777:
4579:, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He, however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption." §535.
1214:
1480:
3778:. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories -- specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms
3410:
2683:
3076:
3599:
4350:(1952) . "§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions".
3186:
2842:
2285:. The order is important because function composition is not necessarily commutative. Having successive transformations applying and composing to the right agrees with the left-to-right reading sequence.
4113:
3786:
form categories, and so the approach via categories fits well with the objective of organizing and understanding
Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics.
3414:
A unary operation always commutes with itself, but this is not necessarily the case for a binary (or higher arity) operation. A binary (or higher arity) operation that commutes with itself is called
2501:
3719:. The axioms of a category are in fact inspired from the properties (and also the definition) of function composition. The structures given by composition are axiomatized and generalized in
2912:
4680:
629:
3772:. It is possible to start differently, by axiomatising not elements of sets but functions between sets. This can be done by using the language of categories and universal constructions.
3110:. The partial composition in only one argument mentioned previously can be instantiated from this more general scheme by setting all argument functions except one to be suitably chosen
3691:
2415:
1204:
683:
1550:
In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the symmetric semigroup is a
3468:
762:
505:
2708:
4763:
723:
703:
549:
529:
3896:
4635:). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log
3870:
2430:
31:
4302:
5162:
5075:
5019:
4988:
4894:
4844:
4814:
4365:
4269:
4238:
4211:
4184:
4153:
4097:
4038:
257:
1476:. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.)
4463:
4261:
3405:{\displaystyle f(g(a_{11},\ldots ,a_{1m}),\ldots ,g(a_{n1},\ldots ,a_{nm}))=g(f(a_{11},\ldots ,a_{n1}),\ldots ,f(a_{1m},\ldots ,a_{nm})).}
2318:" for this, thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in the
467:
212:
5044:
4861:
4662:(xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
2846:
In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of
5157:
4675:
4448:
4290:
1774:
1263:. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example,
85:
3136:-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition.
5141:
3795:
3151:
if it contains all projections and is closed under generalized composition. A clone generally contains operations of various
443:
242:
5129:
4806:
4357:
4145:
1465:
2847:
1442:. In general, transformation monoids can have remarkably complicated structure. One particular notable example is the
1340:
5124:
2375:
197:
3701:
1973:
However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g.,
5167:
4030:
277:
776:
2696:
1402:
554:
365:
2066:
3986:
3965:
3881:
3748:
3625:
3619:
3426:
2331:
1035:
741:
4576:
4321:
1770:
1940:. For trigonometric functions, usually the latter is meant, at least for positive exponents. For example, in
4689:
3140:
1949:
460:
227:
152:
107:
3978:
3886:
3716:
3155:. The notion of commutation also finds an interesting generalization in the multivariate case; a function
2369:
2003:
1207:
1159:
1135:
508:
60:
2678:{\displaystyle f|_{x_{i}=g}=f(x_{1},\ldots ,x_{i-1},g(x_{1},x_{2},\ldots ,x_{n}),x_{i+1},\ldots ,x_{n}).}
5119:
5091:
3891:
3071:{\displaystyle h(x_{1},\ldots ,x_{m})=f(g_{1}(x_{1},\ldots ,x_{m}),\ldots ,g_{n}(x_{1},\ldots ,x_{m})).}
2448:
2282:
1434:
1367:
764:. As a result, all properties of composition of relations are true of composition of functions, such as
423:
4692:, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall: 8–26 .
3727:
as the category-theoretical replacement of functions. The reversed order of composition in the formula
1170:
5067:
5011:
4089:
3969:
2436:
2343:
2323:
2278:
1503:
1355:
638:
408:
182:
93:
3693:
however, the text sequence is reversed to illustrate the different operation sequences accordingly.
4083:
3111:
1511:
1425:
1291:
342:
332:
327:
42:
4940:
4757:
4709:
4701:
3934:
3865:
2062:
1888:
453:
337:
312:
3594:{\displaystyle R\circ S=\{(x,z)\in X\times Z:(\exists y\in Y)((x,y)\in R\,\land \,(y,z)\in S)\}}
662:
1494:. On the left is the original object. Above is shear, then rotate. Below is rotate, then shear.
