Mathematical analysis
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Analysis is the branch of
These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
History
Ancient
Mathematical analysis formally developed in the 17th century during the
Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 BCE.[9]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.[10] In the 12th century, the Indian mathematician Bhāskara II used infinitesimal and used what is now known as Rolle's theorem.[11]
In the 14th century,
Modern
Foundations
The modern foundations of mathematical analysis were established in 17th century Europe.
Modernization
In the 18th century,
Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a
Also, various
Important concepts
Metric spaces
In
Much of analysis happens in some metric space; the most commonly used are the
Formally, a metric space is an ordered pair where is a set and is a
such that for any , the following holds:
- , with equality if and only if (identity of indiscernibles),
- (symmetry), and
- (triangle inequality).
By taking the third property and letting , it can be shown that (non-negative).
Sequences and limits
A sequence is an ordered list. Like a
One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit. Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as n becomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the distance between an and x approaches 0 as n → ∞, denoted
Main branches
Calculus
Real analysis
Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the
Complex analysis
Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis that investigates
Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of
Harmonic analysis
Harmonic analysis is a branch of mathematical analysis concerned with the representation of
Differential equations
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders.[21][22][23] Differential equations play a prominent role in engineering, physics, economics, biology, and other disciplines.
Differential equations arise in many areas of science and technology, specifically whenever a
Measure theory
A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.[24] In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the -dimensional Euclidean space . For instance, the Lebesgue measure of the interval in the
Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set . It must assign 0 to the
Numerical analysis
Numerical analysis is the study of
Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.
Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
Vector analysis
Vector analysis is a branch of mathematical analysis dealing with values which have both magnitude and direction. Some examples of vectors include velocity, force, and displacement. Vectors are commonly associated with scalars, values which describe magnitude.[26]
Scalar analysis
Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have.
Tensor analysis
Other topics
- Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions.
- Harmonic analysis deals with the representation of functions or signals as the superposition of basic waves.
- Geometric analysis involves the use of geometrical methods in the study of partial differential equations and the application of the theory of partial differential equations to geometry.
- Clifford analysis, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions.
- p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
- Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimalsand infinitely large numbers.
- Computable analysis, the study of which parts of analysis can be carried out in a computable manner.
- stochastic processes.
- Set-valued analysis– applies ideas from analysis and topology to set-valued functions.
- Convex analysis, the study of convex sets and functions.
- idempotent semiring, where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A.
- min-plus algebra).
- constructive, rather than classical, logic and set theory.
- Intuitionistic analysis, which is developed from constructive logic like constructive analysis but also incorporates choice sequences.
- Paraconsistent analysis, which is built upon a foundation of paraconsistent, rather than classical, logic and set theory.
- Smooth infinitesimal analysis, which is developed in a smooth topos.
Applications
Techniques from analysis are also found in other areas such as:
Physical sciences
The vast majority of
Functional analysis is also a major factor in quantum mechanics.
Signal processing
When processing signals, such as
Other areas of mathematics
Techniques from analysis are used in many areas of mathematics, including:
- Analytic number theory
- Analytic combinatorics
- Continuous probability
- Differential entropy in information theory
- Differential games
- Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.
- Differentiable manifolds
- Differential topology
- Partial differential equations
Famous Textbooks
- Foundation of Analysis: The Arithmetic of Whole Rational, Irrational and Complex Numbers, by Edmund Landau
- Introductory Real Analysis, by Andrey Kolmogorov, Sergei Fomin[28]
- Differential and Integral Calculus (3 volumes), by
- The Fundamentals of Mathematical Analysis (2 volumes), by
- A Course Of Mathematical Analysis (2 volumes), by Sergey Nikolsky[34][35]
- Mathematical Analysis (2 volumes), by Vladimir Zorich[36][37]
- A Course of Higher Mathematics (5 volumes, 6 parts), by Vladimir Smirnov[38][39][40][41][42]
- Differential And Integral Calculus, by Nikolai Piskunov[43]
- A Course of Mathematical Analysis, by Aleksandr Khinchin[44]
- Mathematical Analysis: A Special Course, by Georgiy Shilov[45]
- Theory of Functions of a Real Variable (2 volumes), by Isidor Natanson[46][47]
- Problems in Mathematical Analysis, by Boris Demidovich[48]
- Problems and Theorems in Analysis (2 volumes), by George Polya, Gabor Szegö[49][50]
- Mathematical Analysis: A Modern Approach to Advanced Calculus, by Tom Apostol[51]
- Principles of Mathematical Analysis, by Walter Rudin[52]
- Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias Stein[53]
- Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, by Lars Ahlfors[54]
- Complex Analysis, by Elias Stein[55]
- Functional Analysis: Introduction to Further Topics in Analysis, by Elias Stein[56]
- Analysis (2 volumes), by Terence Tao[57][58]
- Analysis (3 volumes), by Herbert Amann, Joachim Escher[59][60][61]
- Real and Functional Analysis, by Vladimir Bogachev, Oleg Smolyanov[62]
- Real and Functional Analysis, by Serge Lang[63]
See also
- Constructive analysis
- History of calculus
- Hypercomplex analysis
- Multiple rule-based problems
- Multivariable calculus
- Paraconsistent logic
- Smooth infinitesimal analysis
- Timeline of calculus and mathematical analysis
References
- ^ Edwin Hewitt and Karl Stromberg, "Real and Abstract Analysis", Springer-Verlag, 1965
- Stillwell, John Colin. "analysis | mathematics". Encyclopædia Britannica. Archivedfrom the original on 2015-07-26. Retrieved 2015-07-31.
- ^ ISBN 978-0821826232. Archivedfrom the original on 2016-05-17. Retrieved 2015-11-15.
- ISBN 978-0387953366.
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 + 1⁄22 + 1⁄23 + 1⁄24 + ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing the infinite series 1 + 1⁄4 + 1⁄42 + 1⁄43 + ... = 4⁄3. Both these examples are special cases of the result we express as summation of a geometric series
- ISBN 978-0486204307.
- ISBN 978-1898563990. Archivedfrom the original on 2016-06-11. Retrieved 2015-11-15.
- ISBN 978-0-7923-3463-7. Archived from the original on 2016-06-17. Retrieved 2015-11-15., Chapter, p. 279 Archived 2016-05-26 at the Wayback Machine
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- ^ K. B. Basant, Satyananda Panda (2013). "Summation of Convergent Geometric Series and the concept of approachable Sunya" (PDF). Indian Journal of History of Science. 48: 291–313.
- ISBN 978-0763759957. Archivedfrom the original on 2019-04-21. Retrieved 2015-11-15.
- S2CID 3958488
- ^
Rajagopal, C. T.; Rangachari, M. S. (June 1978). "On an untapped source of medieval Keralese Mathematics". Archive for History of Exact Sciences. 18 (2): 89–102. S2CID 51861422.
- ^ Pellegrino, Dana. "Pierre de Fermat". Archived from the original on 2008-10-12. Retrieved 2008-02-24.
- ^ Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. p. 17.
- ISBN 978-0471180821.
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)
- ISBN 978-0070542358.
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- ISBN 978-0070006577.
- ISBN 978-0070542365.
- ISBN 978-0387972459. Archivedfrom the original on 2020-09-09. Retrieved 2016-02-11.
- ISBN 978-0486603490.
- ISBN 0486495108
- ISBN 978-0821807729.
- ISBN 978-0821869192. Archivedfrom the original on 2019-12-27. Retrieved 2018-10-26.
- ISBN 978-0070287617.
- ^ Borisenko, A. I.; Tarapov, I. E. (1979). Vector and Tensor Analysis with Applications (Dover Books on Mathematics). Dover Books on Mathematics.
- ISBN 978-0139141010.
- ^ "Introductory Real Analysis". 1970.
- ^ "Курс дифференциального и интегрального исчисления. Том I". 1969.
- ^ "Основы математического анализа. Том II". 1960.
- ^ "Курс дифференциального и интегрального исчисления. Том III". 1960.
- ^ The Fundamentals of Mathematical Analysis: International Series in Pure and Applied Mathematics, Volume 1. ASIN 0080134734.
- ^ The Fundamentals of Mathematical Analysis: International Series of Monographs in Pure and Applied Mathematics, Vol. 73-II. ASIN 1483213153.
- ^ "A Course of Mathematical Analysis Vol 1". 1977.
- ^ "A Course of Mathematical Analysis Vol 2". 1987.
- ^ Mathematical Analysis I. ASIN 3662569558.
- ^ Mathematical Analysis II. ASIN 3662569663.
- ^ "A Course of Higher Mathematics Vol 3 1 Linear Algebra". 1964.
- ^ "A Course of Higher Mathematics Vol 2 Advanced Calculus". 1964.
- ^ "A Course of Higher Mathematics Vol 3-2 Complex Variables Special Functions". 1964.
- ^ "A Course of Higher Mathematics Vol 4 Integral and Partial Differential Equations". 1964.
- ^ "A Course of Higher Mathematics Vol 5 Integration and Functional Analysis". 1964.
- ^ "Differential and Integral Calculus". 1969.
- ^ "A Course of Mathematical Analysis". 1960.
- ^ Mathematical Analysis: A Special Course. ASIN 1483169561.
- ^ "Theory of functions of a real variable (Teoria functsiy veshchestvennoy peremennoy, chapters I to IX)". 1955.
- ^ "Theory of functions of a real variable =Teoria functsiy veshchestvennoy peremennoy". 1955.
- ^ "Problems in Mathematical Analysis". 1970.
- ^ Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions. ASIN 3540636404.
- ^ Problems and Theorems in Analysis II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry. ASIN 3540636862.
- ^ Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd Edition. ASIN 0201002884.
- ^ Principles of Mathematical Analysis. ASIN 0070856133.
- ^ Real Analysis: Measure Theory, Integration, and Hilbert Spaces. ASIN 0691113866.
- ISBN 978-0070006577.
- ^ Complex Analysis. ASIN 0691113858.
- ^ Functional Analysis: Introduction to Further Topics in Analysis. ASIN 0691113874.
- ^ Analysis I: Third Edition. ASIN 9380250649.
- ^ Analysis II: Third Edition. ASIN 9380250657.
- ISBN 978-3764371531.
- ISBN 978-3764374723.
- ISBN 978-3764374792.
- ISBN 978-3030382216.
- ISBN 978-1461269380.
Further reading
- The M.I.T. Press / American Mathematical Society.
- ISBN 978-0201002881.
- Binmore, Kenneth George (1981) [1981]. The foundations of analysis: a straightforward introduction. Cambridge University Press.
- M. Dekker.
- ISBN 978-1402006098.
- ISBN 978-8820726751.
- Rombaldi, Jean-Étienne (2004). Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques (in French). ISBN 978-2868836816.
- ISBN 978-0070542358.
- ISBN 978-0070542341.
- ISBN 0521067944. (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992)
- "Real Analysis – Course Notes" (PDF). Archived (PDF) from the original on 2007-04-19.