Timeline of calculus and mathematical analysis

A timeline of calculus and mathematical analysis

1000 to 1500

• 1020 — Abul Wáfa — Discussed the quadrature of the parabola and the volume of the paraboloid.
• 1021 — Ibn al-Haytham completes his Book of Optics, which formulated and solved “Alhazen's problem” geometrically, and developed and proved the earliest general formula for infinitesimal and integral calculus using mathematical induction.
• 1100s — Bhaskara Acharya conceives differential calculus, and also develops Rolle's theorem, Pell's equation, a proof for the Pythagorean Theorem, proves that division by zero is infinity, computes π to 5 decimal places, and calculates the time taken for the earth to orbit the sun to 9 decimal places
• 1300s — Madhava is considered the father of mathematical analysis, who also worked on the power series for p and for sine and cosine functions, and along with other Kerala school mathematicians, founded the important concepts of Calculus
• 1300s — Parameshvara, a Kerala school mathematician, presents a series form of the sine function that is equivalent to its Taylor series expansion, states the mean value theorem of differential calculus, and is also the first mathematician to give the radius of circle with inscribed cyclic quadrilateral
• 1400 — Madhava discovers the series expansion for the inverse-tangent function, the infinite series for arctan and sin, and many methods for calculating the circumference of the circle, and uses them to compute π correct to 11 decimal places

16th century

• 1501 — Nilakantha Somayaji writes the “Tantra Samgraha”, which lays the foundation for a complete system of fluxions (derivatives), and expands on concepts from his previous text, the “Aryabhatiya Bhasya”.
• 1550 — Jyeshtadeva, a Kerala school mathematician, writes the “Yuktibhasa”, the world's first calculus text, which gives detailed derivations of many calculus theorems and formulae.

17th century

• 1629 - Pierre de Fermat develops a rudimentary differential calculus,
• 1634 - Gilles de Roberval shows that the area under a cycloid is three times the area of its generating circle,
• 1658 - Christopher Wren shows that the length of a cycloid is four times the diameter of its generating circle,
• 1665 - Isaac Newton works on the fundamental theorem of calculus and develops his version of infinitesimal calculus,
• 1671 - James Gregory develops a series expansion for the inverse-tangent function (originally discovered by Madhava)
• 1673 - Gottfried Leibniz also develops his version of infinitesimal calculus,
• 1675 - Isaac Newton invents an Newton's method for the computation of functional roots,
• 1691 - Gottfried Leibniz discovers the technique of separation of variables for ordinary differential equations,
• 1696 - Guillaume de L'Hôpital states his rule for the computation of certain limits,
• 1696 - Jakob Bernoulli and Johann Bernoulli solve brachistochrone problem, the first result in the calculus of variations,

18th century

• 1712 - Brook Taylor develops Taylor series,
• 1730 - James Stirling publishes The Differential Method,
• 1734 - Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations,
• 1735 - Leonhard Euler solves the Basel problem, relating an infinite series to π,
• 1739 - Leonhard Euler solves the general homogeneous linear ordinary differential equation with constant coefficients,
• 1748 - Maria Gaetana Agnesi discusses analysis in Instituzioni Analitiche ad Uso della Gioventu Italiana,
• 1762 - Joseph Louis Lagrange discovers the divergence theorem,

19th century

• 1807 - Joseph Fourier announces his discoveries about the trigonometric decomposition of functions,
• 1811 - Carl Friedrich Gauss discusses the meaning of integrals with complex limits and briefly examines the dependence of such integrals on the chosen path of integration,
• 1815 - Siméon-Denis Poisson carries out integrations along paths in the complex plane,
• 1817 - Bernard Bolzano presents the intermediate value theorem---a continuous function which is negative at one point and positive at another point must be zero for at least one point in between,
• 1822 - Augustin-Louis Cauchy presents the Cauchy integral theorem for integration around the boundary of a rectangle in the complex plane,
• 1825 - Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths -- he assumes the function being integrated has a continuous derivative, and he introduces the theory of residues in complex analysis,
• 1825 - André-Marie Ampère discovers Stokes' theorem,
• 1828 - George Green proves Green's theorem,
• 1831 - Mikhail Vasilievich Ostrogradsky rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green,
• 1841 - Karl Weierstrass discovers but does not publish the Laurent expansion theorem,
• 1843 - Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem,
• 1850 - Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points,
• 1850 - George Gabriel Stokes rediscovers and proves Stokes' theorem,
• 1873 - Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points,

20th century

• 1908 - Josip Plemelj solves the Riemann problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky - Plemelj formulae,
• 1966 - Abraham Robinson presents Non-standard analysis.
• 1985 - Louis de Branges de Bourcia proves the Bieberbach conjecture,