he married Louise-Anna Trénel. They had three sons and two daughters.

Bordeaux, 1893–1897

In 1893 Hadamard became a lecturer at the University of Bordeaux, and in 1896 professor of astronomy and rational mechanics. These four years were among the most productive of his life. In 1893 he published his inequality for determinants (Lesson 07). In 1896 he published the proof of the prime number theorem (Lesson 06), the culmination of the programme his Grand Prix memoir had begun. The Belgian Charles de la Vallée Poussin proved the same theorem independently in the same year. The theorem had been conjectured by Adrien-Marie Legendre and Carl Friedrich Gauss a century before.

Also in 1896 he proved the three-circle theorem for analytic functions, and he won the Académie’s Bordin Prize for work on geodesics on surfaces, which led to his 1898 memoir on negative curvature (Lesson 08). His first two sons, Pierre and Étienne, were born in Bordeaux. In 1897 he resigned his chair to return to Paris.

Taylor series and their singularities

For the general reader. A power series converges inside a circle and diverges outside. The size of the circle is set by the nearest singular point of the function. Hadamard gave an exact formula for the radius in terms of the coefficients, and showed that series with very sparse terms cannot be continued beyond their circle at all.

Proof. Let \(L=\limsup|a_n|^{1/n}\). If \(|z|<1/L\), pick \(q\) with \(L|z|<q<1\); then \(|a_n|^{1/n}|z|<q\) for large \(n\), so \(|a_nz^n|<q^n\) and the series converges by comparison. If \(|z|>1/L\), then \(|a_n|^{1/n}|z|>1\) for infinitely many \(n\), so the terms do not tend to 0. Augustin-Louis Cauchy had stated the formula in his 1821 course, but it was little noticed; Hadamard rediscovered it and made it the starting point of his thesis.

Gaps and natural boundaries

A point of the circle of convergence is regular if the function continues analytically to a neighbourhood of it, and singular otherwise. There is always at least one singular point on the circle. If every point is singular, the circle is a natural boundary.

The usual proof, due to Louis Mordell, substitutes \(z=\frac{w^p+w^{p+1}}{2}\), which maps the closed unit disc into itself, touching the circle only at \(w=1\mapsto z=1\), and maps a slightly larger disc onto a region containing a neighbourhood of 1. If \(f\) were regular at 1, the composed series in \(w\) would converge beyond \(|w|=1\). The gaps ensure that the powers \((w^p+w^{p+1})^{n_k}\) do not overlap for suitable \(p\), so the composed series is just a regrouping of the original, and convergence at some \(|w|>1\) forces convergence of \(\sum|c_k|r^{n_k}\) for some \(r>1\), a contradiction. Rotating the argument covers every point of the circle. Eugène Fabry (1896) and George Pólya later showed that much weaker gap conditions suffice.

Entire functions and factorization

For the general reader. A polynomial is determined, up to a constant, by its roots. Entire functions such as \(\sin z\) can have infinitely many roots, and Weierstrass showed how to build a function with prescribed zeros. Hadamard showed that if a function does not grow too fast, its zeros determine it up to a simple exponential factor. This was the tool needed to make Riemann’s ideas about primes rigorous.

Order and canonical products

For an entire function \(f\) let \(M(r)=\max_{|z|=r}|f(z)|\). The order of \(f\) is

So \(e^{z}\) and \(\sin z\) have order 1, \(e^{z^2}\) order 2, and polynomials order 0. Weierstrass’s elementary factors are \(E_0(u)=1-u\) and \(E_p(u)=(1-u)\exp\big(u+\frac{u^2}{2}+\cdots+\frac{u^p}{p}\big)\). If \(a_1,a_2,\dots\) are the non-zero zeros, the genus of the sequence is the least integer \(p\) with \(\sum|a_n|^{-p-1}<\infty\).

Proof idea. Jensen’s formula bounds the number \(n(r)\) of zeros in \(|z|\le r\) by \(O(r^{\rho+\varepsilon})\), which gives \(\sum|a_n|^{-\rho-\varepsilon}<\infty\) and hence the convergence of the canonical product \(\Pi(z)\). The quotient \(g=f/(z^m\Pi)\) is entire without zeros, so \(g=e^{h}\). The heart of the proof is a lower bound for \(|\Pi(z)|\) away from small discs around the zeros. It shows that \(\operatorname{Re}h(z)\le C|z|^{\rho+\varepsilon}\) on a sequence of circles, and the Borel–Carathéodory inequality then forces \(h\) to be a polynomial of degree at most \(\rho\).

