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Encyclopedia > Mathematical analysis

Analysis has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function.[1] It also includes the theories of differentiation, integration and measure, infinite series[2], 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 is equipped with a definition of "nearness" (a topological space) or more specifically "distance" (a metric space). For other uses, see Calculus (disambiguation). ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... The limit of a sequence is one of the oldest concepts in mathematical analysis. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... For a non-technical overview of the subject, see Calculus. ... This article is about the concept of integrals in calculus. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... This article is about the idea of space. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

Contents

Motivation

The motivation for studying mathematical analysis in the wider context of topological or metric spaces is twofold:

  • First, the same basic techniques have proved applicable to a wider class of problems (e.g., the study of function spaces).
  • Second, and just as importantly, a greater understanding of analysis in more abstract spaces frequently proves to be directly applicable to classical problems. For example, in Fourier analysis, functions are expressed in terms of certain infinite series (of complex exponentials or trigonometric functions). Physically, this decomposition amounts to reducing an arbitrary (sound) wave to its frequency components. The "weights" or coefficients of the terms in the Fourier expansion of a function can be thought of as components of a vector in an infinite dimensional space known as a Hilbert space. Study of functions defined in this more general setting thus provides a convenient method of deriving results about the way functions vary in space as well as time or, in more mathematical terms, partial differential equations, where this technique is known as separation of variables.

Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ...

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.[3] 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.[4] In India, the 12th century mathematician Bhaskara conceived of differential calculus, and gave examples of the derivative and differential coefficient, along with a statement of what is now known as Rolle's theorem. Zeno of Elea (IPA:zɛnoʊ, ɛlɛɑː)(circa 490 BC? – circa 430 BC?) was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. ... “Arrow paradox” redirects here. ... Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... Another article concerns Eudoxus of Cyzicus. ... For other uses, see Archimedes (disambiguation). ... The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. ... (11th century - 12th century - 13th century - other centuries) As a means of recording the passage of time, the 12th century was that century which lasted from 1101 to 1200. ... Bhaskara (1114-1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician-astronomer. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... For a non-technical overview of the subject, see Calculus. ... The differential dy In calculus, a differential is an infinitesimally small change in a variable. ... In calculus, Rolles theorem states that if a function f is continuous on a closed interval and differentiable on the open interval , and then there is some number c in the open interval such that . Intuitively, this means that if a smooth curve is equal at two points then...


In the 14th century, the roots of mathematical analysis began with work done by Madhava of Sangamagrama, regarded by some as the "founder of mathematical analysis",[5] who developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine, tangent and arctangent. Along side 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. He further developed infinite continued fractions, term by term integration, and the power series of the radius, diameter, circumference, angle θ,[citation needed] π, and π/4. His followers at the Kerala School further expanded his works, up to the 16th century. This 14th-century statue from south India depicts the gods Shiva (on the left) and Uma (on the right). ... Madhavan (മാധവന്) of Sangamagramam (1350–1425) was a prominent mathematician-astronomer from Kerala, India. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... For other uses, see tangent (disambiguation). ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... This article is about the concept of integrals in calculus. ... This article is about an authentication, authorization, and accounting protocol. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... The circumference is the distance around a closed curve. ... Note: A theta probe is a device for measuring soil moisture. ... Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ...


In Europe, during the latter half of the 17th century, Newton and Leibniz independently developed 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. (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Leibniz redirects here. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ...


In the 18th century, Euler introduced the notion of mathematical function.[6] Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816.[7] In the 19th century, Cauchy helped to put calculus on a firm logical foundation by introducing the concept of the Cauchy sequence. He also 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 modern notion of mathematical rigor, thus founding the field of mathematical analysis (at least in the modern sense). (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... Bernard Bolzano Bernard (Bernhard) Placidus Johann Nepomuk Bolzano (October 5, 1781 – December 18, 1848) was a Bohemian mathematician, theologian, philosopher, logician and antimilitarist of German mother tongue. ... Year 1816 (MDCCCXVI) was a leap year starting on Monday (link will display the full calendar) of the Gregorian calendar (or a leap year starting on Saturday of the 12-day slower Julian calendar). ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... The plot of a Cauchy sequence shown in blue, as versus If the space containing the sequence is complete, the ultimate destination of this sequence, that is, the limit, exists. ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... Simeon Poisson. ... Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ... Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. ... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ...


In the middle of the century Riemann introduced his theory of integration. The last third of the 19th century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the "epsilon-delta" definition of limit. Then, 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 a mathematician creates irrational numbers that serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers. 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. Bernhard Riemann. ... This article is about the concept of integrals in calculus. ... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In mathematics, the word continuum sometimes denotes the real line. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic. ... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... Look up theorem in Wiktionary, the free dictionary. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... Continuous functions are of utmost importance in mathematics and applications. ...


Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be created. 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 axiomatic set theory. Lebesgue solved the problem of measure, and 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. In mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. ... In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. ... Weierstrass function may also refer to the Weierstrass elliptic function () or the Weierstrass sigma, zeta, or eta functions. ... Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube). ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... This article is about the mathematical topic. ... René-Louis Baire (born January 21, 1874, died July 5, 1932) was a French mathematician. ... The Baire category theorem is an important tool in general topology and functional analysis. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999... This article or section is in need of attention from an expert on the subject. ... Henri Léon Lebesgue (June 28, 1875 - July 26, 1941) was a French mathematician, most famous for his theory of integration. ... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... The 1920s is sometimes referred to as the Jazz Age or the Roaring Twenties, usually applied to America. ... Stefan Banach Stefan Banach (March 30, 1892 in Kraków, Austria-Hungary now Poland– August 31, 1945 in Lwów, Soviet Union - occupied Poland), was an eminent Polish mathematician, one of the moving spirits of the Lwów School of Mathematics in pre-war Poland. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...


Subdivisions

Mathematical analysis includes the following subfields.

Classical analysis would normally be understood as any work not using functional analysis techniques, and is sometimes also called hard analysis; it also naturally refers to the more traditional topics. The study of differential equations is now shared with other fields such as dynamical systems, though the overlap with conventional analysis is large. Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Look up Rigour in Wiktionary, the free dictionary. ... For a non-technical overview of the subject, see Calculus. ... This article is about the concept of integrals in calculus. ... For other senses of this word, see sequence (disambiguation). ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... It has been suggested that this article be split into multiple articles. ... P-adic analysis (p-adic analysis) is a branch of mathematics that deals with functions of p-adic numbers. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... Look up Rigour in Wiktionary, the free dictionary. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... The Lorenz attractor is an example of a non-linear dynamical system. ...


Notes

  1. ^ (Whittaker and Watson, 1927, Chapter III)
  2. ^ Edwin Hewitt and Karl Stromberg, "Real and Abstract Analysis", Springer-Verlag, 1965
  3. ^ Stillwell (2004). "Infinite Series", , 170. “Infinite series were present in Greek mathematics, [...]There is no question that Zeno's paradox of the dichotomy (Section 4.1), for example, concerns the decomposition of the number 1 into the infinite series 1/2 + 1/2^2 + 1/2^3 + 1/2^4 + ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing the infinite series 1 + 1/4 + 1/4^2 + 1/4^3 + ... = 4/3. Both these examples are special cases of the result we express as summation of a geometric series” 
  4. ^ (Smith, 1958)
  5. ^ G. G. Joseph (1991). The crest of the peacock, London.
  6. ^ Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America, 17. 
  7. ^ *Cooke, Roger (1997). "Beyond the Calculus", The History of Mathematics: A Brief Course. Wiley-Interscience, 379. ISBN 0471180823. “Real analysis began its growth as an independent subject with the introduction of the modern definition of continuity in 1816 by the Czech mathematician Bernard Bolzano (1781-1848).” 
  8. ^ Carl L. Devito, "Functional Analysis", Academic Press, 1978

References

  • Apostol, Tom M., Mathematical Analysis, 2nd ed. Addison-Wesley, 1974. ISBN 978-0201002881.
  • Nikol'skii, S. M., "Mathematical analysis", in Encyclopaedia of Mathematics, Michiel Hazewinkel (editor), Springer-Verlag (2002). ISBN 1-4020-0609-8.
  • Smith, David E., History of Mathematics, Dover Publications, 1958. ISBN 0-486-20430-8.
  • Stillwell, John (2004). Mathematics and its History, Second Edition, Springer Science + Business Media Inc.. ISBN 0387953361. 
  • Whittaker, E. T. and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1927. ISBN 0521588073.

  Results from FactBites:
 
Mathematical analysis - Wikipedia, the free encyclopedia (782 words)
In the 14th century, mathematical analysis originated with Madhava in South India, who developed the fundamental ideas of the infinite series expansion of a function, the power series, the Taylor series, and the rational approximation of an infinite series.
Mathematical analysis in Europe began in the 17th century, with the possibly independent invention of calculus by Newton and Leibniz.
In the 17th and 18th centuries, analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis and generating functions were developed mostly in applied work.
BIGpedia - Mathematical analysis - Encyclopedia and Dictionary Online (512 words)
Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions.
Historically, analysis originated in the 17th century, with the invention of calculus by Newton and Leibniz.
In the 17th and 18th centuries, analysis topics such as the calculus of variations, differential and partial differential equations, Fourier analysis and generating functions were developed mostly in applied work.
  More results at FactBites »

 
 

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