## The history of mathematical analysis

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- Published: Thursday, 14 January 2016 11:20

History links the origins of mathematical analysis to the names of the great Greeks Eudoxus (near 408-355 BC) and Archimedes (near 287-212 BC) (the first one created the exhaustion method to calculate the area of planar figures, the other one used this method to find not only the area of planar figures, but also the volume of bodies, also he solved the problem of finding functions’ extremum using tangent lines).

These methods have been in the inventory of civilizations of different times in different regions for almost 20 centuries, but it was only at the end of of XVI – the beginning of XVII century that european researchers made the next step in updating ancient Greek methods. Johannes Kepler (1571-1670), Bonaventura Cavalieri (1598-1647) created the method of indivisibles, Evangelista Torricelli (1608-1647) - the kinematic method of tangents, and Pierre de Fermat (1601-1665) rejected the method of tangents for the determination of extreme values of function f (x) and offered a purely algebraic method.

Of course, this was facilitated by Rene Descartes (1596-1650) establishing the connection between algebra and geometry of curves. The second half of the XVII century became a flagstone date for mathematical analysis, when two genius scientists – the Englishman Isaac Newton (1642-1727) and the German Gottfried Leibniz (1646-1716) made a landmark discovery. It was the discovery of differential and integral calculus and ties between them. Although Newton obtained most results in the 60s and 70s, Leibniz was published first (1648). In this work, the foundations of differential calculus are stated and notation is introduced (dx, dy), which remained unchanged to the present day. In 1686 Leibniz’s second work sees the world, where the foundations of integral calculus are stated, and the integral sign is introduced.

The newly discovered methods were applied to the analysis of variables, introduced by geometrical means (composed of certain characters) or analytical expressions or as abstractions of different kinds of continuous mechanical motion (Newton). There was a need to unify objects to which new operations could apply. The concept of function became such generalizing concept. The term "function" first appeared in 1692 in Leibniz’s work, as an expression of dependence of length of segments linked to a curve, from the position of a point on the curve. In 1718 Johann Bernoulli (1667-1748) proposed to regard function as an analytical expression. The great founder of mathematical analysis Leonhard Euler (1707-1783) in his eight-volume course of analysis states that "analysis of the infinitesimal evolves around variables and their functions." Thus the main object of mathematical analysis was clearly defined in the XVIII century. As for methods of function investigation, the principal ones became differentiation (the study of the properties of functions using expansion into formal power series) and integration (finding the primary function).

However, both theoretical research and solution of practical problems required more accurate methods and primarily required an interpretation of substance if the infinitesimal and an explanation of legitimacy of the methods used. Once again a way out was found through interpretation of methods used by ancient Greeks, namely through realization that the boundary transitions effectuated in individual problems may be used to build algebraic limit theorem.

The first attempt was made by Newton (he also introduced a special term - "limes"). Nor did Euler overlook this important problem. However, these efforts were not well perceived by XVIII century mathematicians, primarily because of the absence of an algorithm for computing limits. It was only after Augustin-Louis Cauchy (1789-1857) suggested his own ε-δ instruments and created with it his own course of mathematical analysis, that the limit was recognized as the main method of analysis.

The system of logical justification of mathematical analysis was completed by Karl Weierstrass (1815-1897), who laid its foundation in the form of a strict theory of real numbers. In parallel with the theory of numerical functions, the theories of functions of complex variable, functions of many variables and their generalizations were being developed. The biggest achievement of XX century analysis was the creation of functional analysis, whose main purpose is to study the functions (operators) in which at least one variable takes the value in the infinitely dimensional space. The highest level of abstraction was reached in the analysis of functions defined by the so-called topological spaces.

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