Meta tags:

Headings (most frequently used words):

multivalued, function, contents, motivation, inverses, of, functions, concrete, examples, branch, points, applications, see, also, further, reading, references,

Text of the page (most frequently used words):
the (75), function (53), displaystyle (29), #multivalued (27), functions (26), for (24), #valued (23), and (21), are (15), log (14), complex (13), with (12), from (12), edit (12), single (11), that (11), one (11), this (10), domain (10), branch (10), values (10), tan (10), may (9), set (9), inverse (8), wikipedia (7), root (7), article (7), square (7), all (6), multiple (6), they (6), points (6), real (6), sqrt (6), its (6), page (5), articles (5), analytic (5), analysis (5), these (5), example (5), point (5), main (5), not (5), along (5), has (5), roots (5), plane (5), contents (4), search (4), about (4), citations (4), retrieved (4), relation (4), many (4), mathematics (4), vol (4), also (4), range (4), principal (4), considered (4), left (4), tfrac (4), right (4), two (4), times (4), gamma (4), value (4), hide (4), move (4), sidebar (4), view (3), text (3), available (3), implicit (3), continuous (3), constant (3), history (3), 2024 (3), applications (3), references (3), further (3), see (3), mathematical (3), have (3), logarithm (3), curve (3), called (3), riemann (3), examples (3), over (3), trigonometric (3), arctan (3), which (3), numbers (3), general (3), only (3), given (3), subset (3), mathbf (3), subseteq (3), when (3), ordinary (3), distinguish (3), curves (3), changes (3), more (3), tools (3), languages (2), toggle (2), table (2), contact (2), privacy (2), policy (2), additional (2), terms (2), using (2), non (2), use (2), was (2), february (2), unsourced (2), statements (2), lacking (2), january (2), 2020 (2), short (2), description (2), wikidata (2), mappings (2), category (2), space (2), partial (2), binary (2), restriction (2), injective (2), properties (2), list (2), doi (2), multifunction (2), defects (2), world (2), scientific (2), singapore (2), online (2), gauge (2), fields (2), condensed (2), matter (2), kleinert (2), reading (2), physics (2), theory (2), imaginary (2), restricting (2), through (2), thus (2), case (2), restricted (2), surface (2), since (2), original (2), information (2), often (2), because (2), hyperbolic (2), periodic (2), can (2), antiderivative (2), derivative (2), follows (2), related (2), arg (2), number (2), than (2), zero (2), write (2), concrete (2), any (2), continuation (2), graph (2), defined (2), between (2), theorem (2), then (2), inverses (2), into (2), extend (2), neighbourhood (2), starting (2), depending (2), extended (2), some (2), them (2), motivation (2), learn (2), help (2), appearance (2), upload (2), file (2), links (2), read (2), english (2), create (2), account (2), donate (2), menu (2), add, topic, mobile, cookie, statement, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, under, apply, site, you, agree, registered, trademark, profit, organization, wikimedia, foundation, inc, creative, commons, attribution, sharealike, license, rendered, parsoid, last, edited, 2026, utc, hidden, categories, july, 2013, different, https, org, index, php, title, multivalued_function, oldid, 1339512506, functor, morphism, higher, order, generalizations, composition, constructions, bijective, surjective, measurable, smooth, algebraic, rational, polynomial, linear, identity, classes, types, codomain, specific, repovš, dušan, 1998, pavel, vladimirovič, semenov, dordrecht, kluwer, academic, 39739641, oclc, 7923, 5277, isbn, selections, wolfram, mathworld, ledyaev, yuri, zhu, qiji, 1999, 238, 1023, 1008775413250, 209, volume, theorems, rabadi, anas, zwick, martin, 2004, 920, 1108, 03684920410533967, 906, kybernetes, modified, reconstructability, relations, mit, opencourseware, variables, archive, lib, msu, edu, superflow, vortex, lines, 742, stresses, 743, 1456, 1989, 2008, electrodynamics, gravitation, play, increasingly, important, role, form, basis, crystals, resulting, materials, systems, instance, origin, structures, branches, citation, needed, field, quark, confinement, melting, phase, transitions, superconductors, superfluids, vortices, plasticity, magnetic, monopoles, dirac, variable, arctangent, units, redefined, suitable, interval, found, kind, connects, pairs, reducing, multilayered, layer, cut, come, preserve, their, inputs, reversible, axis, reals, except, arcosh, arsech, whose, fact, integration, consequence, intuitively, several, treat, monotonically, increasing, becomes, cdots, assumed, integers, each, nonzero, three, cube, every, greater, although, note, usually, denotes, open, always, holomorphic, exponential, viewed, gives, conditions, locally, manifolds, differentiable, define, producing, whole, discontinuous, certain, boundary, alternatively, dealing, allows, having, something, everywhere, cost, possible, closed, path, problems, resolved, consider, without, discarding, multiplies, layered, associated, manifold, covering, surfaces, monodromy, let, usual, positive, vary, continuously, extending, negative, gets, opposite, whether, been, upper, lower, half, phenomenon, very, frequent, occurring, logarithms, term, originated, occurs, knows, around, such, situation, doing, finds, depends, chosen, none, new, natural, others, incorporated, taylor, series, global, those, taking, sets, least, context, though, authors, multifunctions, confused, multivariate, variational, includes, how, remove, message, please, precise, introducing, improve, but, lacks, sufficient, corresponding, inline, generalized, free, encyclopedia, item, other, projects, printable, version, download, pdf, print, export, switch, legacy, parser, get, shortened, url, cite, permanent, link, what, here, actions, talk, українська, türkçe, தமிழ், svenska, slovenčina, simple, русский, português, polski, 한국어, қазақша, 日本語, italiano, עברית, galego, français, فارسی, eesti, español, deutsch, чӑвашла, català, العربية, top, personal, special, pages, recent, community, portal, contribute, random, current, events, navigation, jump, content,

Text of the page (random words):
multivalued function wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 motivation 2 inverses of functions 3 concrete examples 4 branch points 5 applications 6 see also 7 further reading 8 references toggle the table of contents multivalued function 24 languages العربية català чӑвашла deutsch español eesti فارسی français galego עברית italiano 日本語 қазақша 한국어 polski português русский simple english slovenčina svenska தமிழ் türkçe українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia generalized mathematical function this article includes a list of general references but lacks sufficient corresponding inline citations please help improve this article by introducing more precise citations january 2020 learn how and when to remove this message this article is about multivalued functions as they are considered in mathematical analysis for set valued functions as considered in variational analysis see set valued function not to be confused with multivariate function multivalued function 1 2 3 a b c d in mathematics a multivalued function 1 multiple valued function 2 many valued function 3 or multifunction 4 is a function that has two or more values in its range for at least one point in its domain 5 it is a set valued function with additional properties depending on context though some authors do not distinguish between set valued functions and multifunctions 6 a multivalued function of sets f x y is a subset γ f x y displaystyle gamma _ f subseteq x times y write f x for the set of those y y with x y γ f if f is an ordinary function it is a multivalued function by taking its graph γ f x f x x x displaystyle gamma _ f x f x x in x they are called single valued functions to distinguish them motivation edit main article global analytic function the term multivalued function originated in complex analysis from analytic continuation it often occurs that one knows the value of a complex analytic function f z displaystyle f z in some neighbourhood of a point z a displaystyle z a this is the case for functions defined by the implicit function theorem or by a taylor series around z a displaystyle z a in such a situation one may extend the domain of the single valued function f z displaystyle f z along curves in the complex plane starting at a displaystyle a in doing so one finds that the value of the extended function at a point z b displaystyle z b depends on the chosen curve from a displaystyle a to b displaystyle b since none of the new values is more natural than the others all of them are incorporated into a multivalued function for example let f z z displaystyle f z sqrt z be the usual square root function on positive real numbers one may extend its domain to a neighbourhood of z 1 displaystyle z 1 in the complex plane and then further along curves starting at z 1 displaystyle z 1 so that the values along a given curve vary continuously from 1 1 displaystyle sqrt 1 1 extending to negative real numbers one gets two opposite values for the square root for example i for 1 depending on whether the domain has been extended through the upper or the lower half of the complex plane this phenomenon is very frequent occurring for n th roots logarithms and inverse trigonometric functions to define a single valued function from a complex multivalued function one may distinguish one of the multiple values as the principal value producing a single valued function on the whole plane which is discontinuous along certain boundary curves alternatively dealing with the multivalued function allows having something that is everywhere continuous at the cost of possible value changes when one follows a closed path monodromy these problems are resolved in the theory of riemann surfaces to consider a multivalued function f z displaystyle f z as an ordinary function without discarding any values one multiplies the domain into a many layered covering space a manifold which is the riemann surface associated to f z displaystyle f z inverses of functions edit if f x y is an ordinary function then its inverse is the multivalued function γ f 1 y x displaystyle gamma _ f 1 subseteq y times x defined as γ f viewed as a subset of x y when f is a differentiable function between manifolds the inverse function theorem gives conditions for this to be single valued locally in x for example the complex logarithm log z is the multivalued inverse of the exponential function e z c c with graph γ log z z w w log z c c displaystyle gamma _ log z z w w log z subseteq mathbf c times mathbf c times it is not single valued given a single w with w log z we have log z w 2 π i z displaystyle log z w 2 pi i mathbf z given any holomorphic function on an open subset of the complex plane c its analytic continuation is always