Mathematical analysis

• 1. Introduction

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  Mathematical analysis begins with a paradox about a runner who can never reach the finish line. To cross a distance, the runner must first cover half of it, then half of what remains, then half again, forever. Zeno's paradox of the dichotomy hides an infinite geometric sum inside it. Strictly speaking, the point of the paradox is to deny that such a sum can exist at all. Yet the tools to tame that infinity would take more than two thousand years to mature. Analysis is the branch of mathematics that deals with continuous functions, limits, differentiation, integration, infinite sequences, and series. How did a discipline born from calculus learn to measure the area of a circle, fill the gaps between rational numbers, and study functions so strange that mathematicians called them monsters? The answers run from ancient Greece to a Banach space.

• 2. Ancient Foundations

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  Eudoxus and Archimedes used the method of exhaustion to compute the area and volume of regions and solids. They worked with limits and convergence in an explicit but informal way, long before any formal definition existed. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work that stayed lost until it was rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui applied the method of exhaustion in the 3rd century CE to find the area of a circle. Jain literature suggests that Hindus possessed formulae for the sum of arithmetic and geometric series as early as the 4th century BCE. Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra. Zu Chongzhi found the volume of a sphere in the 5th century, by a method later called Cavalieri's principle. In the 12th century, the Indian mathematician Bhāskara II worked with infinitesimals and used what is now known as Rolle's theorem.

• 3. The Kerala Series

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  Madhava of Sangamagrama, working in the 14th century, developed infinite series expansions for sine, cosine, tangent, and arctangent. These expansions are now called Taylor series, after a later European naming. Madhava did more than write the series down. He estimated the magnitude of the error left over when the series is cut short, and he gave a rational approximation of some infinite series. His followers at the Kerala School of Astronomy and Mathematics extended this work up to the 16th century. Their results sat at the edge of calculus centuries before Europe arrived at the same ideas.

• 4. The Modern Foundations

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  Fermat and Descartes developed analytic geometry, the precursor to modern calculus. Fermat's method of adequality let him find the maxima and minima of functions and the tangents of curves. Descartes published La Géométrie in 1637 and introduced the Cartesian coordinate system. That publication is considered the establishment of mathematical analysis. A few decades later, Newton and Leibniz independently developed infinitesimal calculus. Applied work through the 18th century pushed calculus into new territory: the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. Mathematicians used these techniques to approximate discrete problems by continuous ones.

• 5. Putting Calculus On Firm Ground

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  Euler introduced the notion of a mathematical function in the 18th century. Bernard Bolzano gave the modern definition of continuity in 1816, but his work stayed obscure until the 1870s. In 1821, Cauchy began putting calculus on a firm logical foundation. He rejected the principle of the generality of algebra, which earlier mathematicians, particularly Euler, had used freely. Cauchy required an infinitesimal change in x to correspond to an infinitesimal change in y. He introduced the Cauchy sequence and started the formal theory of complex analysis. Poisson, Liouville, and Fourier studied partial differential equations and harmonic analysis. Weierstrass and others developed the epsilon-delta definition of limit, which founded the modern field. Around the same time, Riemann introduced his theory of integration and advanced complex analysis.

• 6. Building The Real Numbers

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  Toward the end of the 19th century, mathematicians grew uneasy that they had assumed a continuum of real numbers without proof. Dedekind constructed the real numbers using Dedekind cuts, which formally define irrational numbers and fill the gaps between rational numbers. Simon Stevin had already developed this continuum in terms of decimal expansions. Attempts to refine the theorems of Riemann integration led to study of the size of the set of discontinuities of real functions. Pathological objects then came under investigation, including nowhere continuous functions, continuous but nowhere differentiable functions, and space-filling curves. Mathematicians called them monsters. Jordan developed his theory of measure, Cantor developed naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using axiomatic set theory. Lebesgue improved measure theory and introduced Lebesgue integration, a major advance over Riemann's. Hilbert introduced Hilbert spaces to solve integral equations, and in the 1920s Banach created functional analysis.

• 7. The Branches Of Analysis

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  Real analysis, traditionally the theory of functions of a real variable, deals with the real numbers and real-valued functions, including convergence, limits, continuity, and smoothness. Complex analysis investigates functions of complex numbers and reaches into algebraic geometry, number theory, hydrodynamics, thermodynamics, electrical engineering, and quantum field theory. Because the real and imaginary parts of an analytic function satisfy Laplace's equation, complex analysis applies widely to two-dimensional problems in physics. Functional analysis studies vector spaces carrying limit-related structure, such as an inner product, norm, or topology, and the linear operators acting on them. Its roots lie in spaces of functions and transformations like the Fourier transform. Harmonic analysis represents functions and signals as superpositions of basic waves, with applications from music theory to tidal analysis and neuroscience. Free or noncommutative analysis, a sub-branch of functional analysis, works with noncommutative variables and connects to free probability theory.

• 8. Measuring And Approximating

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  A measure assigns a number to each suitable subset of a set, interpreted as its size, generalizing length, area, and volume. The Lebesgue measure of the interval in the real numbers equals its everyday length, specifically 1. A measure must assign 0 to the empty set and be countably additive, so a large subset split into disjoint smaller pieces has a measure equal to the sum of those pieces. Trying to size every subset consistently yields only trivial examples like the counting measure. The fix is to define measure only on measurable subsets, which form a sigma-algebra closed under countable unions, intersections, and complements. Non-measurable sets in Euclidean space are badly mixed up with their complement, and their existence is a non-trivial consequence of the axiom of choice. Numerical analysis takes a different stance entirely. It does not seek exact answers, because exact answers are often impossible in practice. Instead it pursues approximate solutions while keeping reasonable bounds on errors, reaching from celestial mechanics to the simulation of living cells for medicine and biology.