725:
second. Intuitively, reverse composition is a chaining process in which the output of function
5071:
5040:
5015:
5005:
4984:
4890:
4840:
4834:
4810:
4800:
4777:
4383:
4361:
4265:
4234:
4207:
4180:
4149:
4139:
4093:
4034:
4024:
3946:
3901:
3854:
3791:
3752:
3442:
2156:
2074:
2058:
1779:
1563:
1551:
1336:
413:
317:
307:
292:
287:
5061:
4980:
4976:
4255:
4228:
2302:, in keeping with the order the symbols occur in postfix notation, thus making the notation "
1502:(and thus invertible), then the set of all possible combinations of these functions forms a
5137:
4932:
4693:
4294:
4201:
4170:
3859:
3708:
3697:
2206:
1862:
1838:
1303:
1213:
747:
490:
403:
380:
137:
2330:, it is correct to use the semicolon for function composition as well (see the article on
1514:, essentially says that any group is in fact just a subgroup of a permutation group (up to
4745:
4584:
3930:
3906:
3875:
3760:
3720:
3608:), function composition satisfies the definition for relation composition. A small circle
3605:
3430:
3148:
2418:
2327:
2254:
2152:
1540:
1507:
398:
375:
322:
4870:
4599:—Three principal notations have been used to denote, say, the square of sin
4314:
4727:
4347:
2258:
1945:
1743:
1727:
1347:
1226:
1222:
708:
688:
534:
514:
418:
4301:. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 .
1479:
30:
This article is about the mathematical concept. For the computer science concept, see
5151:
4917:
4886:
4713:
3828:
1484:
1443:
1295:
49:
38:
4944:
4141:
Mathematics across the Iron
Curtain: A History of the Algebraic Theory of Semigroups
2695:, composition degenerates into a (partial) valuation, whose result is also known as
4830:
3756:
3415:
3176:
2183:
1941:
814:
122:
17:
4619:, though the first is least likely to be misinterpreted. In the case of sin
4356:. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, USA:
4299:
A Collection of
Examples of the Applications of the Calculus of Finite Differences
1101:
of the latter. Moreover, it is often convenient to tacitly restrict the domain of
4966:
4351:
4058:
3848:
1536:
1515:
1242:
1218:
1129:
1031:
838:
481:
4575:, etc., "as he then supposed for the first time. The work of a German Analyst,
4781:
4387:
3950:
3769:
2837:{\displaystyle f|_{x_{i}=b}=f(x_{1},\ldots ,x_{i-1},b,x_{i+1},\ldots ,x_{n}).}
2319:
2266:
2160:
1332:
1328:
1298:(surjective) functions is always onto. It follows that the composition of two
1076:. Since the parentheses do not change the result, they are generally omitted.
302:
297:
4972:
4936:
4176:
3982:
1499:
1351:
1299:
347:
4697:
4885:(NB. This is the updated and free version of book originally published by
4587:'s books, to remove the chief objection to them; Peirce wrote: "cos
1331:
of compositions involving differentiable functions can be found using the
1294:(injective) functions is always one-to-one. Similarly, the composition of
3996:
3724:
3712:
2070:
1406:. Then one can form chains of transformations composed together, such as
428:
167:
97:
3622:, as well as functions. When used to represent composition of functions
2061:
becomes a continuous parameter; in this case, such a system is called a
2057:
Under additional restrictions, this idea can be generalized so that the
433:
4705:
3604:
Considering a function as a special case of a binary relation (namely
4583:— The use of Herschel's notation underwent a slight change in
1573:
1429:
1098:
1097:; in a wider sense, it is sufficient that the former be an improper
4295:"Part III. Section I. Examples of the Direct Method of Differences"
4802:
Making
Mathematics Come to Life: A Guide for Teachers and Students
4651:) are of frequent occurrence in analysis. The notation sin
3152:
3133:
1539:) forms a group with respect to function composition. This is the
1478:
1212:
775:
4085:
Learning to Reason: An
Introduction to Logic, Sets, and Relations
4491:), but he justifies his own notation by pointing out that since
2435:
Function composition appears in one form or another in numerous
1777:. Repeated composition of such a function with itself is called
2069:. Iterated functions and flows occur naturally in the study of
1229:, in different orders, show a non-commutativity of composition.
4555:.=c. Some years later Herschel explained that in 1813 he used
3832:
1346:
Composition of functions is sometimes described as a kind of
3463:
are two binary relations, then their composition amounts to
1400:
having the same domain and codomain; these are often called
1350:
on a function space, but has very different properties from
1306:
of a composition (assumed invertible) has the property that
214:
4786:
Encyclopédie des sciences mathématiques pures et appliquées
4678:(1813) . "On a Remarkable Application of Cotes's Theorem".
4257:
1510:
by these functions. A fundamental result in group theory,
3964:
to denote function compositions must not be confused with
4681:
4623:
two interpretations suggest themselves; first, sin
2205:
During the mid-20th century, some mathematicians adopted
685:, applies the operation in the opposite order, applying
37:"Ring operator" redirects here. Not to be confused with
5039:. AMS Mathematical Surveys and Monographs. p. xv.
4752:. Vol. I (new ed.). Boston, USA. p. 203.
3933:, where a subset relation is modelled explicitly by an
4659:) has been widely used and is now the prevailing one.
4543: V=∫ V, we may write similarly sin.
4203:
Algebraic Theory of
Automata Networks: An introduction
3862:, a formal axiomatization of the composition operation
3768:
The standard "foundation" for mathematics starts with
2478:
in some computer engineering contexts, and is denoted
2080:
To avoid ambiguity, some mathematicians choose to use
1594:
may compose with itself; this is sometimes denoted as
780:
Concrete example for the composition of two functions.
740:
The composition of functions is a special case of the
3831:
article for similar-appearing
Unicode characters. In
3628:
3471:
3189:
2915:
2711:
2504:
2378:
1354:
multiplication of functions (e.g. composition is not
1173:
750:
711:
691:
665:
557:
537:
517:
493:
4581:
Persistence of rival notations for inverse function.
4479:, but what is usually written thus, arc (cos.=
4968:
Universal
Algebra: Fundamentals and Selected Topics
4254:Ganyushkin, Olexandr; Mazorchuk, Volodymyr (2008).
4172:
Semigroups: An Introduction to the Structure Theory
2288:
Mathematicians who use postfix notation may write "
143:
4918:"Logic Minimization Algorithms for VLSI Synthesis"
4615:. The prevailing notation at present is sin
3851:– a graphical technique for functional composition
3685:
3593:
3404:
3070:
2836:
2677:
2417:Composition operators are studied in the field of
2409:
1198:
756:
717:
697:
677:
623:
543:
523:
499:
3878:, distribution of a function of a random variable
3700:and Cayley's theorem has its analogue called the
139:
4670:
4668:
4285:
4283:
4281:
1944:, this superscript notation represents standard
259:
4960:
4958:
4956:
4954:
4483:)." He admits that some authors use cos.
4018:
4016:
3696:The composition is defined in the same way for
4788:(in French). Vol. I. p. 195. Part I.
4475:must not be understood to signify 1/cos.
4342:
4340:
4338:
4336:
4334:
4332:
4330:
2034:has a unique solution for some natural number
1017:is the pressure around the plane at time
4200:Dömösi, Pál; Nehaniv, Chrystopher L. (2005).
2084:to denote the compositional meaning, writing
785:Composition of functions on a finite set: If
461:
8:
3588:
3484:
2451:. The function resulting when some argument
2308:" ambiguous. Computer scientists may write "
115:
3897:Infinite compositions of analytic functions
624:{\displaystyle h(x):=(g\circ f)(x)=g(f(x))}
48:"∘" redirects here. For the character, see
4762:: CS1 maint: location missing publisher (
3620:infix notation of composition of relations
1980:In some cases, when, for a given function
1891:(in particular for real or complex-valued
468:
454:
124:
56:
3686:{\displaystyle (g\circ f)(x)\ =\ g(f(x))}
3627:
3563:
3559:
3470:
3384:
3362:
3331:
3312:
3278:
3256:
3225:
3206:
3188:
3053:
3034:
3021:
2999:
2980:
2967:
2945:
2926:
2914:
2822:
2797:
2772:
2753:
2726:
2721:
2716:
2710:
2663:
2638:
2622:
2603:
2590:
2565:
2546:
2519:
2514:
2509:
2503:
2383:
2377:
1189:
1172:
749:
710:
690:
664:
556:
536:
516:
492:
4468:
4317:
1372:Suppose one has two (or more) functions
4734:(in French). Vol. IV. p. 229.
