Introductio in analysin infinitorum •
Leonhard Euler (1748) The eminent historian of mathematics
Carl Boyer once called Euler's
Introductio in analysin infinitorum the greatest modern textbook in mathematics. Published in two volumes, this book more than any other work succeeded in establishing
analysis as a major branch of mathematics, with a focus and approach distinct from that used in geometry and algebra. Notably, Euler identified functions rather than curves to be the central focus in his book. Logarithmic, exponential, trigonometric, and transcendental functions were covered, as were expansions into partial fractions, evaluations of for a positive integer between 1 and 13, infinite series and infinite product formulas, In this work, Euler proved that every rational number can be written as a finite continued fraction, that the continued fraction of an irrational number is infinite, and derived continued fraction expansions for and \textstyle\sqrt{e}. and served as a summary of the
Kerala School's achievements in infinite series,
trigonometry and
mathematical analysis, most of which were earlier discovered by the 14th century mathematician
Madhava. Some of its important developments includes
infinite series and
Taylor series expansion of some trigonometry functions and π approximation.
Calculus Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illi calculi genus •
Gottfried Leibniz (1684) Leibniz's first publication on differential calculus, containing the now familiar notation for differentials as well as rules for computing the derivatives of powers, products and quotients. ====
Philosophiae Naturalis Principia Mathematica==== •
Isaac Newton (1687) The
Philosophiae Naturalis Principia Mathematica (
Latin: "mathematical principles of natural philosophy", often
Principia or
Principia Mathematica for short) is a three-volume work by
Isaac Newton published on 5 July 1687. Perhaps the most influential scientific book ever published, it contains the statement of
Newton's laws of motion forming the foundation of
classical mechanics as well as his
law of universal gravitation, and derives
Kepler's laws for the motion of the
planets (which were first obtained empirically). Here was born the practice, now so standard we identify it with science, of explaining nature by postulating mathematical axioms and demonstrating that their conclusion are observable phenomena. In formulating his physical theories, Newton freely used his unpublished work on calculus. When he submitted Principia for publication, however, Newton chose to recast the majority of his proofs as geometric arguments.
Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum •
Leonhard Euler (1755) Published in two books, Euler's textbook on differential calculus presented the subject in terms of the function concept, which he had introduced in his 1748
Introductio in analysin infinitorum. This work opens with a study of the calculus of
finite differences and makes a thorough investigation of how differentiation behaves under substitutions. Also included is a systematic study of
Bernoulli polynomials and the
Bernoulli numbers (naming them as such), a demonstration of how the Bernoulli numbers are related to the coefficients in the
Euler–Maclaurin formula and the values of ζ(2n), a further study of
Euler's constant (including its connection to the
gamma function), and an application of partial fractions to differentiation.
Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe •
Bernhard Riemann (1867) Written in 1853, Riemann's work on trigonometric series was published posthumously. In it, he extended Cauchy's definition of the integral to that of the
Riemann integral, allowing some functions with dense subsets of discontinuities on an interval to be integrated (which he demonstrated by an example). He also stated the
Riemann series theorem, and developed the Riemann localization principle.
Intégrale, longueur, aire •
Henri Lebesgue (1901) Lebesgue's
doctoral dissertation, summarizing and extending his research to date regarding his development of
measure theory and the
Lebesgue integral.
Complex analysis Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse • Bernhard Riemann (1851) Riemann's doctoral dissertation introduced the notion of a
Riemann surface,
conformal mapping, simple connectivity, the
Riemann sphere, the Laurent series expansion for functions having poles and branch points, and the
Riemann mapping theorem.
Functional analysis Théorie des opérations linéaires •
Stefan Banach (1932; originally published 1931 in
Polish under the title
Teorja operacyj.) • The first mathematical monograph on the subject of
linear metric spaces, bringing the abstract study of
functional analysis to the wider mathematical community. The book introduced the ideas of a
normed space and the notion of a so-called
B-space, a
complete normed space. The
B-spaces are now called
Banach spaces and are one of the basic objects of study in all areas of modern mathematical analysis. Banach also gave proofs of versions of the
open mapping theorem,
closed graph theorem, and
Hahn–Banach theorem.
Produits Tensoriels Topologiques et Espaces Nucléaires • Grothendieck's thesis introduced the notion of a
nuclear space,
tensor products of locally convex topological vector spaces, and the start of Grothendieck's work on tensor products of Banach spaces.
Alexander Grothendieck also wrote a textbook on
topological vector spaces: •
Sur certains espaces vectoriels topologiques •
Fourier analysis Mémoire sur la propagation de la chaleur dans les corps solides •
Joseph Fourier (1807) Introduced
Fourier analysis, specifically
Fourier series. Key contribution was to not simply use
trigonometric series, but to model
all functions by trigonometric series: {{blockquote|\varphi(y)=a\cos\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}+a''\cos5\frac{\pi y}{2}+\cdots. Multiplying both sides by \cos(2i+1)\frac{\pi y}{2}, and then integrating from y=-1 to y=+1 yields: a_i=\int_{-1}^1\varphi(y)\cos(2i+1)\frac{\pi y}{2}\,dy.}} When Fourier submitted his paper in 1807, the committee (which included
Lagrange,
Laplace,
Malus and
Legendre, among others) concluded:
...the manner in which the author arrives at these equations is not exempt of difficulties and [...] his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. Making Fourier series rigorous, which in detail took over a century, led directly to a number of developments in analysis, notably the rigorous statement of the integral via the
Dirichlet integral and later the
Lebesgue integral.
Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données •
Peter Gustav Lejeune Dirichlet (1829, expanded German edition in 1837) In his habilitation thesis on Fourier series, Riemann characterized this work of Dirichlet as "
the first profound paper about the subject". This paper gave the first rigorous proof of the convergence of
Fourier series under fairly general conditions (piecewise continuity and monotonicity) by considering partial sums, which Dirichlet transformed into a particular
Dirichlet integral involving what is now called the
Dirichlet kernel. This paper introduced the nowhere continuous
Dirichlet function and an early version of the
Riemann–Lebesgue lemma.
On convergence and growth of partial sums of Fourier series •
Lennart Carleson (1966) Settled
Lusin's conjecture that the Fourier expansion of any L^2 function converges
almost everywhere. ==
Geometry==