Mathematical analysis

The study of the Mandelbrot group, which is a fractal object with statistical self-similarity, involves various areas of the mathematical analysis: the analysis of convergence, the theory of measurement, geometry and the theory of probability and statistics.

Numerical assemblies

Mathematical analysis is a branch of mathematics that studies numerical sets (real numbers, complex numbers) from both an algebraic and a topological point of view, as well as the functions between these sets and derived constructions. It begins to develop from the beginning of the rigorous formulation of limits and studies concepts such as continuity, integration and different types of derivation.

One of the differences between algebra and analysis is that the latter resorts to constructions involving sequences of an infinite number of elements, while algebra is usually finitistic.

History

Archimedes used the exhaustion method to calculate the area within a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.

In the Old Age

Greek mathematicians such as Eudoxus of Cnidos and Archimedes made informal use of the concepts of limit and convergence when they used the exhaustive method to calculate the area and volume of regions and solids. In fact, the number π was approximated using the exhaustive method. In 12th century India the mathematician Bhaskara conceived elements of the differential calculus, as well as the concept of what we now know as Rolle's theorem.

The first results of analysis were implicitly present in the early days of ancient Greek mathematics. For example, an infinite geometric sum is implicit in Zeno's dichotomy paradox. [4] 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 Exhaustive Method to calculate the area and volume of regions and solids.[5] Explicit use of infinitesimals appears in Archimedes' Method of Mechanical Theorems, a work rediscovered in the 20th century. [6] In Asia, the Chinese mathematician Liu Hui used the exhaustion method in the III century AD. C. to find the area of a circle. [7] From Jain literature, it appears that the Hindus were in possession of the formulas for addition of arithmetic and geometry as early as the 18th century IV a. C. [8] Ācārya Bhadrabāhu uses the sum of a sum of a geometric series in his Kalpasūtra in 433 BC [9] In Hindu mathematics, particular cases of arithmetic and have been found to occur implicitly in Vedic literature since the year 2000 BCE c. In the 14th century, the Indian mathematician Madhava developed fundamental ideas such as expansion of infinite series, power series, series Taylor and the rational approximation of infinite series. He further developed the Taylor series of trigonometric functions—sine, cosine, tangent—and estimated the magnitude of miscalculations by truncating these series. He also developed infinite continued fractions, term-by-term integration, and the power series of pi. His disciples of the Kerala School continued his work until the 16th century .

In Europe, in the 17th century century, the modern foundations of mathematical analysis were laid, in which Newton and Leibniz invented calculus. We now know that Newton developed the calculus some ten years before Leibniz. The latter did so in 1675 and published his work in 1684, approximately twenty years before Newton decided to do the same with his work. Newton had only communicated the news to a few of his colleagues and Halley's instigation for Newton to publish his work earlier was useless. This attitude served as the basis for creating an unpleasant controversy over the patronage of the idea; a discussion that could have been avoided if another great mathematician, Fermat, had not also had the inexplicable habit of not making his work public. In a letter from Fermat to Roberval, dated 22 October 1636, both analytic geometry and mathematical analysis are clearly described. Descartes also developed analytic geometry independently. In that century and in the 18th century, certain topics of analysis such as the calculus of variations, differential equations, and equations in partial derivatives, Fourier analysis, and generating functions were developed primarily for application work. Calculus techniques were successfully applied in the approximation of discrete problems by continuous ones.

In the Middle Ages

In the 5th century, Zu Chongzhi established a method later called Cavalieri's Principle for finding the volume of a sphere.[10] The Indian mathematician Bhaskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. [eleven].

In the 14th century, Madhava of Sangamagrama developed expansions of infinite series, such as the power series and the Taylor, of functions such as sine, cosine, tangent, and arctangent[12] Along with his development of the Taylor series of trigonometric functions, he also estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series. Followers of his at the Kerala School of Astronomy and Mathematics extended his works up to the 16th century .

In the Modern Age

Approximation of a discontinuous "square wave" function through a series of continuous and differential trigonometric functions.

Basics

The modern foundations of mathematical analysis were laid in Europe in the 17th century, when Descartes and Fermat independently developed analytical geometry, precursor of modern calculus. Fermat's adequacy method allowed him to determine the maxima and minima of the functions and the tangents of the curves.[13] The publication of Descartes' La Géométrie in 1637, which introduced the Cartesian coordinate system, is considered the establishment of mathematical analysis. A few decades later, Newton and Leibniz independently developed calculus, which grew, with the encouragement of applied work that continued through the eighteenth century, into topics of analysis such as the calculus of variations, ordinary and partial differential equations., Fourier analysis and generating functions. During this period, calculus techniques were applied to approximate discrete problems to continuous ones.

Modernization

Throughout the 18th century the definition of the concept of a function was the subject of debate among mathematicians. In the 19th century, Cauchy was the first to establish calculus on a firm logical foundation by using the concept of sequence of Cauchy. He also initiated the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis.

In the middle of that century, Riemann introduced his theory of integration. In the last third of the century XIX Weierstrass leads to the arithmetization of the analysis, since he thought that geometric reasoning was misleading by nature, and introduces the ε-δ definition of limit. Then mathematicians began to wonder if they were assuming the existence of some continuum of real numbers without proving its existence. Dedekind then constructs the real numbers using Dedekind cuts. Around the same time, attempts to refine the Riemann integration theorems led to the study of the "size" of discontinuity sets of real functions.

Also, "monster" functions (functions continuous nowhere, functions continuous but not differentiable anywhere, Space-Filling Curve, Peano Curve) began to emerge. In this context, Jordan developed his measurement theory, Cantor developed what is now called set theory, and Baire proves Baire's category theorem. At the beginning of the XX century, calculus is formalized using set theory. Lebesgue solves the measurement problem and Hilbert introduces Hilbert spaces to solve integral equations. The idea of normed vector spaces was in the making and in the 1920s Banach created functional analysis.

Important concepts

Metric space

In mathematics, a metric space' is a set in which a notion of distance (called a metric) is defined between the elements of the set.

Much of the analysis is done in some metric space; the most used are the real line, the complex plane, the Euclidean space, other vector spaces and the integers. Examples of metricless analysis include measure theory (which describes size rather than distance) and functional analysis, which studies topological vector spaces that need not have any sense of distance.

Formally, a metric space is an orderly pair ${displaystyle (M,d)}$ where ${displaystyle M}$ It's a set and ${displaystyle d}$ is a metric ${displaystyle M}$, that is, a function

${displaystyle dcolon Mtimes Mrightarrow mathbb {R} }$

for any ${displaystyle x,y,zin M}$ the following remains:

1. ${displaystyle d(x,y)=0}$ Yes and only if ${displaystyle x=y}$ (Identity of indiscerns),
2. ${displaystyle d(x,y)=d(y,x)}$ (symmetry), and
3. ${displaystyle d(x,z)leq d(x,y)+d(y,z)}$ (Triangular inequality).

Taking the third property and leaving ${displaystyle z=x}$, you can prove that ${displaystyle d(x,y)geq 0}$ (non-negative).

Sequences and limits

A sequence is an ordered list. Like a set, it contains members (also called elements or terms). Unlike a set, order matters and the exact same elements can appear multiple times at different positions in the sequence. More precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.

One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit. Continuing informally, a (simple infinite) sequence has a limit if it approaches some point x, called the limit, of n when n it gets very big. That is, for an abstract sequence (a_n) (with n going from 1 to infinity understood) the distance between a_n and x approaches 0 as n → ∞ is indicated as follows:

${displaystyle lim _{nto infty }a_{n}=x. !$

Main branches

Actual analysis

Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis that deals with real numbers and functions of real value of a real variable. In particular, it is concerned with the analytic properties of real functions and of sequences, including the convergence and limits of sequences of real numbers, the computation of real numbers, and the continuity, smoothness, and related properties of real-valued functions.

Complex analysis

Complex analysis, traditionally known as theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and particularly, quantum field theory.

Complex analysis is concerned especially with analytic functions of complex variables (or, more generally, with 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 vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, topology)., etc.) and the linear operators that act on these spaces and respect these structures in an appropriate sense. The historical roots of functional analysis are found in the study of the spaces of functions and in the formulation of the properties of the transformations of functions such as the Fourier transformation as transformations that define continuous, unitary operators, etc. between function spaces. This point of view turned out to be particularly useful for the study of differentials and integral equations.

