A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.
Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.
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.

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

- Mathematical analysis

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

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

5 related topics

Alpha

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.

These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.

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

Archimedes Thoughtful
by Domenico Fetti (1620)

Archimedes

Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Approximation of pi; defining and investigating the spiral that now bears his name; and devising a system using exponentiation for expressing very large numbers.

Archimedes Thoughtful
by Domenico Fetti (1620)
The Death of Archimedes (1815) by Thomas Degeorge
Cicero Discovering the Tomb of Archimedes (1805) by Benjamin West
The Archimedes' screw can raise water efficiently.
Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse.
Archimedes calculates the side of the 12-gon from that of the hexagon and for each subsequent doubling of the sides of the regular polygon.
A proof that the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure from Quadrature of the Parabola.
Frontpage of Archimedes' Opera, in Greek and Latin, edited by David Rivault (1615).
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases.
Ostomachion is a dissection puzzle found in the Archimedes Palimpsest.
In 1906, the Archimedes Palimpsest revealed works by Archimedes thought to have been lost.
The Fields Medal carries a portrait of Archimedes.
Artistic interpretation of Archimedes' mirror used to burn Roman ships.
Painting by Giulio Parigi, c. 1599.
Bronze statue of Archimedes in Berlin

Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and volume of a sphere; area of an ellipse; the area under a parabola; the volume of a segment of a paraboloid of revolution; the volume of a segment of a hyperboloid of revolution; and the area of a spiral.

He expressed the solution to the problem as an infinite geometric series with the common ratio 1⁄4:

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).

He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.

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

Sequence

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 ''.

In particular, sequences are the basis for series, which are important in differential equations and analysis.

Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional.

Trigonometric functions

Angle of a right-angled triangle to ratios of two side lengths.

Angle of a right-angled triangle to ratios of two side lengths.

Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional.
Signs of trigonometric functions in each quadrant. The mnemonic "all science teachers (are) crazy" lists the functions which are positive from quadrants I to IV. This is a variation on the mnemonic "All Students Take Calculus".
The unit circle, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.
Graphs of sine, cosine and tangent
The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.
Animation for the approximation of cosine via Taylor polynomials.
A Lissajous curve, a figure formed with a trigonometry-based function.
An animation of the additive synthesis of a square wave with an increasing number of harmonics

Modern definitions express trigonometric functions as infinite series or as solutions of differential equations.

However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles.