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Hugo Hadwiger(23 December 1908 inKarlsruhe, Germany– 29 October 1981 inBern, Switzerland)[1]was aSwissmathematician, known for his work ingeometry,combinatorics, andcryptography.
Although born inKarlsruhe, Germany, Hadwiger grew up inBern, Switzerland.[2]He did his undergraduate studies at theUniversity of Bern, where... | https://en.wikipedia.org/wiki/Hugo_Hadwiger |
Mathematical morphology(MM) is a theory and technique for the analysis and processing ofgeometricalstructures, based onset theory,lattice theory,topology, andrandom functions. MM is most commonly applied todigital images, but it can be employed as well ongraphs,surface meshes,solids, and many other spatial structures.
... | https://en.wikipedia.org/wiki/Morphological_image_processing |
Inmathematics, atopological vector space(also called alinear topological spaceand commonly abbreviatedTVSort.v.s.) is one of the basic structures investigated infunctional analysis.
A topological vector space is avector spacethat is also atopological spacewith the property that the vector space operations (vector addit... | https://en.wikipedia.org/wiki/Topological_vector_space |
In themathematicalstudy offunctional analysis, theBanach–Mazur distanceis a way to define adistanceon the setQ(n){\displaystyle Q(n)}ofn{\displaystyle n}-dimensionalnormed spaces. With this distance, the set ofisometryclasses ofn{\displaystyle n}-dimensional normed spaces becomes acompact metric space, called theBanach... | https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_compactum |
Infunctional analysisand related areas ofmathematics, acontinuous linear operatororcontinuous linear mappingis acontinuouslinear transformationbetweentopological vector spaces.
An operator between twonormed spacesis abounded linear operatorif and only if it is a continuous linear operator.
Suppose thatF:X→Y{\displays... | https://en.wikipedia.org/wiki/Continuous_linear_operator |
Inoperator theory, abounded operatorT:X→Ybetweennormed vector spacesXandYis said to be acontractionif itsoperator norm||T|| ≤ 1. This notion is a special case of the concept of acontraction mapping, but every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight in... | https://en.wikipedia.org/wiki/Contraction_(operator_theory) |
Inmathematics, anormed vector spaceornormed spaceis avector spaceover therealorcomplexnumbers on which anormis defined.[1]A norm is a generalization of the intuitive notion of "length" in the physical world. IfV{\displaystyle V}is a vector space overK{\displaystyle K}, whereK{\displaystyle K}is a field equal toR{\displ... | https://en.wikipedia.org/wiki/Normed_space |
Inmathematics,operator theoryis the study oflinear operatorsonfunction spaces, beginning withdifferential operatorsandintegral operators. The operators may be presented abstractly by their characteristics, such asbounded linear operatorsorclosed operators, and consideration may be given tononlinear operators. The study... | https://en.wikipedia.org/wiki/Operator_theory |
In themathematicalfield offunctional analysisthere are several standardtopologieswhich are given to the algebraB(X)ofbounded linear operatorson aBanach spaceX.
Let(Tn)n∈N{\displaystyle (T_{n})_{n\in \mathbb {N} }}be a sequence of linear operators on the Banach spaceX{\displaystyle X}. Consider the statement that(Tn)n∈... | https://en.wikipedia.org/wiki/Topologies_on_the_set_of_operators_on_a_Hilbert_space |
Inmathematics, more specificallyfunctional analysisandoperator theory, the notion ofunbounded operatorprovides an abstract framework for dealing withdifferential operators, unboundedobservablesinquantum mechanics, and other cases.
The term "unbounded operator" can be misleading, since
In contrast tobounded operators,... | https://en.wikipedia.org/wiki/Unbounded_operator |
Inmathematics, more specifically infunctional analysis, aBanach space(/ˈbɑː.nʌx/,Polish pronunciation:[ˈba.nax]) is acompletenormed vector space. Thus, a Banach space is a vector space with ametricthat allows the computation ofvector lengthand distance between vectors and is complete in the sense that aCauchy sequenceo... | https://en.wikipedia.org/wiki/Banach_space |
Inmathematics, acontraction mapping, orcontractionorcontractor, on ametric space(M,d) is afunctionffromMto itself, with the property that there is somereal number0≤k<1{\displaystyle 0\leq k<1}such that for allxandyinM,
The smallest such value ofkis called theLipschitz constantoff. Contractive maps are sometimes calle... | https://en.wikipedia.org/wiki/Contraction_mapping |
Infunctional analysis, theHahn–Banach theoremis a central result that allows the extension ofbounded linear functionalsdefined on avector subspaceof somevector spaceto the whole space. The theorem also shows that there are sufficientcontinuouslinear functionals defined on everynormed vector spacein order to study the... | https://en.wikipedia.org/wiki/Hahn-Banach_theorem |
Inmathematics, anormed vector spaceornormed spaceis avector spaceover therealorcomplexnumbers on which anormis defined.[1]A norm is a generalization of the intuitive notion of "length" in the physical world. IfV{\displaystyle V}is a vector space overK{\displaystyle K}, whereK{\displaystyle K}is a field equal toR{\displ... | https://en.wikipedia.org/wiki/Normed_vector_space |
Inmathematics, alinear form(also known as alinear functional,[1]aone-form, or acovector) is alinear map[nb 1]from avector spaceto itsfieldofscalars(often, thereal numbersor thecomplex numbers).
IfVis a vector space over a fieldk, the set of all linear functionals fromVtokis itself a vector space overkwith addition and... | https://en.wikipedia.org/wiki/Linear_functional |
Inmathematics, afunctionf{\displaystyle f}issuperadditiveiff(x+y)≥f(x)+f(y){\displaystyle f(x+y)\geq f(x)+f(y)}for allx{\displaystyle x}andy{\displaystyle y}in thedomainoff.{\displaystyle f.}
Similarly, asequencea1,a2,…{\displaystyle a_{1},a_{2},\ldots }is calledsuperadditiveif it satisfies theinequalityan+m≥an+am{\di... | https://en.wikipedia.org/wiki/Superadditivity |
Āryabhata's sine tableis a set of twenty-four numbers given in the astronomical treatiseĀryabhatiyacomposed by the fifth centuryIndian mathematicianand astronomerĀryabhata(476–550 CE), for the computation of thehalf-chordsof a certain set of arcs of a circle. The set of numbers appears in verse 12 in Chapter 1Dasagit... | https://en.wikipedia.org/wiki/Aryabhata%27s_sine_table |
Brahmagupta's interpolation formulais a second-order polynomialinterpolation formuladeveloped by theIndianmathematicianandastronomerBrahmagupta(598–668CE) in the early 7th centuryCE. TheSanskritcouplet describing the formula can be found in the supplementary part ofKhandakadyakaa work ofBrahmaguptacompleted in 665 CE.[... | https://en.wikipedia.org/wiki/Brahmagupta%27s_interpolation_formula |
Inmathematics, theEisenstein integers(named afterGotthold Eisenstein), occasionally also known[1]asEulerian integers(afterLeonhard Euler), are thecomplex numbersof the form
whereaandbareintegersand
is aprimitive(hence non-real)cube root of unity.
The Eisenstein integers form atriangular latticein thecomplex plane, i... | https://en.wikipedia.org/wiki/Eisenstein_integer |
In the mathematical field ofcomplex analysis,elliptic functionsare special kinds ofmeromorphicfunctions, that satisfy two periodicity conditions. They are named elliptic functions because they come fromelliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the calculation... | https://en.wikipedia.org/wiki/Elliptic_function |
In mathematicsAbel elliptic functionsare a special kind ofelliptic functions, that were established by the Norwegian mathematicianNiels Henrik Abel. He published his paper "Recherches sur les Fonctions elliptiques" inCrelle's Journalin 1827.[1]It was the first work on elliptic functions that was actually published.[2]A... | https://en.wikipedia.org/wiki/Abel_elliptic_functions |
Inmathematics, theJacobi elliptic functionsare a set of basicelliptic functions. They are found in the description of themotion of a pendulum, as well as in the design of electronicelliptic filters. Whiletrigonometric functionsare defined with reference to a circle, the Jacobi elliptic functions are a generalization w... | https://en.wikipedia.org/wiki/Jacobi_elliptic_functions |
Inmathematics, theWeierstrass elliptic functionsareelliptic functionsthat take a particularly simple form. They are named forKarl Weierstrass. This class of functions is also referred to as℘-functionsand they are usually denoted by the symbol ℘, a uniquely fancyscriptp. They play an important role in the theory of elli... | https://en.wikipedia.org/wiki/Weierstrass_elliptic_function |
TheLee conformal world in a tetrahedronis apolyhedral,conformalmap projectionthat projects the globe onto atetrahedronusingDixon elliptic functions. It is conformal everywhere except for the four singularities at the vertices of the polyhedron. Because of the nature of polyhedra, this map projection can betessellatedin... | https://en.wikipedia.org/wiki/Lee_conformal_world_in_a_tetrahedron |
Incomplex analysis, aSchwarz–Christoffel mappingis aconformal mapof theupper half-planeor the complexunit diskonto the interior of asimple polygon. Such a map isguaranteed to existby theRiemann mapping theorem(stated byBernhard Riemannin 1851); the Schwarz–Christoffel formula provides an explicit construction. They we... | https://en.wikipedia.org/wiki/Schwarz%E2%80%93Christoffel_mapping |