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. 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 when a deterministic relationship is known or postulated involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives). This is illustrated in classical mechanics, where the movement of a body is described by its position and speed by varying the value of time. Newton's Laws allow (given position, velocity, acceleration, and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called the equation of motion) can be solved explicitly.

Measure Theory

One measure about a set is a systematic way of assigning a number to each appropriate subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area and volume. A particularly important example is the measurement of Lebesgue in an euclidian space, which assigns the conventional length, area and volume of euclidian geometry to appropriate subsets of euclidian space ${displaystyle n}$- dimensional. ${displaystyle mathbb {R} ^{n}$. For example, the Lebesgue measurement of the interval ${displaystyle left[0.1right]}$ in real numbers is its length in the everyday meaning of the word - specifically, 1.

Technically, a measure is a function that assigns a real number not negative or +∞ to (certs) subsets of a set ${displaystyle X}$. It must assign 0 to the empty set and be (contiguously) additive: the measure of a "great" subset that can be broken down into a finite (or accounting) number of "small" subsets, is the sum of the measures of the "smallest" subsets. In general, if you want to associate a size consistent a each subset of a given set while satisfying the other axioms of a measure, only trivial examples such as the counting measure are found. This problem was solved by defining the measure only in a sub-collection of all subsets; so-called subsets measurablewhich are required to form a ${displaystyle sigma }$. This means that the accounting union, the accounting intersection and the complement of measurable subsets are measurable. Non-measurable sets in an Euclidean space, on which the Lebesgue measure cannot be consistently defined, are necessarily complicated in the sense of being mis-mixed with its complement. Indeed, its existence is a non-trivial consequence of the axiom of choice.

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for problems in mathematical analysis (as opposed to discrete mathematics).

Modern numerical analysis does not look for 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 physical sciences, but in the 21st century, the life sciences, and even the arts have embraced elements of scientific calculation. 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 living cell simulation for medicine and biology.

Vector analysis

Tensor Analysis

Subdivisions

There has been much discussion about how many and which branches would make up the analysis, since as the discipline develops, different branches that were previously independent end up forming part of the same body and sometimes independent branches seem to emerge. The mathematical analysis includes the following fields:

• Real analysis, that is, the formally rigorous study of the derivatives and integrals of the actual-valued functions, including the study of limits and series.
  • Ordinary differential equations and partial derivatives.
  • Differential geometry, which extends the methods of real analysis on euclid spaces to more general topological spaces.
  • Integration and theory of the measure, which generalizes the concept of integral and measuring calculation.
  • Theory of probability, which largely shares formalism with the theory of measure, from the axiomization of Kolmogórov.
  • Numerical analysis, commissioned to design algorithms for, through simple mathematical numbers and rules, simulate more complex mathematical processes applied to the real world.
  • Non-standard analysis, which investigates certain hyperreal numbers and their functions, and gives a rigorous treatment of infinitesimal and infinitely large numbers.

• Non-real analysiswhich extends the actual analysis to different bodies of the actual numbers.
  • Complex analysis, which studies functions that go from the complex plane to itself and that are complex-differentiable, the holomorphic functions.
  • P-addic analysis, analysis in the context of the p-addic numbers, which differs interestingly and surprisingly from their real and complex counterpart.
• Functional analysiswhich studies spaces and functions and introduces concepts such as Banach spaces and Hilbert spaces.
  • Harmonic analysis, which deals with the Fourier series and their abstractions^{[chuckles]required]} and subarmonic analytical additions.
  • Analytical geometry, or geometry of the coordinates, which corresponds to n-uplas with points and sets of n-uplas with geometric places.
• Topology
  • Differential topology, which generalizes the actual and complex analysis of several variables to more general topological spaces than ${displaystyle scriptstyle mathbb {R} ^{n}$ or ${displaystyle scriptstyle mathbb {C} ^{n}$.
  • Algebraic topology
  • Lie Groups
• Other areas:
  • Algebraic geometry