In mathematics, the Hankel transform expresses any given function f ( r ) as the weighted sum of an infinite number of Bessel functions of the first kind J < sub > ν </ sub >( kr ).

* Bessel function in mathematics

It can be applied in the study of classical concepts of mathematics, such as real numbers, complex variables, trigonometric functions, and algorithms, or of non-classical concepts like constructivism, harmonics, infinity, and vectors.

The term can also be applied to some degree to functions in mathematics, referring to the anatomy of curves.

Sets are of great importance in mathematics ; in fact, in modern formal treatments, most mathematical objects ( numbers, relations, functions, etc.

Calculus ( Latin, calculus, a small stone used for counting ) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.

A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis.

Although a " bijection " seems a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets.

In mathematics and computer science, currying is the technique of transforming a function that takes multiple arguments ( or an n-tuple of arguments ) in such a way that it can be called as a chain of functions each with a single argument ( partial application ).

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated.

For instance, instead of a hardware multiplier, a calculator might implement floating point mathematics with code in ROM, and compute trigonometric functions with the CORDIC algorithm because CORDIC does not require hardware floating-point.

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects ( for example, numbers and functions ) from all the traditional areas of mathematics ( such as algebra, analysis and topology ) in a single theory, and provides a standard set of axioms to prove or disprove them.

* Gramian matrix, used in mathematics to test for linear independence of functions

In mathematics, holomorphic functions are the central objects of study in complex analysis.

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms.

With amplification, logic gates can be cascaded in the same way that Boolean functions can be composed, allowing the construction of a physical model of all of Boolean logic, and therefore, all of the algorithms and mathematics that can be described with Boolean logic.

It was first formulated by Alonzo Church as a way to formalize mathematics through the notion of functions, in contrast to the field of set theory.

Recursive functions are a fundamental concept within computer science and mathematics.

In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational functions are defined in terms of each other.

The Ionian School of philosophers were the first natural philosophers ( φυσιολόγοι: physiologoi ) who tried to explain phenomena according to non-supernatural laws, and Pythagoras introduced the abstract mathematical-relations which formed the basis of the science of mathematics.

George W. Snedecor, the head of Iowa State's Statistics Department, was very likely the first user of an electronic digital computer to solve real world mathematics problems.

André-Marie Ampère took his first regular job in 1799 as a mathematics teacher, which gave him the financial security to marry Carron and father his first child, Jean-Jacques, the next year.

Many of the refugee children being hidden in Chambon attended Cévenol and it was at this school that Grothendieck apparently first became fascinated with mathematics.

" The Four Books on Measurement " were published at Nuremberg in 1525 and was the first book for adults on mathematics in German, as well as being cited later by Galileo and Kepler.

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in a paper on the method of least squares published in 1880.

The mathematics of crystal structures developed by Bravais, Federov and others was used to classify crystals by their symmetry group, and tables of crystal structures were the basis for the series International Tables of Crystallography, first published in 1935.

An innovator in mathematics, statistics, philosophy, research methodology, and various sciences, Peirce considered himself, first and foremost, a logician.

He was the son of Sarah Hunt Mills and Benjamin Peirce, himself a professor of astronomy and mathematics at Harvard University and perhaps the first serious research mathematician in America.

According to Mark Bollman, a mathematics and calculator historian and associate professor of mathematics at Albion College, the " Construction Master is the first in a long and profitable line of CI construction calculators " which carried them through the 1980s, 1990s, and to the present.

The invention of Cartesian coordinates in the 17th century by René Descartes ( Latinized name: Cartesius ) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.

Knuth was elected as a Fellow ( first class of Fellows ) of the Society for Industrial and Applied Mathematics in 2009 for his outstanding contributions to mathematics.

Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

In his first works Husserl tries to combine mathematics, psychology and philosophy with a main goal to provide a sound foundation for mathematics.

Later, in the first volume of his Logical Investigations, the Prolegomena of Pure Logic, Husserl, while attacking the psychologistic point of view in logic and mathematics, also appears to reject much of his early work, although the forms of psychologism analysed and refuted in the Prolegomena did not apply directly to his Philosophy of Arithmetic.

He was the first to use the word " group " () as a technical term in mathematics to represent a group of permutations.

Thales ( 635-543 BC ) of Miletus ( now in southwestern Turkey ), was the first to whom deduction in mathematics is attributed.

In 1934, she earned a Ph. D. in mathematics from Yale under the direction of Øystein Ore .< ref > Though some books, including Kurt Beyer's Grace Hopper and the Invention of the Information Age, reported that Hopper was the first woman to earn a Yale PhD in mathematics, the first of ten women prior to 1934 was Charlotte Cynthia Barnum ( 1860-1934 ).

In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

In mathematics, the arithmetic – geometric mean ( AGM ) of two positive real numbers and is defined as follows:

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

In mathematics, any vector space, V, has a corresponding dual vector space ( or just dual space for short ) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors.

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it.

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

Many objects in mathematics have natural group actions defined on them.

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set.

In mathematics, the Klein bottle () is a non-orientable surface, informally, a surface ( a two-dimensional manifold ) in which notions of left and right cannot be consistently defined.

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

In mathematics, the Hofstadter Female and Male sequences are an example of a pair of integer sequences defined in a mutually recursive manner.

In set theory and its applications throughout mathematics, a class is a collection of sets ( or sometimes other mathematical objects ) which can be unambiguously defined by a property that all its members share.

If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort.

A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right.

As discussed below, the definition given above turned out to be inadequate for formal mathematics ; instead, the notion of a " set " is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo – Fraenkel axioms.

In pure mathematics, a vector is defined more generally as any element of a vector space.

When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as geometric, spatial, or Euclidean vectors.

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set ( P, ≤) is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.

In mathematics, the L < sup > p </ sup > spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity.

In mathematics, Catalan's constant G, which occasionally appears in estimates in combinatorics, is defined by

In mathematics, the domain of definition or simply the domain of a function is the set of " input " or argument values for which the function is defined.

In mathematics, a singular function is any function ƒ defined on the interval b that has the following properties: