List of things named after Leonhard Euler. In mathematics and physics, there are a large number of topics named in honor of Leonhard Euler, many of which include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Unfortunately, many of these entities have been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula.
Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. Physicists and mathematicians sometimes jest that, in an effort to avoid naming everything after Euler, discoveries and theorems are named after the "first person after Euler to discover it".[1][2] Euler's conjectures[edit] Euler's equations[edit] Euler's formulas[edit] Euler's functions[edit] Euler's identities[edit] Euler's numbers[edit] Euler's theorems[edit] Euler's laws[edit] Other things named after Euler[edit] Topics by field of study[edit] Selected topics from above, grouped by subject. Graph theory[edit] Contributions of Leonhard Euler to mathematics.
Mathematical notation[edit] Euler introduced much of the mathematical notation in use today, such as the notation f(x) to describe a function and the modern notation for the trigonometric functions. He was the first to use the letter e for the base of the natural logarithm, now also known as Euler's number. The use of the Greek letter to denote the ratio of a circle's circumference to its diameter was also popularized by Euler (although it did not originate with him).[1] He is also credited for inventing the notation i to denote Complex analysis[edit] A geometric interpretation of Euler's formula Euler made important contributions to complex analysis. . , the complex exponential function satisfies This has been called "The most remarkable formula in mathematics " by Richard Feynman. [3] Euler's identity is a special case of this: This identity is particularly remarkable as it involves e, , i, 1, and 0, arguably the five most important constants in mathematics.
Analysis[edit] for any positive real. Bernoulli polynomials. In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions. Bernoulli polynomials Representations[edit] Explicit formula[edit] for n ≥ 0, where bk are the Bernoulli numbers. Generating functions[edit] The generating function for the Bernoulli polynomials is The generating function for the Euler polynomials is Representation by a differential operator[edit] The Bernoulli polynomials are also given by cf. integrals below.
Representation by an integral operator[edit] . And. Bernoulli polynomials. List of things named after Leonhard Euler. Www.gbv.de/dms/goettingen/229406319.pdf. Multivalued function. In mathematics, a multivalued function (short form: multifunction; other names: many-valued function, set-valued function, set-valued map, multi-valued map, multimap, correspondence, carrier) is a left-total relation; that is, every input is associated with at least one output. Examples[edit] Every real number greater than zero has two square roots.
The square roots of 4 are in the set {+2,−2}. The square root of 0 is 0.Each complex number except zero has two square roots, three cube roots, and in general n nth roots. The nth root of 0 is 0.The complex logarithm function is multiple-valued. The values assumed by for real numbers and are for all integers .Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic. We have As a consequence, arctan(1) is intuitively related to several values: π/4, 5π/4, −3π/4, and so on. The indefinite integral can be considered as a multivalued function. Set-valued analysis[edit] Types of multivalued functions[edit] Euler's formula. This article is about Euler's formula in complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic.
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics History[edit] It was Johann Bernoulli who noted that[3] And since the above equation tells us something about complex logarithms.
Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand complex logarithms. Meanwhile, Roger Cotes, in 1714, discovered that ("ln" is the natural logarithm with base e).[4] where the real part the imaginary part . . Euler's formula. Carl B Boyer: "Foremost Modern Textbook" Muḥammad ibn Mūsā al-Khwārizmī. Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī[note 1][pronunciation?] (Arabic: عَبْدَالله مُحَمَّد بِن مُوسَى اَلْخْوَارِزْمِي), earlier transliterated as Algoritmi or Algaurizin, (c. 780 – c. 850) was a Persian[1][5] mathematician, astronomer and geographer during the Abbasid Empire, a scholar in the House of Wisdom in Baghdad.
Some words reflect the importance of al-Khwarizmi's contributions to mathematics. "Algebra" is derived from al-jabr, one of the two operations he used to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the Latin form of his name.[7] His name is also the origin of (Spanish) guarismo[8] and of (Portuguese) algarismo, both meaning digit. Life He was born in a Persian[1][5] family, and his birthplace is given as Chorasmia[9] by Ibn al-Nadim. Few details of al-Khwārizmī's life are known with certainty. Al-Tabari gave his name as Muhammad ibn Musa al-Khwārizmī al-Majousi al-Katarbali (محمد بن موسى الخوارزميّ المجوسـيّ القطربّـليّ). D. Contributions. Muḥammad ibn Mūsā al-Khwārizmī. Floyd-Warshall Algorithm.
Genetic Algorithm Tutorial. Genetic Algorithms in Plain English Introduction The aim of this tutorial is to explain genetic algorithms sufficiently for you to be able to use them in your own projects. This is a stripped-down to-the-bare-essentials type of tutorial. I'm not going to go into a great deal of depth and I'm not going to scare those of you with math anxiety by throwing evil equations at you every few sentences. In fact, I'm not going to throw any nasty equations at you at all!