4012:
3977:, introduced by Hans Maurer (1901) and
3918:
3871:Function composition (computer science)
2431:Function composition (computer science)
2334:for further details on this notation).
2186:, omit the composition symbol, writing
1030:The composition of functions is always
971:If an airplane's altitude at time
442:
389:
356:
276:
244:
235:
91:
84:
59:
32:Function composition (computer science)
5060:Hilton, Peter; Wu, Yel-Chiang (1989).
4860:Barr, Michael; Wells, Charles (1998).
4755:
4026:How to Prove It: A Structured Approach
2002:, that function can be defined as the
1109:produces only values in the domain of
1089:is only meaningful if the codomain of
4863:Category Theory for Computing Science
4535:for log. log. log.
4262:Springer Science & Business Media
2372:which maps functions to functions as
2182:Many mathematicians, particularly in
1287:. The picture shows another example.
263:
248:
229:
218:
203:
184:
173:
154:
128:
109:
7:
4464:Philosophical Transactions of London
4461:, etc., was published by him in the
4316:(NB. Inhere, Herschel refers to his
4052:
4050:
2447:Partial composition is possible for
1898:), there is a risk of confusion, as
199:
190:
4382:We note here the symbolism used by
4360:. pp. 108, 176–179, 336, 346.
4353:A History of Mathematical Notations
3163:is said to commute with a function
1521:The set of all bijective functions
1079:In a strict sense, the composition
986:, and the air pressure at altitude
27:Operation on mathematical functions
4597:Powers of trigonometric functions.
4451:'s notation for inverse functions,
3520:
1794:is defined as the identity map on
808:= {(1, 2), (2, 1), (3, 2), (4, 3)}
797:= {(1, 2), (2, 3), (3, 1), (4, 2)}
790:= {(1, 1), (2, 3), (3, 1), (4, 2)}
25:
2322:the ⨾ character is used for left
2065:, specified through solutions of
1506:; and one says that the group is
5092:"Saunders Mac Lane - Quotations"
4676:Herschel, John Frederick William
4291:Herschel, John Frederick William
3429:can be generalized to arbitrary
2410:{\displaystyle C_{g}f=f\circ g.}
2253:. This can be more natural than
1490:and a clockwise rotation by 45°
1199:{\displaystyle g(x)={\sqrt {x}}}
169:
160:
4631:; second, sin (sin
4523:, we ought to write sin.
4305:from the original on 2020-08-04
4114:"3.4: Composition of Functions"
3825:∘, ∘
3132:can be seen as a single vector/
1865:power of the inverse function:
1775:John Frederick William Herschel
1339:of such functions are given by
1113:. For example, the composition
1034:—a property inherited from the
813:Composition of functions on an
765:
86:History of the function concept
50:Degree symbol § Lookalikes
5142:Wolfram Demonstrations Project
4925:IEEE Transactions on Computers
4799:Ivanov, Oleg A. (2009-01-01).
4467:, for the year 1813. He says (
4138:Hollings, Christopher (2014).
3796:Mathematics: Form and Function
3680:
3677:
3671:
3665:
3650:
3644:
3641:
3629:
3585:
3576:
3564:
3550:
3538:
3535:
3532:
3517:
3499:
3487:
3396:
3393:
3355:
3340:
3305:
3299:
3290:
3287:
3249:
3234:
3199:
3193:
3062:
3059:
3027:
3005:
2973:
2960:
2951:
2919:
2828:
2746:
2717:
2669:
2628:
2583:
2539:
2510:
1747:can be defined inductively by
1183:
1177:
669:
618:
615:
609:
603:
594:
588:
585:
573:
567:
561:
1:
4916:Bryant, R. E. (August 1986).
4807:American Mathematical Society
4471:): "This notation cos.
4358:Open court publishing company
4227:Carter, Nathan (2009-04-09).
4146:American Mathematical Society
3995:pre-superscript notation for
3080:This is sometimes called the
1847:, negative functional powers
1466:full transformation semigroup
551:, and returns a new function
5163:Basic concepts in set theory
5037:Symmetric Inverse Semigroups
4906:ISO/IEC 13568:2002(E), p. 23
4750:Curves, Functions and Forces
4023:Velleman, Daniel J. (2006).
2848:primitive recursive function
2466:is replaced by the function
2099:-th iterate of the function
1543:, also sometimes called the
744:, sometimes also denoted by
731:feeds the input of function
5125:Encyclopedia of Mathematics
4869:. p. 6. Archived from
4169:Grillet, Pierre A. (1995).
3876:Function of random variable
2470:is called a composition of
2326:. Since all functions are
1769:, a notation introduced by
1498:If the transformations are
5184:
5063:A Course in Modern Algebra
5004:Tourlakis, George (2012).
4965:Bergman, Clifford (2011).
4527:for sin. sin.
4031:Cambridge University Press
3925:The strict sense is used,
2428:
2341:
2281:and the composition is by
2257:in many cases, such as in
1561:
1365:
678:{\displaystyle f\mapsto g}
444:List of specific functions
47:
36:
29:
4839:. Springer. p. 118.
4647:and log (log
1905:could also stand for the
1302:is also a bijection. The
810:, as shown in the figure.
5138:Composition of Functions
3987:David Patterson Ellerman
3966:Rudolf von Bitter Rucker
3882:Functional decomposition
3759:. These structures form
3749:composition of relations
2425:In programming languages
2332:composition of relations
2140:. For the same purpose,
1726:More generally, for any
1438:or (much more seldom) a
1036:composition of relations
742:composition of relations
5140:" by Bruce Atwood, the
4937:10.1109/tc.1986.1676819
4732:Formulaire mathématique
4690:Royal Society of London
4082:Rodgers, Nancy (2000).
3806:The composition symbol
3770:sets and their elements
3183:, and vice versa i.e.:
2294:", meaning first apply
1950:trigonometric functions
1424:. Such chains have the
5158:Functions and mappings
4698:10.1098/rstl.1813.0005
4118:Mathematics LibreTexts
3979:Reuben Louis Goodstein
3887:Functional square root
3799:
3702:Wagner–Preston theorem
3687:
3618:has been used for the
3595:
3406:
3072:
2838:
2679:
2449:multivariate functions
2443:Multivariate functions
2411:
2348:Given a function
2110:, as in, for example,
2004:functional square root
1998:has a unique solution
1909:-fold product of
1887:takes its values in a
1495:
1341:Faà di Bruno's formula
1230:
1206:can be defined on the
1200:
781:
758:
757:{\displaystyle \circ }
719:
699:
679:
625:
545:
525:
501:
500:{\displaystyle \circ }
5068:John Wiley & Sons
5035:Lipscomb, S. (1997).
5012:John Wiley & Sons
5007:Theory of Computation
4603:, namely, (sin
4322:Hans Heinrich Bürmann
4090:John Wiley & Sons
4063:mathworld.wolfram.com
3892:Higher-order function
3765:
3688:
3596:
3407:
3082:generalized composite
3073:
2883:, the composition of
2839:
2691:is a simple constant
2680:
2437:programming languages
2412:
2283:matrix multiplication
2178:Alternative notations
2024:More generally, when
1771:Hans Heinrich Bürmann
1482:
1435:transformation monoid
1368:Transformation monoid
1216:
1201:
1093:equals the domain of
1050:are composable, then
779:
759:
720:
700:
680:
631:. Thus, the function
626:
546:
526:
502:
5120:"Composite function"
4836:Discrete Mathematics
4092:. pp. 359–362.
3723:with the concept of
3715:is the prototypical
3626:
3606:functional relations
3469:
3187:
3112:projection functions
2913:
2709:
2502:
2376:
2356:composition operator
2344:Composition operator
2338:Composition operator
2324:relation composition
1975:tan = arctan ≠ 1/tan
1504:transformation group
1217:Compositions of two
1171:
748:
709:
689:
663:
659:, sometimes denoted
555:
535:
515:
491:
486:composition operator
4539:. Just as we write
4380:Iterated logarithms
4233:. MAA. p. 95.
