used the
method of exhaustion to compute the
area inside a circle by finding the area of
regular polygons with more and more sides. This was an early but informal example of a
limit, one of the most basic concepts in mathematical analysis.
Ancient Mathematical analysis formally developed in the 17th century during the
Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of
ancient Greek mathematics. For instance, an
infinite geometric sum is implicit in
Zeno's paradox of the dichotomy. In Asia, the
Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle. From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the
arithmetic and
geometric series as early as the 4th century BCE.
Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in .
Medieval Zu Chongzhi established a method that would later be called
Cavalieri's principle to find the volume of a
sphere in the 5th century. In the 12th century, the
Indian mathematician Bhāskara II used infinitesimal and used what is now known as
Rolle's theorem. In the 14th century,
Madhava of Sangamagrama developed
infinite series expansions, now called
Taylor series, of functions such as
sine,
cosine,
tangent and
arctangent. Alongside his development of Taylor series of
trigonometric functions, he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. His followers at the
Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century.
Modern Foundations The modern foundations of mathematical analysis were established in 17th century Europe. Descartes's publication of
La Géométrie in 1637, which introduced the
Cartesian coordinate system, is considered to be the establishment of mathematical analysis. It would be a few decades later that
Newton and
Leibniz independently developed
infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the
calculus of variations,
ordinary and
partial differential equations,
Fourier analysis, and
generating functions. During this period, calculus techniques were applied to approximate
discrete problems by continuous ones.
Modernization In the 18th century,
Euler introduced the notion of a
mathematical function. Real analysis began to emerge as an independent subject when
Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become widely known until the 1870s. In 1821,
Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the
generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and
infinitesimals. Thus, his definition of continuity required an infinitesimal change in
x to correspond to an infinitesimal change in
y. He also introduced the concept of the
Cauchy sequence, and started the formal theory of
complex analysis.
Poisson,
Liouville,
Fourier and others studied partial differential equations and
harmonic analysis. The contributions of these mathematicians and others, such as
Weierstrass, developed the
(ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time,
Riemann introduced his theory of
integration, and made significant advances in complex analysis. Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a
continuum of
real numbers without proof.
Dedekind then constructed the real numbers by
Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a
complete set: the continuum of real numbers, which had already been developed by
Simon Stevin in terms of
decimal expansions. Around that time, the attempts to refine the
theorems of
Riemann integration led to the study of the "size" of the set of
discontinuities of real functions. Also, various
pathological objects, (such as
nowhere continuous functions, continuous but
nowhere differentiable functions, and
space-filling curves), commonly known as "monsters", began to be investigated. In this context,
Jordan developed his theory of
measure,
Cantor developed what is now called
naive set theory, and
Baire proved the
Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic
set theory.
Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as
Lebesgue integration, which proved to be a big improvement over Riemann's.
Hilbert introduced
Hilbert spaces to solve
integral equations. The idea of
normed vector space was in the air, and in the 1920s
Banach created
functional analysis. == Important concepts ==