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  Analysis (Greek, ‘freeing up’), in mathematics, is the study of calculus and related topics. For a long time after calculus was revolutionized by Newton and Leibniz in the 17th century, when they discovered the fundamental theorem of calculus, it was regarded as something of a ‘poor relation’ of other branches of mathematics, something to be used only if absolutely necessary. The reason is that it was a subject which was not set up in any kind of rigorous way, and which used methods of proof depending on ‘infinitesimals’ and other vague and loosely defined concepts. (An infinitesimal was a number smaller in size than any other, but not actually zero; the existence of this kind of number seemed extremely unlikely.) The way that both the differential and integral calculi were derived also depended so strongly on infinitesimals (at least in Leibniz\'s formulation) that it did not appear that mathematicians would ever succeed in removing it. At any earlier time this would not have mattered much, but this was a period in which mathematicians began to seek ever increasing rigour. Michel Rolle (1652 - 1719) even taught that the calculus was a series of ingenious fallacies.

In the early years of the 19th century, Karl Weierstrass, Bernard Bolzano and Augustin-Louis Cauchy brought rigour to the subject by closer examination of the basic principles, and Weierstrass finally evolved a definition of the concept of ‘limit’ which did not involve the infinitesimal. Previously, the limit of a function at a point a was defined to be the value of that function infinitesimally close to a; the question was, how can a function be defined infinitesimally close to a point? By Weierstrass\'s new definition, the limit of the function f at the point a is the number y if for every t there is a w such that if the distance between x and a is less than w then the difference of f(x) from y is less than t. From this definition it becomes easy to define integration and differentiation; for example, the derivative of the function f at the point a is the limit at a (if it exists) of the quotient of the distance of f(x) from f(a) by the distance from x to f.

This success prompted an enormous growth in the subject of analysis in the 19th century. There were several different ways in which these ideas were generalized and developed. First, mathematicians began to experiment with generalizations of the concepts of the distance between two numbers and continuity (a function f is continuous at a if the limit of f at a is equal to f(a)); this led to the study of metrics (functions generalizing distance) and hence to topology and functional analysis (a marriage of the techniques of analysis and topology with those of linear algebra).

Second, Cauchy extended the specific generalization of analysis of real numbers to that of complex numbers. He proved the fundamental theorems of the new field of complex analysis, many of which seem extremely counter-intuitive, so that complex analysis is extremely powerful but a difficult technique to use. SMcL



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