In mathematics, acomplex numberis an element of anumber systemthat extends thereal numberswith a specific element denotedi, called theimaginary unitand satisfying the equationi2=−1{\displaystyle i^{2}=-1}; every complex number can be expressed in the forma+bi{\displaystyle a+bi}, whereaandbare real numbers. Because no ... | https://en.wikipedia.org/wiki/Complex_number |
Inmathematics,Euler's identity[note 1](also known asEuler's equation) is theequalityeiπ+1=0{\displaystyle e^{i\pi }+1=0}where
Euler's identity is named after the Swiss mathematicianLeonhard Euler. It is a special case ofEuler's formulaeix=cosx+isinx{\displaystyle e^{ix}=\cos x+i\sin x}when evaluated forx=π{\displays... | https://en.wikipedia.org/wiki/Euler%27s_identity |
Inintegral calculus,Euler's formulaforcomplex numbersmay be used to evaluateintegralsinvolvingtrigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namelyeix{\displaystyle e^{ix}}ande−ix{\displaystyle e^{-ix}}and then integrated. This techni... | https://en.wikipedia.org/wiki/Integration_using_Euler%27s_formula |
Thehistory ofLorentz transformationscomprises the development oflinear transformationsforming theLorentz grouporPoincaré grouppreserving theLorentz interval−x02+⋯+xn2{\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}}and theMinkowski inner product−x0y0+⋯+xnyn{\displaystyle -x_{0}y_{0}+\cdots +x_{n}y_{n}}.
Inmathematics, tran... | https://en.wikipedia.org/wiki/History_of_Lorentz_transformations |
Inmathematicsandphysics, many topics arenamed in honorof Swiss mathematicianLeonhard Euler(1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Man... | https://en.wikipedia.org/wiki/List_of_topics_named_after_Leonhard_Euler |
Inmathematics, thePythagorean theoremorPythagoras' theoremis a fundamental relation inEuclidean geometrybetween the three sides of aright triangle. It states that the area of thesquarewhose side is thehypotenuse(the side opposite theright angle) is equal to the sum of the areas of the squares on the other two sides.
T... | https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclidean_geometry |
The numbereis amathematical constantapproximately equal to 2.71828 that is thebaseof thenatural logarithmandexponential function. It is sometimes calledEuler's number, after the Swiss mathematicianLeonhard Euler, though this can invite confusion withEuler numbers, or withEuler's constant, a different constant typically... | https://en.wikipedia.org/wiki/E_(mathematical_constant) |
Ingeometry, theequal incircles theoremderives from a JapaneseSangaku, and pertains to the following construction: a series of rays are drawn from a given point to a given line such that the inscribed circles of the triangles formed by adjacent rays and the base line are equal. In the illustration the equal blue circles... | https://en.wikipedia.org/wiki/Equal_incircles_theorem |
Thehyperbolastic functions, also known ashyperbolastic growth models, aremathematical functionsthat are used in medicalstatistical modeling. These models were originally developed to capture the growth dynamics of multicellular tumor spheres, and were introduced in 2005 by Mohammad Tabatabai, David Williams, and Zoran ... | https://en.wikipedia.org/wiki/Hyperbolastic_functions |
Inmathematics, theinverse hyperbolic functionsareinversesof thehyperbolic functions, analogous to theinverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cot... | https://en.wikipedia.org/wiki/Inverse_hyperbolic_function |
The following is a list ofintegrals(anti-derivativefunctions) ofhyperbolic functions. For a complete list of integral functions, seelist of integrals.
In all formulas the constantais assumed to be nonzero, andCdenotes theconstant of integration. | https://en.wikipedia.org/wiki/List_of_integrals_of_hyperbolic_functions |
Inmathematics,Poinsot's spiralsare twospiralsrepresented by thepolar equations
where csch is thehyperbolic cosecant, and sech is thehyperbolic secant.[1]They are named after the French mathematicianLouis Poinsot.
