# Get PDF Limits, Limits Everywhere: The Tools of Mathematical Analysis

It is in this way that the exponential function , the logarithm , the trigonometric functions and their inverses are extended to functions of a complex variable. Fourier series decomposes periodic functions or periodic signals into the sum of a possibly infinite set of simple oscillating functions, namely sines and cosines or complex exponentials.

### Module Overview

Integration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface. The basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as the method of exhaustion. Generally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed.

By considering approximations consisting of a larger and larger "infinite" number of smaller and smaller "infinitesimal" pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value.

The spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann or Darboux sums converge to a common value as thinner and thinner rectangular slices "refinements" are considered. Though the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind.

Compared to the Riemann integral, the more sophisticated Lebesgue integral allows area or length, volume, etc.

Calculus 1 Lecture 1.1: An Introduction to Limits

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. When the chosen tags give the maximum respectively, minimum value of each interval, the Riemann sum is known as the upper respectively, lower Darboux sum.

A function is Darboux integrable if the upper and lower Darboux sums can be made to be arbitrarily close to each other for a sufficiently small mesh. Although this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal.

In fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former. The fundamental theorem of calculus asserts that integration and differentiation are inverse operations in a certain sense. Lebesgue integration is a mathematical construction that extends the integral to a larger class of functions; it also extends the domains on which these functions can be defined.

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The concept of a measure , an abstraction of length, area, or volume, is central to the definition of the Lebesgue integral and is important to the study of probability theory. For a construction of the Lebesgue integral, the main article on Lebesgue integration should be consulted. Distributions or generalized functions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation , integration and sequences of functions.

By definition, real analysis focuses on the real numbers , often including positive and negative infinity to form the extended real line. Real analysis is closely related to complex analysis , which studies broadly the same properties of complex numbers. In complex analysis, it is natural to define differentiation via holomorphic functions , which have a number of useful properties, such as repeated differentiability, expressability as power series , and satisfying the Cauchy integral formula.

In real analysis, it is usually more natural to consider differentiable , smooth , or harmonic functions , which are more widely applicable, but may lack some more powerful properties of holomorphic functions.

However, results such as the fundamental theorem of algebra are simpler when expressed in terms of complex numbers. Techniques from the theory of analytic functions of a complex variable are often used in real analysis — such as evaluation of real integrals by residue calculus.

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4. Various ideas from real analysis can be generalized from the real line to broader or more abstract contexts. These generalizations link real analysis to other disciplines and subdisciplines, in many cases playing an important role in their development as distinct areas of mathematics. For instance, generalization of ideas like continuous functions and compactness from real analysis to metric spaces and topological spaces connects real analysis to the field of general topology , while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the study of Banach spaces , and Hilbert spaces as topics of importance in functional analysis.

Georg Cantor 's investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to naive set theory. The study of issues of convergence for sequences of functions eventually gave rise to Fourier analysis as a subdiscipline of mathematical analysis. Investigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of holomorphic functions and the inception of complex analysis as another distinct subdiscipline of analysis.

On the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract measure spaces , a fundamental concept in measure theory. Finally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of vector calculus , whose further generalization and formalization played an important role in the evolution of the concepts of differential forms and smooth differentiable manifolds in differential geometry and other closely related areas of geometry and topology.

Main article: Limit mathematics. Main article: Uniform convergence. Main article: Compactness. Main article: Continuous function.

Main article: Uniform continuity. The notion of limit and convergence are two key ideas on which rest most of modern Analysis. This module aims to present these notions building on the experience gained by students in first year Calculus module. The context of our study is: limits and convergences of sequences and series of real numbers, and sequences and series of functions.

These classical results will be applied to derive properties of continues and differentiable functions. The module will introduce tools that are of importance in applications, for instance, power series expansions of functions, etc. Completeness of the set of the reals. The Weierstrass M-test. A Whole Lot of Numbers 2. Let's Get Real 3.

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The Joy of Inequality 4. Bounds for Glory 6. Wonderful Numbers 8. Infinite Products 9. Continued Fractions How Infinite Can You Get? Constructing the Real Numbers Where to Next in Analysis?

## Limits, Limits Everywhere: The Tools of Mathematical Analysis

The Calculus The Binomial Theorem 2. The Language of Set Theory 3. Proof by Mathematical Induction 4.