Application to the zeta function

Riemann’s \(\xi(s)=\tfrac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) is entire of order 1, and its zeros are the non-trivial zeros \(\varrho\) of \(\zeta\). Hadamard’s theorem gives

which Riemann had asserted without proof. It also shows that there are infinitely many non-trivial zeros, and taking logarithmic derivatives leads to von Mangoldt’s explicit formula linking primes to zeros.

The prime number theorem

For the general reader. The primes thin out as numbers grow. Near \(x\), roughly one number in \(\log x\) is prime. Gauss guessed this as a teenager, and Riemann in 1859 showed it is governed by the zeros of the zeta function. In 1896 Hadamard and de la Vallée Poussin proved it, by showing that no zero lies on the line \(\operatorname{Re}s=1\).

The non-vanishing on \(\operatorname{Re}s=1\)

For \(\operatorname{Re}s>1\), the Euler product gives \(\log\zeta(s)=\sum_p\sum_{k\ge1}\frac{1}{k\,p^{ks}}\). With \(s=\sigma+it\), \(\operatorname{Re}\,p^{-ks}=p^{-k\sigma}\cos(kt\log p)\). The inequality \(3+4\cos\theta+\cos2\theta\ge0\), used in this way by Franz Mertens in 1898, gives

for \(\sigma>1\). Suppose \(\zeta(1+it)=0\) for some \(t\ne0\). As \(\sigma\to1^+\), \(\zeta(\sigma)^3\sim(\sigma-1)^{-3}\), \(|\zeta(\sigma+it)|^4=O((\sigma-1)^4)\), and \(\zeta(\sigma+2it)\) stays bounded. The product tends to 0, a contradiction. Hadamard’s own 1896 argument was different and more involved, but rested on the same Euler-product positivity.

From zeros to primes

The logarithmic derivative is \(-\frac{\zeta'}{\zeta}(s)=\sum_n\Lambda(n)n^{-s}\), where \(\Lambda(p^k)=\log p\) and \(\Lambda=0\) otherwise, so \(\psi(x)=\sum_{n\le x}\Lambda(n)\). The function \(-\frac{\zeta'}{\zeta}(s)-\frac{1}{s-1}\) is analytic on \(\operatorname{Re}s\ge1\) because \(\zeta\) has a simple pole at 1 and no zeros on the line. A Tauberian argument then gives \(\psi(x)\sim x\). Hadamard used contour integration and his factorization of \(\xi\) to control \(\zeta'/\zeta\); a short modern route is Donald Newman’s 1980 Tauberian theorem. De la Vallée Poussin went further in 1899 and proved an error term \(\pi(x)=\operatorname{Li}(x)+O\big(xe^{-c\sqrt{\log x}}\big)\).

Maximal determinants

For the general reader. A determinant measures the volume of the box spanned by the rows of a matrix. If the rows have fixed lengths, the box is largest when they are perpendicular. Hadamard turned this into a sharp inequality in 1893 and asked which matrices of \(\pm1\) entries reach the bound. These Hadamard matrices are now used in coding theory, signal processing and quantum computing.

Proof. If the rows are dependent, both sides behave correctly. Otherwise apply Gram–Schmidt: \(A=LQ\) where \(Q\) has orthonormal rows and \(L\) is lower triangular with diagonal entries \(\ell_{ii}=\|a_i-\text{proj}_{\langle a_1,\dots,a_{i-1}\rangle}a_i\|\le\|a_i\|\). Then \(|\det A|=\prod\ell_{ii}\le\prod\|a_i\|\), with equality exactly when each \(a_i\) is already orthogonal to the previous rows. If \(|a_{ij}|\le1\), then \(\|a_i\|\le\sqrt n\).

Hadamard matrices

A real \(n\times n\) matrix \(H\) with entries \(\pm1\) attains the bound \(n^{n/2}\) exactly when \(HH^{\mathsf T}=nI\). Such matrices are called Hadamard matrices. James Joseph Sylvester had constructed examples of order \(2^k\) in 1867 by the doubling rule

Hadamard found examples of orders 12 and 20 and showed that the order must be 1, 2 or divisible by 4. Raymond Paley in 1933 constructed many more from quadratic residues.

Geodesics on negatively curved surfaces

For the general reader. On a saddle-shaped surface, two paths that start almost together spread apart exponentially fast. In 1898 Hadamard studied the shortest paths, the geodesics, on surfaces curved like this everywhere. He showed that their long-term behaviour is unpredictable in practice, even though it is completely determined. This was one of the first mathematical examples of what is now called chaos, and his method of coding paths by sequences of symbols began symbolic dynamics.

The 1898 memoir

In Les surfaces à courbures opposées et leurs lignes géodésiques (1898), Hadamard considered complete surfaces in space with negative curvature everywhere, built like a central piece with several infinite flaring ends. He showed that every closed curve that cannot be shrunk to a point can be deformed into exactly one closed geodesic. He classified geodesics by the sequence of ends and handles they pass around, and showed that every admissible sequence is realized. Small changes in initial direction lead to completely different sequences. Pierre Duhem pointed out in 1906 that this made long-term prediction physically meaningless, an early discussion of sensitive dependence on initial conditions.

Proof. If two geodesics met twice, they would bound a geodesic bigon \(D\) with interior angles \(\alpha,\beta\in(0,\pi)\). The Gauss–Bonnet formula gives \(\int_D K\,dA+(\pi-\alpha)+(\pi-\beta)=2\pi\), so \(\int_DK\,dA=\alpha+\beta>0\), contradicting \(K\le0\).

For surfaces the theorem goes back to Hans von Mangoldt (1881) and Hadamard (1898). Élie Cartan proved it in all dimensions in 1928. The key fact is that there are no conjugate points: a Jacobi field \(J\) along a geodesic, satisfying \(J''+K\,J=0\) on a surface, with \(J(0)=0\) and \(J'(0)\ne0\), never vanishes again when \(K\le0\).

Coding and symbolic dynamics

Hadamard’s idea of describing a geodesic by the infinite sequence of regions it visits was developed by Marston Morse in 1921 and by Morse and Gustav Hedlund in 1938 into symbolic dynamics. Eberhard Hopf proved in 1936 that the geodesic flow of a compact surface of constant negative curvature is ergodic, and Dmitri Anosov extended this to variable negative curvature in 1967.

Well-posed problems and the Cauchy problem

For the general reader. A mathematical model of a physical system is useful only if its solution exists, is unique and changes little when the measured data change a little. Hadamard made this a definition in 1902 and showed with a simple example that the Laplace equation fails the test when posed as an initial-value problem. His lectures at Yale in 1922 became the standard treatment of the initial-value problem for wave equations.

The Cauchy–Kovalevskaya theorem guarantees a unique local solution of an analytic Cauchy problem, but says nothing about condition (iii). Hadamard argued that physical data are never known exactly, so only stable problems describe physics. Hyperbolic equations with data on a spacelike surface are well posed; elliptic equations with Cauchy data are not.

Hadamard’s example

Consider \(u_{xx}+u_{yy}=0\) for \(y>0\) with Cauchy data \(u(x,0)=0\), \(u_y(x,0)=\frac{1}{n}\sin nx\). The unique solution is

As \(n\to\infty\) the data tend to zero uniformly, together with any finite number of derivatives if \(1/n\) is replaced by \(e^{-\sqrt n}\). But for any fixed \(y>0\), \(\sup_x|u_n(x,y)|=\sinh(ny)/n^2\to\infty\). The zero data have the zero solution, and arbitrarily small changes in the data produce arbitrarily large changes in the solution.

The method of descent and the finite part

The solution of the wave equation \(u_{tt}=c^2\Delta u\) in three dimensions with \(u(\cdot,0)=0\), \(u_t(\cdot,0)=\psi\) is Kirchhoff’s formula \(u(x,t)=t\,\overline{\psi}(x,ct)\), where \(\overline\psi(x,r)\) is the mean of \(\psi\) over the sphere of radius \(r\) centred at \(x\). Hadamard’s method of descent treats a two-dimensional problem as a three-dimensional one whose data do not depend on the third coordinate. Integrating over the sphere then gives Poisson’s formula in the plane, in which the solution at \(x\) depends on all data in the disc \(|y-x|\le ct\), not just on its boundary circle. This is why sharp signals propagate cleanly in three dimensions but leave a trailing wake in two.

For the general second-order hyperbolic equation, Hadamard constructed the fundamental solution from integrals that diverge at the light cone. He defined their finite part (partie finie) by discarding the divergent terms in a systematic way, an idea later absorbed into Laurent Schwartz’s theory of distributions.

Functionals and domain variation

For the general reader. Some quantities depend not on a number but on a whole function or a whole shape: the length of a curve, the energy of a membrane, the pitch of a drum. Hadamard called such quantities functionals and studied how they change when the function or shape is perturbed. His students Fréchet and Lévy turned this into functional analysis.