This tutorial is designed to be read through twice... so don't worry if little of it makes sense the first time you study it. (A reader, Daniel, has kindly translated this tutorial into German. (Another reader, David Lewin, has translated the tutorial into French. First, a Biology Lesson Every organism has a set of rules, a blueprint so to speak, describing how that organism is built up from the tiny building blocks of life. When two organisms mate they share their genes. Now let's zip a few thousand generations into the future. The Language of Algebra - Main.
The Language of Algebra - Main. Set theory. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice.
Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. History[edit] Mathematical topics typically emerge and evolve through interactions among many researchers. Cantor's work initially polarized the mathematicians of his day. Basic concepts and notation[edit] Some ontology[edit] Sets alone. Arity. In logic, mathematics, and computer science, the arity Examples[edit] The term "arity" is rarely employed in everyday usage.
For example, rather than saying "the arity of the addition operation is 2" or "addition is an operation of arity 2" one usually says "addition is a binary operation". In general, the naming of functions or operators with a given arity follows a convention similar to the one used for n-based numeral systems such as binary and hexadecimal. One combines a Latin prefix with the -ary ending; for example: A nullary function takes no arguments.A unary function takes one argument.A binary function takes two arguments.A ternary function takes three arguments.An n-ary function takes n arguments.
Nullary[edit] Unary[edit] Binary[edit] Most operators encountered in programming are of the binary form. Ternary[edit] with arbitrary precision. N-ary[edit] Variable arity[edit] In computer science, a function accepting a variable number of arguments is called variadic. Other names[edit] Cartesian product. Cartesian product of the sets and The simplest case of a Cartesian product is the Cartesian square, which returns a set from two sets. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). A Cartesian product of n sets can be represented by an array of n dimensions, where each element is an n-tuple.
The Cartesian product is named after René Descartes,[1] whose formulation of analytic geometry gave rise to the concept. Examples[edit] A deck of cards[edit] An illustrative example is the standard 52-card deck. Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}. Suits × Ranks returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}. A two-dimensional coordinate system[edit] . , where For example: , or. Signature (logic) In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure.
In model theory, signatures are used for both purposes. Signatures play the same role in mathematics as type signatures in computer programming. They are rarely made explicit in more philosophical treatments of logic. Formally, a (single-sorted) signature can be defined as a triple σ = (Sfunc, Srel, ar), where Sfunc and Srel are disjoint sets not containing any other basic logical symbols, called respectively function symbols (examples: +, ×, 0, 1) andrelation symbols or predicates (examples: ≤, ∈), and a function ar: Sfunc Srel → which assigns a non-negative integer called arity to every function or relation symbol.
A signature with no function symbols is called a relational signature, and a signature with no relation symbols is called an algebraic signature. Symbol types S. Model theory. This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang and Keisler (1990):[1] universal algebra + logic = model theory. Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997): although model theorists are also interested in the study of fields. In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science.
Branches of model theory[edit] This article focuses on finitary first order model theory of infinite structures. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. And or. First-order logic. A theory about some topic is usually first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions which map from that domain into it, finitely many predicates defined on that domain, and a recursive set of axioms which are believed to hold for those things.
Sometimes "theory" is understood in a more formal sense, which is just a set of sentences in first-order logic. The adjective "first-order" distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.[1] In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets.
First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Introduction[edit] . X in . Well-formed formula. Introduction[edit] A key use of formulae is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true? ", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven. Although the term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence being expressed, with the marks being a token instance of formula.
It is not necessary for the existence of a formula that there be any actual tokens of it. A formal language may thus have an infinite number of formulas regardless whether each formula has a token instance. Propositional calculus[edit] The formulas of propositional calculus, also called propositional formulas,[2] are expressions such as . <form> | (<form> (((p q. Well-formed formula. Atomic formula. Atomic formula in first-order logic[edit] The well-formed terms and propositions of ordinary first-order logic have the following syntax: Terms: that is, a term is recursively defined to be a constant c (a named object from the domain of discourse), or a variable x (ranging over the objects in the domain of discourse), or an n-ary function f whose arguments are terms tk.
Functions map tuples of objects to objects. Propositions: An atomic formula or atom is simply a predicate applied to a tuple of terms; that is, an atomic formula is a formula of the form P (t1, …, tn) for P a predicate, and the tk terms. All other well-formed formulae are obtained by composing atoms with logical connectives and quantifiers. For example, the formula ∀x. When all of the terms in an atom are ground terms, then the atom is called a ground atom or ground predicate.
See also[edit] References[edit] Domain of discourse. Simple Gates" List of numbers. Mathematical constant. Theory of Thought. Decoding Euler's Identity. Golden ratio. Mathematical constant. Mathematical constant.