| Mathematical Analysis |
Article Index for Mathematical |
Website Links For Mathematical |
Information AboutMathematical Analysis |
| CATEGORIES ABOUT MATHEMATICAL ANALYSIS | |
| mathematical analysismathematical analysis | |
| mathematics | |
| analysis | |
|
MOTIVATION The motivation for studying mathematical analysis in the wider context of topological or metric spaces is twofold:
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 .1 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.(Smith, 1958) 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 . 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",G. G. Joseph (1991). ''The crest of the peacock'', London. 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 Function s 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 Fraction s, term by term Integration , and the power series of the Radius , Diameter , Circumference , angle θ , π , and π/4. His followers at the Kerala School further expanded his works, up to the 16th Century . 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 Equation s, Fourier Analysis , and Generating Function s. During this period, calculus techniques were applied to approximate Discrete Problems by continuous ones.
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 s 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 Curve s) 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 Space s to solve Integral Equation s. 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 Equation s is now shared with other fields such as Dynamical System s, though the overlap with conventional analysis is large. NOTES REFERENCES
|
|
|