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Encyclopedia > Mathematical analysis

## Contents

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

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 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 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 â€“ 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 &#8212; 19th century &#8212; 20th century &#8212; 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 â€“ 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 (WeierstraÃŸ) (October 31, 1815 â€“ 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). ...

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] â€“ 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&#8211;2000 in the sense of the Gregorian calendar (1900&#8211;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, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, 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 KrakÃ³w, Austria-Hungary now Polandâ€“ August 31, 1945 in LwÃ³w, Soviet Union - occupied Poland), was an eminent Polish mathematician, one of the moving spirits of the LwÃ³w 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.

## 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.
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