a multivalued function concrete examples edit every real number greater than zero has two real square roots so that square root may be considered a multivalued function for example we may write 4 2 2 2 displaystyle sqrt 4 pm 2 2 2 although zero has only one square root 0 0 displaystyle sqrt 0 0 note that x displaystyle sqrt x usually denotes only the principal square root of x displaystyle x each nonzero complex number has two square roots three cube roots and in general n n th roots the only n th root of 0 is 0 the complex logarithm function is multiple valued the values assumed by log a b i displaystyle log a bi for real numbers a displaystyle a and b displaystyle b are log a 2 b 2 i arg a b i 2 π n i displaystyle log sqrt a 2 b 2 i arg a bi 2 pi ni for all integers n displaystyle n inverse trigonometric functions are multiple valued because trigonometric functions are periodic we have tan π 4 tan 5 π 4 tan 3 π 4 tan 4 n 1 π 4 1 displaystyle tan left tfrac pi 4 right tan left tfrac 5 pi 4 right tan left tfrac 3 pi 4 right tan left tfrac 4n 1 pi 4 right cdots 1 as a consequence arctan 1 is intuitively related to several values π 4 5 π 4 3 π 4 and so on we can treat arctan as a single valued function by restricting the domain of tan x to π 2 x π 2 a domain over which tan x is monotonically increasing thus the range of arctan x becomes π 2 y π 2 these values from a restricted domain are called principal values the antiderivative can be considered as a multivalued function the antiderivative of a function is the set of functions whose derivative is that function the constant of integration follows from the fact that the derivative of a constant function is 0 inverse hyperbolic functions over the complex domain are multiple valued because hyperbolic functions are periodic along the imaginary axis over the reals they are single valued except for arcosh and arsech these are all examples of multivalued functions that come about from non injective functions since the original functions do not preserve all the information of their inputs they are not reversible often the restriction of a multivalued function is a partial inverse of the original function branch points edit main article branch point multivalued functions of a complex variable have branch points for example for the n th root and logarithm functions 0 is a branch point for the arctangent function the imaginary units i and i are branch points using the branch points these functions may be redefined to be single valued functions by restricting the range a suitable interval may be found through use of a branch cut a kind of curve that connects pairs of branch points thus reducing the multilayered riemann surface of the function to a single layer as in the case with real functions the restricted range may be called the principal branch of the function applications edit in physics multivalued functions play an increasingly important role they form the mathematical basis for dirac s magnetic monopoles for the theory of defects in crystals and the resulting plasticity of materials for vortices in superfluids and superconductors and for phase transitions in these systems for instance melting and quark confinement they are the origin of gauge field structures in many branches of physics citation needed see also edit relation mathematics function mathematics binary relation set valued function further reading edit h kleinert multivalued fields in condensed matter electrodynamics and gravitation world scientific singapore 2008 also available online h kleinert gauge fields in condensed matter vol i superflow and vortex lines 1 742 vol ii stresses and defects 743 1456 world scientific singapore 1989 also available online vol i and vol ii references edit multivalued function archive lib msu edu retrieved 2024 10 25 multiple valued functions complex variables with applications mathematics mit opencourseware retrieved 2024 10 25 al rabadi anas zwick martin 2004 01 01 modified reconstructability analysis for many valued functions and relations kybernetes 33 5 6 906 920 doi 10 1108 03684920410533967 ledyaev yuri zhu qiji 1999 09 01 implicit multifunction theorems set valued analysis volume 7 3 209 238 doi 10 1023 a 1008775413250 multivalued function wolfram mathworld retrieved 10 february 2024 repovš dušan 1998 continuous selections of multivalued mappings pavel vladimirovič semenov dordrecht kluwer academic isbn 0 7923 5277 7 oclc 39739641 v t e function history list of specific functions types by domain codomain x 𝔹 𝔹 x 𝔹ⁿ 𝔹 x ℤ ℤ x x ℝ ℝ x ℝⁿ x x ℂ ℂ x ℂⁿ x classes properties constant identity linear polynomial rational algebraic analytic smooth continuous measurable injective surjective bijective constructions restriction composition λ inverse generalizations relation binary relation set valued multivalued partial implicit space higher order morphism functor category retrieved from https en wikipedia org w index php title multivalued_function oldid 1339512506 category functions and mappings hidden categories articles with short description short description is different from wikidata articles lacking in text citations from january 2020 all articles lacking in text citations all articles with unsourced statements articles with unsourced statements from july 2013 this page was last edited on 20 february 2026 at 23 04 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents multivalued function 24 languages add topic