Mathematical analysis

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.
Archimedes used the method of exhaustion to compute the area inside 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.

Branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

- Mathematical analysis
A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.

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Alpha

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Mathematics

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
The quadratic formula expresses concisely the solutions of all quadratic equations
Rubik's cube: the study of its possible moves is a concrete application of group theory
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

Archimedes used the method of exhaustion to calculate the area under a parabola.

Calculus

Mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

Mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

Archimedes used the method of exhaustion to calculate the area under a parabola.
Alhazen, 11th-century Arab mathematician and physicist
Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.
Maria Gaetana Agnesi
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.

In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits.

A symbol for the set of real numbers

Real number

Value of a continuous quantity that can represent a distance along a line .

Value of a continuous quantity that can represent a distance along a line .

A symbol for the set of real numbers
Real numbers

The completeness property of the reals is the basis on which calculus, and, more generally mathematical analysis are built.

The failure of a function to be continuous at a point is quantified by its oscillation.

Continuous function

Function such that a continuous variation of the argument induces a continuous variation of the value of the function.

Function such that a continuous variation of the argument induces a continuous variation of the value of the function.

The failure of a function to be continuous at a point is quantified by its oscillation.
The graph of a cubic function has no jumps or holes. The function is continuous.
For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.
A right-continuous function
A left-continuous function

Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers.

Schematic depiction of a function described metaphorically as a "machine" or "black box" that for each input yields a corresponding output

Function (mathematics)

Called the domain of the function and the set Y is called the codomain of the function.

Called the domain of the function and the set Y is called the codomain of the function.

Schematic depiction of a function described metaphorically as a "machine" or "black box" that for each input yields a corresponding output
The red curve is the graph of a function, because any vertical line has exactly one crossing point with the curve.
A function that associates any of the four colored shapes to its color.
The function mapping each year to its US motor vehicle death count, shown as a line chart
The same function, shown as a bar chart
Graph of a linear function
Graph of a polynomial function, here a quadratic function.
Graph of two trigonometric functions: sine and cosine.
Together, the two square roots of all nonnegative real numbers form a single smooth curve.
A composite function g(f(x)) can be visualized as the combination of two "machines".
A simple example of a function composition
Another composition. In this example, {{math|1=(g ∘ f )(c) = #}}.

Typically, this occurs in mathematical analysis, where "a function from X to Y " often refers to a function that may have a proper subset of X as domain.

An illustration of Desargues' theorem, a result in Euclidean and projective geometry

Geometry

Geometry is, with arithmetic, one of the oldest branches of mathematics.

Geometry is, with arithmetic, one of the oldest branches of mathematics.

An illustration of Desargues' theorem, a result in Euclidean and projective geometry
A European and an Arab practicing geometry in the 15th century
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
An illustration of Euclid's parallel postulate
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A tiling of the hyperbolic plane
Differential geometry uses tools from calculus to study problems involving curvature.
A thickening of the trefoil knot
Quintic Calabi–Yau threefold
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems.

Portrait by Jakob Emanuel Handmann (1753)

Leonhard Euler

Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus.

Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus.

Portrait by Jakob Emanuel Handmann (1753)
1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, in English written as "v − e + f = 2".
Euler's grave at the Alexander Nevsky Monastery
Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges.
An Euler diagram
Euler portrait on the sixth series of the 10 Franc banknote
Euler portrait on the seventh series of the 10 Franc banknote
Illustration from Solutio problematis... a. 1743 propositi published in Acta Eruditorum, 1744
The title page of Euler's Methodus inveniendi lineas curvas.
Euler's 1760 world map.
Euler's 1753 map of Africa.

Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.

In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.

The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.
David Hilbert
The overtones of a vibrating string. These are eigenfunctions of an associated Sturm–Liouville problem. The eigenvalues 1, 1⁄2, 1⁄3, ... form the (musical) harmonic series.
The path of a billiard ball in the Bunimovich stadium is described by an ergodic dynamical system.
Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top)
Spherical harmonics, an orthonormal basis for the Hilbert space of square-integrable functions on the sphere, shown graphed along the radial direction
The orbitals of an electron in a hydrogen atom are eigenfunctions of the energy.

. The proof is basic in mathematical analysis, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space).

The Mandelbrot set, a fractal

Complex analysis

The Mandelbrot set, a fractal

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0.

Measure (mathematics)

Generalization and formalization of geometrical measures and other common notions, such as mass and probability of events.

Generalization and formalization of geometrical measures and other common notions, such as mass and probability of events.

Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0.
Countable additivity of a measure μ: The measure of a countable disjoint union is the same as the sum of all measures of each subset.

Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures.