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

Mathematics

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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, 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

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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.

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

Continuous function

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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)

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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

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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.

A symbol for the set of real numbers

Real number

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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.

Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.

Series (mathematics)

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In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.

The study of series is a major part of calculus and its generalization, mathematical analysis.

The Mandelbrot set, a fractal

Complex analysis

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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)

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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.

An infinite sequence of real numbers (in blue). This sequence is neither increasing, decreasing, convergent, nor Cauchy. It is, however, bounded.

Sequence

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Enumerated collection of objects in which repetitions are allowed and order matters.

Enumerated collection of objects in which repetitions are allowed and order matters.

An infinite sequence of real numbers (in blue). This sequence is neither increasing, decreasing, convergent, nor Cauchy. It is, however, bounded.
A tiling with squares whose sides are successive Fibonacci numbers in length.
The plot of a convergent sequence (an) is shown in blue. From the graph we can see that the sequence is converging to the limit zero as n increases.
The plot of a Cauchy sequence (Xn), shown in blue, as Xn versus n. In the graph the sequence appears to be converging to a limit as the distance between consecutive terms in the sequence gets smaller as n increases.  In the real numbers every Cauchy sequence converges to some limit.

In mathematical analysis, a sequence is often denoted by letters in the form of ''.