Thisgeometry-relatedarticle is astub. You can help Wikipedia byexpanding it. | https://en.wikipedia.org/wiki/Poinsot%27s_spirals |
In mathematics, anelliptic Gauss sumis an analog of aGauss sumdepending on anelliptic curvewith complex multiplication. Thequadratic residuesymbol in a Gauss sum is replaced by a higher residue symbol such as a cubic or quartic residue symbol, and the exponential function in a Gauss sum is replaced by anelliptic functi... | https://en.wikipedia.org/wiki/Elliptic_Gauss_sum |
Inmathematics, thelemniscate constantϖis atranscendentalmathematical constant that is the ratio of theperimeterofBernoulli's lemniscateto itsdiameter, analogous to the definition ofπfor the circle.[1]Equivalently, the perimeter of the lemniscate(x2+y2)2=x2−y2{\displaystyle (x^{2}+y^{2})^{2}=x^{2}-y^{2}}is2ϖ. The lemnis... | https://en.wikipedia.org/wiki/Lemniscate_constant |
ThePeirce quincuncial projectionis theconformal map projectionfrom the sphere to an unfolded squaredihedron, developed byCharles Sanders Peircein 1879.[1]Eachoctantprojects onto anisosceles right triangle, and these are arranged into a square. The namequincuncialrefers to this arrangement: the north pole at the center ... | https://en.wikipedia.org/wiki/Peirce_quincuncial_projection |
Levi ben Gershon(1288 – 20 April 1344), better known by hisGraecizedname asGersonides, or by his Latinized nameMagister Leo Hebraeus,[1]or inHebrewby the abbreviation of first letters asRaLBaG,[2]was a medieval French Jewishphilosopher, Talmudist,mathematician,physicianandastronomer/astrologer. He was born atBagnolsinL... | https://en.wikipedia.org/wiki/Gersonides |
Inspherical trigonometry, thehalf side formularelates the angles and lengths of the sides ofspherical triangles, which are triangles drawn on thesurface of a sphereand so have curved sides and do not obey the formulas forplane triangles.[1]
For a triangle△ABC{\displaystyle \triangle ABC}on a sphere, the half-side form... | https://en.wikipedia.org/wiki/Half-side_formula |
Spherical trigonometryis the branch ofspherical geometrythat deals with the metrical relationships between thesidesandanglesofspherical triangles, traditionally expressed usingtrigonometric functions. On thesphere,geodesicsaregreat circles. Spherical trigonometry is of great importance for calculations inastronomy,geod... | https://en.wikipedia.org/wiki/Spherical_triangles |
Intrigonometry, thelaw of cosines(also known as thecosine formulaorcosine rule) relates the lengths of the sides of atriangleto thecosineof one of itsangles. For a triangle with sidesa{\displaystyle a},b{\displaystyle b}, andc{\displaystyle c}, opposite respective anglesα{\displaystyle \alpha },β{\displaystyle... | https://en.wikipedia.org/wiki/Law_of_cosines |
Intrigonometry, thelaw of tangentsortangent rule[1]is a statement about the relationship between the tangents of two angles of atriangleand the lengths of the opposing sides.
In Figure 1,a,b, andcare the lengths of the three sides of the triangle, andα,β, andγare the anglesoppositethose three respective sides. The la... | https://en.wikipedia.org/wiki/Law_of_tangents |
Intrigonometry, thelaw of cotangentsis a relationship among the lengths of the sides of atriangleand thecotangentsof the halves of the three angles.[1][2]
Just as three quantities whose equality is expressed by thelaw of sinesare equal to the diameter of thecircumscribed circleof the triangle (or to its reciprocal, de... | https://en.wikipedia.org/wiki/Law_of_cotangents |
Intrigonometry,Mollweide's formulais a pair of relationships between sides and angles in a triangle.[1][2]
A variant in more geometrical style was first published byIsaac Newtonin 1707 and then byFriedrich Wilhelm von Oppel[de]in 1746.Thomas Simpsonpublished the now-standard expression in 1748.Karl Mollweiderepublishe... | https://en.wikipedia.org/wiki/Mollweide%27s_formula |
Solution of triangles(Latin:solutio triangulorum) is the maintrigonometricproblem of finding the characteristics of atriangle(angles and lengths of sides), when some of these are known. The triangle can be located on aplaneor on asphere. Applications requiring triangle solutions includegeodesy,astronomy,construction, a... | https://en.wikipedia.org/wiki/Solution_of_triangles |
Surveyingorland surveyingis the technique, profession, art, and science of determining theterrestrialtwo-dimensionalorthree-dimensionalpositions ofpointsand thedistancesandanglesbetween them. These points are usually on the surface of the Earth, and they are often used to establish maps and boundaries forownership, loc... | https://en.wikipedia.org/wiki/Surveying |
Aristarchus's inequality(after the GreekastronomerandmathematicianAristarchus of Samos; c. 310 – c. 230 BCE) is a law oftrigonometrywhich states that ifαandβareacute angles(i.e. between 0 and a right angle) andβ<αthen
Ptolemyused the first of these inequalities while constructinghis table of chords.[1]
The proof is a... | https://en.wikipedia.org/wiki/Aristarchus%27s_inequality |
This article is a summary ofdifferentiation rules, that is, rules for computing thederivativeof afunctionincalculus.
Unless otherwise stated, all functions are functions ofreal numbers(R{\textstyle \mathbb {R} }) that return real values, although, more generally, the formulas below apply wherever they arewell defined,... | https://en.wikipedia.org/wiki/Table_of_derivatives#Derivatives_of_trigonometric_functions |
In mathematics, the values of thetrigonometric functionscan be expressed approximately, as incos(π/4)≈0.707{\displaystyle \cos(\pi /4)\approx 0.707}, or exactly, as incos(π/4)=2/2{\displaystyle \cos(\pi /4)={\sqrt {2}}/2}. Whiletrigonometric tablescontain many approximate values, the exact values for certain angles ... | https://en.wikipedia.org/wiki/Exact_trigonometric_values |
Theexternal secantfunction (abbreviatedexsecant, symbolizedexsec) is atrigonometric functiondefined in terms of thesecantfunction:
exsecθ=secθ−1=1cosθ−1.{\displaystyle \operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.}
It was introduced in 1855 by Americancivil engineerCharles Haslett, who ... | https://en.wikipedia.org/wiki/Exsecant |
Inspherical trigonometry, thelaw of cosines(also called thecosine rule for sides[1]) is a theorem relating the sides and angles ofspherical triangles, analogous to the ordinarylaw of cosinesfrom planetrigonometry.
Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by thegreat circlesc... | https://en.wikipedia.org/wiki/Spherical_law_of_cosines |
The following is a list ofintegrals(antiderivativefunctions) oftrigonometric functions. For antiderivatives involving both exponential and trigonometric functions, seeList of integrals of exponential functions. For a complete list of antiderivative functions, seeLists of integrals. For the special antiderivatives invol... | https://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions |
Intrigonometry, it is common to usemnemonicsto help remembertrigonometric identitiesand the relationships between the varioustrigonometric functions.
Thesine,cosine, andtangentratios in a right triangle can be remembered by representing them as strings of letters, for instance SOH-CAH-TOA in English:
One way to remem... | https://en.wikipedia.org/wiki/Mnemonics_in_trigonometry |
Pentagramma mirificum(Latinfor "miraculous pentagram") is astar polygonon asphere, composed of fivegreat circlearcs, all of whoseinternal anglesareright angles. This shape was described byJohn Napierin his 1614 bookMirifici Logarithmorum Canonis Descriptio(Description of the Admirable Table of Logarithms) along withrul... | https://en.wikipedia.org/wiki/Pentagramma_mirificum |
Inmathematics, thePythagorean theoremorPythagoras' theoremis a fundamental relation inEuclidean geometrybetween the three sides of aright triangle. It states that the area of thesquarewhose side is thehypotenuse(the side opposite theright angle) is equal to the sum of the areas of the squares on the other two sides.
T... | https://en.wikipedia.org/wiki/Pythagorean_theorem |
Intrigonometry,tangent half-angle formulasrelate the tangent of half of an angle to trigonometric functions of the entire angle.[1]
The tangent of half an angle is thestereographic projectionof the circle through the point at angleπ{\textstyle \pi }radiansonto the line through the angles±π2{\textstyle \pm {\frac {\pi ... | https://en.wikipedia.org/wiki/Tangent_half-angle_formula |
In mathematics, the values of thetrigonometric functionscan be expressed approximately, as incos(π/4)≈0.707{\displaystyle \cos(\pi /4)\approx 0.707}, or exactly, as incos(π/4)=2/2{\displaystyle \cos(\pi /4)={\sqrt {2}}/2}. Whiletrigonometric tablescontain many approximate values, the exact values for certain angles ... | https://en.wikipedia.org/wiki/Trigonometric_number |
Trigonometry(fromAncient Greekτρίγωνον(trígōnon)'triangle'andμέτρον(métron)'measure')[1]is a branch ofmathematicsconcerned with relationships betweenanglesand side lengths of triangles. In particular, thetrigonometric functionsrelate the angles of aright trianglewithratiosof its side lengths. The field emerged in theHe... | https://en.wikipedia.org/wiki/Trigonometry |
Amongst the lay public of non-mathematicians and non-scientists,trigonometryis known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in thetheory of music; still other uses are more technical, such as innumber theory. The mathematical topic... | https://en.wikipedia.org/wiki/Uses_of_trigonometry |
Theversineorversed sineis atrigonometric functionfound in some of the earliest (SanskritAryabhatia,[1]Section I)trigonometric tables. The versine of an angle is 1 minus itscosine.
