Mathematical analysis,
which mathematicians refer to simply as
analysis
, has its
beginnings in the rigorous formulation of
infinitesimal calculus
. It is a branch of
pure
mathematics
that includes the theories of
differentiation
,
integration
and
measure
,
limits,
infinite series
,
[1]
and
analytic functions
. These theories are often studied in the context
of
real numbers
,
complex numbers
, and real and complex
functions
. However, they can also
be defined and studied in any
space
of mathematical objects that has a definition
of
nearness
(a
topological space
) or, more specifically,
distance
(a
metric space
).
Contents
[
hide
]
1
History
2
Subdivisions
3
Topological spaces, metric spaces
4
Calculus of finite differences, discrete calculus or discrete analysis
5
See also
6
Notes
7
References
8
External links
[
edit
]
History
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
.
[2]
Later,
Greek mathematicians
such as
Eudoxus
and
Archimedes
made more
explicit, but informal, use of the concepts of limits and convergence when they used
the
method of exhaustion
to compute the area and volume of regions and solids.
[3]
In
India
,
the 12th century mathematician
Bhāskara II
gave examples of the
derivative
and used what is
now known as
Rolle's theorem
.
In the 14th century,
Madhava of Sangamagrama
developed
infinite series
expansions, like
the
power series
and the
Taylor series
, of functions such
as
sine
,
cosine
,
tangent
and
arctangent
. Alongside his development of the Taylor series of
the
trigonometric functions
, he also estimated the magnitude of the error terms created by
truncating these series and gave a rational approximation of an infinite series. His followers at
the
Kerala school of astronomy and mathematics
further expanded his works, up to the 16th
century.
In Europe, during the later half of the 17th century,
Newton
and
Leibniz
independently
developed
infinitesimal calculus
, which grew, with the stimulus of applied work that continued
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View Full Documentthrough 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.
In the 18th century,
Euler
introduced the notion of
mathematical function
.
[4]
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 Spring '11
 Stone
 Differential Equations, Mathematical analysis

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