Functionals

Vito Volterra had studied “functions of lines” from the 1880s. Hadamard introduced the word fonctionnelle and, in 1903, showed that every continuous linear functional \(U\) on \(C[a,b]\) is a limit of integrals: \(U(f)=\lim_{n\to\infty}\int_a^bf(x)\,k_n(x)\,dx\) for suitable continuous kernels \(k_n\). Frigyes Riesz sharpened this in 1909 to the representation \(U(f)=\int_a^bf\,d\alpha\) by a function \(\alpha\) of bounded variation. In his Leçons sur le calcul des variations (1910), Hadamard treated the calculus of variations as the differential calculus of functionals. Fréchet, his student, defined the derivative of a functional in abstract spaces; Lévy, another student, wrote Leçons d’analyse fonctionnelle (1922), with a preface by Hadamard.

Hadamard’s variational formula (1908)

In a prize memoir on clamped elastic plates, Hadamard asked how the Green’s function of a domain changes when its boundary moves. Let \(\Omega\subset\mathbb R^2\) be smooth and let \(G(a,b)\) satisfy \(-\Delta_bG(a,b)=\delta_a\) in \(\Omega\), \(G=0\) on \(\partial\Omega\). Move each boundary point \(\xi\) outward along the normal by \(\delta n(\xi)\).

The formula is the ancestor of shape derivatives in optimal design and of Loewner-type evolution equations in complex analysis. For eigenvalues of the Dirichlet Laplacian the analogous formula is \(\delta\lambda=-\int_{\partial\Omega}\big(\partial_nu\big)^2\,\delta n\,ds\) for a normalized eigenfunction \(u\): enlarging a drum lowers its pitch.

Paris, Dreyfus, war and the seminar, 1897–1939

Hadamard returned to Paris in 1897 to teach at the Sorbonne and the Collège de France. That year the Dreyfus affair broke open. Captain Alfred Dreyfus, convicted of treason in 1894 on forged evidence, was married to Lucie Hadamard, a relative of Jacques. Hadamard became one of the scientists most active in the campaign to reopen it. Dreyfus was finally rehabilitated in 1906. The affair made Hadamard a public man. He sat on the council of the Ligue des droits de l’homme, founded in 1898 in the affair’s wake, and remained engaged in causes of justice, Jewish life and peace for the rest of his life.

His academic career rose steadily. He was president of the Société mathématique de France in 1906. In 1909 he took over the chair of mechanics at the Collège de France, which he held until 1935, and in 1912 he succeeded Camille Jordan as professor of analysis at the École polytechnique. On 9 December 1912 he was elected to the Académie des sciences in place of Henri Poincaré, who had died that summer. In 1920 he succeeded Paul Appell in the chair of analysis at the École centrale.

The First World War cost him his two eldest sons. Pierre was killed in May 1916 and Étienne at Verdun in July 1916. Afterwards Hadamard worked for international reconciliation and peace.

From 1920 he ran a seminar at the Collège de France. Each year he chose recent papers from around the world and assigned them to young mathematicians, who presented them to the group. Hadamard took part in every discussion. André Weil, who wrote his thesis under him in 1928, was among the young mathematicians formed by it, and its format influenced later Paris seminars. Hadamard lectured widely abroad: at Yale in 1922, in the Soviet Union in 1930 and 1934, in Brazil, and in China in 1936, at Tsinghua University. He became a Foreign Member of the Royal Society in 1932.

Exile, old age and legacy, 1940–1963

When German troops approached Paris in 1940, Hadamard, now 74 and a prominent Jewish public figure, fled south with his family. With help from friends abroad they reached the United States, where he held a visiting post at Columbia University and taught at the École libre des hautes études in New York. He could not obtain a permanent position. His third son, Mathieu, was killed in Libya serving in the French forces. In 1944 Hadamard moved to London, and in 1945 he returned to Paris.

In 1945 Princeton University Press published An Essay on the Psychology of Invention in the Mathematical Field, written during his American exile. Following Poincaré’s 1908 account of his own discoveries, Hadamard described invention in four stages: preparation, incubation, illumination and verification. He reported that his own thinking used vague images rather than words, and he included in an appendix a letter from Albert Einstein describing similar experience. The book is still read by psychologists of creativity.

He remained active in old age. In 1950 there were official attempts, linked to his left-wing sympathies, to keep him from the International Congress of Mathematicians in Cambridge, Massachusetts. He attended, and the Congress elected him honorary president. He received the gold medal of the CNRS in 1956 and the Grand Cross of the Légion d’honneur. In December 1962 the Académie des sciences marked fifty years of his membership. That year his grandson Étienne died in a mountaineering accident, a loss from which he did not recover. He died in Paris on 17 October 1963, aged 97.

Legacy

• Analytic number theory: the prime number theorem, and the theory of entire functions that made Riemann’s programme rigorous.
• Complex analysis: the Cauchy–Hadamard formula, the gap theorem, the factorization theorem and the three-circle theorem.
• Combinatorics and engineering: Hadamard matrices, used in error-correcting codes, experimental design and the Walsh–Hadamard transform; the Hadamard gate of quantum computing is the \(2\times2\) Hadamard matrix scaled by \(1/\sqrt2\).
• Geometry and dynamics: the Cartan–Hadamard theorem and the first study of chaotic geodesic flows.
• Partial differential equations: well-posedness, the method of descent, the finite part and the theory of the Cauchy problem.
• Students: Maurice Fréchet, Paul Lévy, Szolem Mandelbrojt, André Weil, Georges Bouligand and Maurice Janet wrote doctorates under him.

Connections

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

Teachers and advisors

• Émile Picard 1892
• Jules Tannery 1892

Students

• André Weil 1928
• Maurice Fréchet 1906
• Paul Lévy 1911
• Georges Bouligand 1914
• Maurice Janet 1920
• Szolem Mandelbrojt 1923

Influenced by

• Bernhard Riemann
• Charles Hermite
• Henri Poincaré

Influenced

• Ivar Fredholm
• Élie Cartan

Colleagues and collaborators

• Albert Einstein
• Henri Lebesgue
• L. E. J. Brouwer
• Vito Volterra
• Émile Borel

Rivals

• Charles de la Vallée Poussin

Also mentioned in this life

• Augustin-Louis Cauchy
• Gaston Darboux
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Carl Friedrich Gauss
• Carl Gustav Jacob Jacobi
• Camille Jordan
• Gustav Kirchhoff
• Sofia Kovalevskaya
• Pierre-Simon Laplace
• Adrien-Marie Legendre
• Siméon Denis Poisson
• Jean-Victor Poncelet
• Frigyes Riesz
• Karl Weierstrass

Fields

• complex analysis 13 lives
• analytic number theory 4 lives
• differential geometry 15 lives
• partial differential equations 8 lives
• functional analysis 11 lives

On CoursesHub

• Complex Analysis · Mathematics for Physics
• Differential Geometry
• ODEs / PDEs / Green Functions · Physical Mathematics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Jacques Hadamard, MacTutor History of Mathematics Archive.
2. Académie des sciences, Notice sur la vie et l’œuvre de Jacques Hadamard.
3. M. L. Cartwright, Jacques Hadamard 1865–1963, Biographical Memoirs of Fellows of the Royal Society 11 (1965).
4. V. Maz’ya and T. Shaposhnikova, Jacques Hadamard: A Universal Mathematician, American Mathematical Society, 1998.
5. Jacques Hadamard, Wikipedia; Jacques Hadamard, Wikipédia (French).
6. Jacques Hadamard, Mathematics Genealogy Project.

Mathematics

1. J. Hadamard, Essai sur l’étude des fonctions données par leur développement de Taylor, J. Math. Pures Appl. (4) 8 (1892).
2. J. Hadamard, Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann, J. Math. Pures Appl. (4) 9 (1893).
3. J. Hadamard, Résolution d’une question relative aux déterminants, Bull. Sci. Math. 17 (1893).
4. J. Hadamard, Sur la distribution des zéros de la fonction \(\zeta(s)\) et ses conséquences arithmétiques, Bull. Soc. Math. France 24 (1896).
5. J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodésiques, J. Math. Pures Appl. (5) 4 (1898).
6. J. Hadamard, Sur les problèmes aux dérivées partielles et leur signification physique, Princeton University Bulletin 13 (1902).
7. J. Hadamard, Leçons sur le calcul des variations, Hermann, 1910.
8. J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press, 1923.
9. J. Hadamard, An Essay on the Psychology of Invention in the Mathematical Field, Princeton University Press, 1945.
10. D. Zagier, Newman’s short proof of the prime number theorem, Amer. Math. Monthly 104 (1997).
11. K. J. Horadam, Hadamard Matrices and Their Applications, Princeton University Press, 2007.