There are several related functions, most notably thecoversineandhaversine. The latter, half a versine, is of particular importance in theh... | https://en.wikipedia.org/wiki/Versine |
Theversineorversed sineis atrigonometric functionfound in some of the earliest (SanskritAryabhatia,[1]Section I)trigonometric tables. The versine of an angle is 1 minus itscosine.
There are several related functions, most notably thecoversineandhaversine. The latter, half a versine, is of particular importance in theh... | https://en.wikipedia.org/wiki/Haversine |
Mādhava of Sangamagrāma(Mādhavan)[4](c.1340– c.1425) was an Indianmathematicianandastronomerwho is considered to be the founder of theKerala school of astronomy and mathematicsin theLate Middle Ages. Madhava made pioneering contributions to the study ofinfinite series,calculus,trigonometry,geometryandalgebra. He was th... | https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama |
Madhava's correction termis a mathematical expression attributed toMadhava of Sangamagrama(c. 1340 – c. 1425), the founder of theKerala school of astronomy and mathematics, that can be used to give a better approximation to the value of the mathematical constantπ(pi) than the partial sum approximation obtained by trunc... | https://en.wikipedia.org/wiki/Madhava%27s_correction_term |
Inmathematics, theLaurent seriesof acomplex functionf(z){\displaystyle f(z)}is a representation of that function as apower serieswhich includes terms of negative degree. It may be used to express complex functions in cases where aTaylor seriesexpansion cannot be applied. The Laurent series was named after and first pub... | https://en.wikipedia.org/wiki/Laurent_series |
Inmathematics,Puiseux seriesare a generalization ofpower seriesthat allow for negative and fractional exponents of theindeterminate. For example, the series
is a Puiseux series in the indeterminatex. Puiseux series were first introduced byIsaac Newtonin 1676[1]and rediscovered byVictor Puiseuxin 1850.[2]
The definit... | https://en.wikipedia.org/wiki/Puiseux_series |
Madhava's correction termis a mathematical expression attributed toMadhava of Sangamagrama(c. 1340 – c. 1425), the founder of theKerala school of astronomy and mathematics, that can be used to give a better approximation to the value of the mathematical constantπ(pi) than the partial sum approximation obtained by trunc... | https://en.wikipedia.org/wiki/Madhava%27s_value_of_%CF%80 |
Thetable of chords, created by theGreekastronomer, geometer, and geographerPtolemyinEgyptduring the 2nd century AD, is atrigonometric tablein Book I, chapter 11 of Ptolemy'sAlmagest,[1]a treatise onmathematical astronomy. It is essentially equivalent to a table of values of thesinefunction. It was the earliest trigon... | https://en.wikipedia.org/wiki/Ptolemy%27s_table_of_chords |
Ingeometry, asimplex(plural:simplexesorsimplices) is a generalization of the notion of atriangleortetrahedronto arbitrarydimensions. The simplex is so-named because it represents the simplest possiblepolytopein any given dimension. For example,
Specifically, ak-simplexis ak-dimensionalpolytopethat is theconvex hullof ... | https://en.wikipedia.org/wiki/Simplex |
Inmathematics, thecross productorvector product(occasionallydirected area product, to emphasize its geometric significance) is abinary operationon twovectorsin a three-dimensionalorientedEuclidean vector space(named hereE{\displaystyle E}), and is denoted by the symbol×{\displaystyle \times }. Given twolinearly indepen... | https://en.wikipedia.org/wiki/Cross_product |
Inmathematics, theseven-dimensional cross productis abilinear operationonvectorsinseven-dimensional Euclidean space. It assigns to any two vectorsa,binR7{\displaystyle \mathbb {R} ^{7}}a vectora×balso inR7{\displaystyle \mathbb {R} ^{7}}.[1]Like thecross productin three dimensions, the seven-dimensional product isa... | https://en.wikipedia.org/wiki/Seven-dimensional_cross_product |
Inmathematics, in particularabstract algebra, agraded ringis aringsuch that the underlyingadditive groupis adirect sum of abelian groupsRi{\displaystyle R_{i}}such thatRiRj⊆Ri+j{\displaystyle R_{i}R_{j}\subseteq R_{i+j}}. Theindex setis usually the set of nonnegativeintegersor the set of integers, but can be anymonoi... | https://en.wikipedia.org/wiki/Graded_algebra |
