# Mathematical analysis: Wikis

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# Encyclopedia

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure 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 has a definition of nearness (a topological space) or, more specifically, distance (a metric space).

## 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 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 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.

In the 18th century, Euler introduced the notion of mathematical function.[5] Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816.[6] 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).

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.

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 an 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.

## 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.

## Topological spaces, metric spaces

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

• Firstly, the same basic techniques have proved applicable to a wider class of problems (e.g., the study of function spaces).
• Secondly, 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 a certain infinite sum of trigonometric functions. Thus Fourier analysis might be used to decompose a sound into a unique combination of pure tones of various pitches. 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.
• Thirdly, the conditions needed to prove the particular result are stated more explicitly. The analyst then becomes more aware exactly what aspect of the assumption is needed to prove the theorem.

## 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". pp. 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 12 + 122 + 123 + 124 + ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing the infinite series 1 + 14 + 142 + 143 + ... = 43. Both these examples are special cases of the result we express as summation of a geometric series"
4. ^ (Smith, 1958)
5. ^ Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. pp. 17.
6. ^ *Cooke, Roger (1997). "Beyond the Calculus". The History of Mathematics: A Brief Course. Wiley-Interscience. pp. 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)."
7. ^ Carl L. Devito, "Functional Analysis", Academic Press, 1978

## References

• Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill Publishing Co.; 3Rev Ed edition (September 1, 1976), ISBN 978-0070856134.
• 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 ed.). Springer Science + Business Media Inc.. ISBN 0387953361.
• Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, fourth edition, Cambridge University Press, 1927. ISBN 0521588073.
• Jean-Étienne Rombaldi, Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques
• K.G. Binmore, The foundations of analysis: a straightforward introduction
• Richard Johnsonbaugh & W. E. Pfaffenberger, Foundations of mathematical analysis
• Aleksandrov, A. D., Kolmogorov, A. N., Lavrent'ev, M. A. (Ed.), Translated by S. H. Gould, K. A. Hirsch and T. Bartha. Translation edited by S. H. Gould; "Mathematics, its Content, Methods, and Meaning", The M.I.T Press; Published in cooperation with the American Mathematical Society, Second Edition, Fourth Printing, 1984 Cambridge, Massachusetts, Library of Congress Card Number: 64-7547.

# Simple English

Mathematical analysis is a part of mathematics. It is often shortened to analysis. It looks at functions, sequences and series. These have useful properties and characteristics that can be used in engineering. The mathematical analysis is about continuous functions, differential calculus and integration.[1]

Gottfried Wilhelm Leibniz and Isaac Newton developed most of the basis of mathematical analysis.

## Parts of mathematical analysis

### Limits

An example for mathematical analysis is limits. Limits are used to see what happens very close to things. Limits can also be used to see what happens when things get very big. For example, $\frac\left\{1\right\}\left\{n\right\}$ is never zero, but as n gets bigger $\frac\left\{1\right\}\left\{n\right\}$ gets close to zero. The limit of $\frac\left\{1\right\}\left\{n\right\}$ as n gets bigger is zero. It is usually said "The limit of $\frac\left\{1\right\}\left\{n\right\}$ as n goes to infinity is zero". It is written as $\lim_\left\{n\to\infty\right\} \frac\left\{1\right\}\left\{n\right\}=0$.

The counterpart would be $\left\{2\right\} \times \left\{n\right\}$. When the $\left\{n\right\}$ gets bigger, the limit goes to infinity. It is written as $\lim_\left\{n\to\infty\right\}$

\times{n}=\infty.

The fundamental theorem of algebra can be proven from some basic results in complex analysis. It says that every polynomial $f\left(x\right)$ with real or complex coefficients has a complex root. A root is a number x which gives a solution $f\left(x\right)=0$. Some of these roots may be the same.

### Differential calculus

The function $f\left(x\right) = \left\{m\right\}\left\{x\right\} + \left\{c\right\}$ is a line. The $\left\{m\right\}$ shows the slope of the function and the $\left\{c\right\}$ the position of the function on the ordinate. With two points on the line, it is possible to calculate the slope $\left\{m\right\}$ with:

$m = \frac\left\{y_1 - y_0\right\}\left\{x_1 - x_0\right\}$.

A function of the form $f\left(x\right) = x^2$, which is not linear, cannot be calculated like above. It is only possible to calculate the slope by using tangents and secants. The tangent has two points and when the two points get closer, it turns into a secant.

The new formula is $m = \frac\left\{f\left(x_1\right) - f\left(x_0\right)\right\}\left\{x_1 - x_0\right\}$.

This is called difference quotient. The $x_1$ gets now closer to $x_0$. This can be expressed with the following formula:

$f\text{'}\left(x_0\right) = \lim_\left\{x\rightarrow x_0\right\}\frac\left\{f\left(x\right) - f\left(x_0\right)\right\}\left\{x - x_0\right\}$.

The result is called derivative or slope.

### Integration

The integration is about the calculation of areas.

$\int_\left\{a\right\}^\left\{b\right\} f\left(x\right)\, \mathrm\left\{d\right\}x$

The $a$ is the point where the area should start and the $b$ where the area ends.

## Other pages

Some topics in analysis are:

• Calculus
• Functional analysis
• Complex analysis

Some useful ideas in analysis are:

## References

1. Hartmut Seeger. Mathematik. Königswinter: Tandem Verlag. p. 17. ISBN 9 783833 107870.