On adifferentiable manifold, theexterior derivativeextends the concept of thedifferentialof a function todifferential formsof higher degree. The exterior derivative was first described in its current form byÉlie Cartanin 1899. The resulting calculus, known asexterior calculus, allows for a natural, metric-independent g... | https://en.wikipedia.org/wiki/Exterior_derivative |
Differential geometryis amathematicaldiscipline that studies thegeometryof smooth shapes and smooth spaces, otherwise known assmooth manifolds. It uses the techniques ofsingle variable calculus,vector calculus,linear algebraandmultilinear algebra. The field has its origins in the study ofspherical geometryas far back a... | https://en.wikipedia.org/wiki/Differential_geometry |
Inmathematics(particularlymultivariable calculus), avolume integral(∭) is anintegralover a3-dimensionaldomain; that is, it is a special case ofmultiple integrals. Volume integrals are especially important inphysicsfor many applications, for example, to calculatefluxdensities, or to calculate mass from a corresponding d... | https://en.wikipedia.org/wiki/Volume_integral |
Inmathematics, the concept of ameasureis a generalization and formalization ofgeometrical measures(length,area,volume) and other common notions, such asmagnitude,mass, andprobabilityof events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Me... | https://en.wikipedia.org/wiki/Measure_(mathematics) |
Aproduct integralis anyproduct-based counterpart of the usualsum-basedintegralofcalculus. The product integral was developed by the mathematicianVito Volterrain 1887 to solve systems oflinear differential equations.[1][2]
The classicalRiemann integralof afunctionf:[a,b]→R{\displaystyle f:[a,b]\to \mathbb {R} }can be d... | https://en.wikipedia.org/wiki/Product_integral |
Inmathematics, tables oftrigonometric functionsare useful in a number of areas. Before the existence ofpocket calculators,trigonometric tableswere essential fornavigation,scienceandengineering. The calculation ofmathematical tableswas an important area of study, which led to the development of thefirst mechanical compu... | https://en.wikipedia.org/wiki/Generating_trigonometric_tables |
Inmathematics, aNewtonian series, named afterIsaac Newton, is a sum over asequencean{\displaystyle a_{n}}written in the form
where
is thebinomial coefficientand(s)n{\displaystyle (s)_{n}}is thefalling factorial. Newtonian series often appear in relations of the form seen inumbral calculus.
The generalizedbinomial t... | https://en.wikipedia.org/wiki/Table_of_Newtonian_series |
Inmathematics, aunit vectorin anormed vector spaceis avector(often aspatial vector) oflength1. A unit vector is often denoted by a lowercase letter with acircumflex, or "hat", as inv^{\displaystyle {\hat {\mathbf {v} }}}(pronounced "v-hat"). The termnormalized vectoris sometimes used as a synonym forunit vector.
Theno... | https://en.wikipedia.org/wiki/Unit_vector |
Ananti-aliasing filter(AAF) is afilterused before asignal samplerto restrict thebandwidthof asignalto satisfy theNyquist–Shannon sampling theoremover theband of interest. Since the theorem states that unambiguous reconstruction of the signal from its samples is possible when thepower of frequenciesabove theNyquist freq... | https://en.wikipedia.org/wiki/Anti-aliasing_filter |
Inmathematics, aBorwein integralis anintegralwhose unusual properties were first presented by mathematiciansDavid BorweinandJonathan Borweinin 2001.[1]Borwein integrals involve products ofsinc(ax){\displaystyle \operatorname {sinc} (ax)}, where thesinc functionis given bysinc(x)=sin(x)/x{\displaystyle \operatorname ... | https://en.wikipedia.org/wiki/Borwein_integral |
Inmathematics, there are severalintegralsknown as theDirichlet integral, after the German mathematicianPeter Gustav Lejeune Dirichlet, one of which is theimproper integralof thesinc functionover the positive real number line.
∫0∞sinxxdx=π2.{\displaystyle \int _{0}^{\infty }{\frac {\sin x}{x}}\,dx={\frac {\pi }{2}}.}
... | https://en.wikipedia.org/wiki/Dirichlet_integral |
Lanczos filteringandLanczos resamplingare two applications of a certain mathematical formula. It can be used as alow-pass filteror used to smoothlyinterpolatethe value of adigital signalbetween itssamples. In the latter case, it maps each sample of the given signal to a translated and scaled copy of theLanczos kernel,... | https://en.wikipedia.org/wiki/Lanczos_resampling |
Infunctional analysis, theShannon wavelet(orsinc wavelets) is a decomposition that is defined by signal analysis by idealbandpass filters. Shannon wavelet may be either ofrealorcomplextype.
Shannon wavelet is not well-localized (noncompact) in the time domain, but its Fourier transform is band-limited (compact suppor... | https://en.wikipedia.org/wiki/Shannon_wavelet |
Insignal processing, asinc filtercan refer to either asinc-in-timefilterwhoseimpulse responseis asinc functionand whosefrequency responseis rectangular, or to asinc-in-frequencyfilter whose impulse response is rectangular and whose frequency response is a sinc function. Calling them according to which domain the filter... | https://en.wikipedia.org/wiki/Sinc_filter |
Innumerical analysisandapplied mathematics,sinc numerical methodsare numerical techniques[1]for finding approximate solutions ofpartial differential equationsandintegral equationsbased on the translates ofsincfunction and Cardinal function C(f,h) which is an expansion of f defined by
where the step size h>0 and where ... | https://en.wikipedia.org/wiki/Sinc_numerical_methods |
Thetrigonometric functions(especiallysineandcosine) for complexsquare matricesoccur in solutions of second-order systems ofdifferential equations.[1]They are defined by the sameTaylor seriesthat hold for the trigonometric functions ofcomplex numbers:[2]
withXnbeing thenthpowerof the matrixX, andIbeing theidentity matr... | https://en.wikipedia.org/wiki/Trigonometric_functions_of_matrices |
TheWhittaker–Shannon interpolation formulaorsinc interpolationis a method to construct acontinuous-timebandlimitedfunction from a sequence of real numbers. The formula dates back to the works ofE. Borelin 1898, andE. T. Whittakerin 1915, and was cited from works ofJ. M. Whittakerin 1935, and in the formulation of theN... | https://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula |
TheWinkel tripel projection(Winkel III), a modified azimuthal[1]map projectionof theworld, is one ofthree projectionsproposed by German cartographer Oswald Winkel (7 January 1874 – 18 July 1953) in 1921. The projection is thearithmetic meanof theequirectangular projectionand theAitoff projection:[2]The nametripel(Germa... | https://en.wikipedia.org/wiki/Winkel_tripel_projection |
Adiscrete cosine transform(DCT) expresses a finite sequence ofdata pointsin terms of a sum ofcosinefunctions oscillating at differentfrequencies. The DCT, first proposed byNasir Ahmedin 1972, is a widely used transformation technique insignal processinganddata compression. It is used in mostdigital media, includingdigi... | https://en.wikipedia.org/wiki/Discrete_cosine_transform |
Inmathematics, thelogarithmic integral functionorintegral logarithmli(x) is aspecial function. It is relevant in problems ofphysicsand hasnumber theoreticsignificance. In particular, according to theprime number theorem, it is a very goodapproximationto theprime-counting function, which is defined as the number ofprime... | https://en.wikipedia.org/wiki/Logarithmic_integral |
Inmathematics,physicsandengineering, thesinc function(/ˈsɪŋk/SINK), denoted bysinc(x), has two forms, normalized and unnormalized.[1]
In mathematics, the historicalunnormalized sinc functionis defined forx≠ 0bysinc(x)=sinxx.{\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}.}
Alternatively, the unnormalize... | https://en.wikipedia.org/wiki/Tanc_function |
Inmathematics,hyperbolic functionsare analogues of the ordinarytrigonometric functions, but defined using thehyperbolarather than thecircle. Just as the points(cost, sint)form acircle with a unit radius, the points(cosht, sinht)form the right half of theunit hyperbola. Also, similarly to how the derivatives ofsin(t)and... | https://en.wikipedia.org/wiki/Tanhc_function |
Inmathematics,physicsandengineering, thesinc function(/ˈsɪŋk/SINK), denoted bysinc(x), has two forms, normalized and unnormalized.[1]
In mathematics, the historicalunnormalized sinc functionis defined forx≠ 0bysinc(x)=sinxx.{\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}.}
Alternatively, the unnormalize... | https://en.wikipedia.org/wiki/Sinhc_function |
Inmathematics,hyperbolic functionsare analogues of the ordinarytrigonometric functions, but defined using thehyperbolarather than thecircle. Just as the points(cost, sint)form acircle with a unit radius, the points(cosht, sinht)form the right half of theunit hyperbola. Also, similarly to how the derivatives ofsin(t)and... | https://en.wikipedia.org/wiki/Coshc_function |
Acrestpoint on awaveis the highest point of the wave. A crest is a point on asurface wavewhere thedisplacementof the medium is at a maximum. Atroughis the opposite of a crest, so theminimumor lowest point of the wave.
When the crests and troughs of twosine wavesof equalamplitudeandfrequencyintersect or collide, while ... | https://en.wikipedia.org/wiki/Crest_(physics) |
Inmathematics, theexponential functionis the uniquereal functionwhich mapszerotooneand has aderivativeeverywhere equal to its value. The exponential of a variablex{\displaystyle x}is denotedexpx{\displaystyle \exp x}orex{\displaystyle e^{x}}, with the two notations used interchangeably. It is calledexponentialbe... | https://en.wikipedia.org/wiki/Complex_exponential |
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