4230:Visual Group Theory
4206:. SIAM. p. 8.
4057:Weisstein, Eric W.
3953:'s (1907) notation
2858:-ary function, and
2368:is defined as that
2067:Schröder's equation
1470:symmetric semigroup
1426:algebraic structure
1362:Composition monoids
1290:The composition of
1271:| + 3 = |
1245:with each other if
657:Reverse composition
43:operator assistance
18:Composition of maps
4778:Pringsheim, Alfred
4688:(Part 1). London:
4643:⋅ log
4639:, where log
4627:⋅ sin
3935:inclusion function
3866:Flow (mathematics)
3753:converse relations
3711:with functions as
3683:
3591:
3416:medial or entropic
3402:
3139:A set of finitary
3068:
2834:
2675:
2407:
2048:can be defined as
2010:, then written as
1780:function iteration
1496:
1440:composition monoid
1337:Higher derivatives
1231:
1196:
837:is the set of all
782:
754:
715:
695:
675:
621:
541:
521:
497:
278:Classes/properties
5168:Binary operations
5077:978-0-471-50405-4
5021:978-1-118-31533-0
4990:978-1-4398-5129-6
4895:978-0-13-120486-7
4846:978-1-4419-8047-2
4816:978-0-8218-4808-1
4809:. pp. 217–.
4547:=arc (sin.=
4367:978-1-60206-714-1
4271:978-1-84800-281-4
4240:978-0-88385-757-1
4213:978-0-89871-569-9
4186:978-0-8247-9662-4
4155:978-1-4704-1493-1
4099:978-0-471-37122-9
4040:978-1-139-45097-3
3947:Alfred Pringsheim
3902:Iterated function
3855:Combinatory logic
3792:Saunders Mac Lane
3761:dagger categories
3698:partial functions
3661:
3655:
3143:on some base set
2157:Alfred Pringsheim
2075:dynamical systems
1564:Iterated function
1558:Functional powers
1552:regular semigroup
1545:composition group
1493:
1489:
1483:Composition of a
1194:
1123:of the functions
718:{\displaystyle g}
698:{\displaystyle f}
544:{\displaystyle g}
524:{\displaystyle f}
478:
477:
390:Generalizations
16:(Redirected from
5175:
5133:
5106:
5105:
5103:
5102:
5088:
5082:
5081:
5057:
5051:
5050:
5032:
5026:
5025:
5001:
4995:
4994:
4962:
4949:
4948:
4922:
4913:
4907:
4904:
4898:
4884:
4882:
4881:
4875:
4868:
4857:
4851:
4850:
4827:
4821:
4820:
4796:
4790:
4789:
4774:
4768:
4767:
4761:
4753:
4746:Peirce, Benjamin
4742:
4736:
4735:
4724:
4718:
4717:
4672:
4663:
4661:
4487:for (cos.
4375:
4374:
4344:
4325:
4313:
4311:
4310:
4287:
4276:
4275:
4251:
4245:
4244:
4224:
4218:
4217:
4197:
4191:
4190:
4166:
4160:
4159:
4135:
4129:
4128:
4126:
4125:
4110:
4104:
4103:
4079:
4073:
4072:
4070:
4069:
4054:
4045:
4044:
4020:
4000:
3994:
3976:
3963:
3944:
3938:
3923:
3860:Composition ring
3838:
3835:, it is written
3826:
3822:
3819:
3816:
3814:
3809:
3746:
3709:category of sets
3692:
3690:
3689:
3684:
3659:
3653:
3617:
3600:
3598:
3597:
3592:
3462:
3448:
3431:binary relations
3411:
3409:
3408:
3403:
3392:
3391:
3370:
3369:
3339:
3338:
3317:
3316:
3286:
3285:
3264:
3263:
3233:
3232:
3211:
3210:
3131:
3109:
3077:
3075:
3074:
3069:
3058:
3057:
3039:
3038:
3026:
3025:
3004:
3003:
2985:
2984:
2972:
2971:
2950:
2949:
2931:
2930:
2908:
2904:
2886:
2882:
2864:
2861:
2857:
2853:
2843:
2841:
2840:
2835:
2827:
2826:
2808:
2807:
2783:
2782:
2758:
2757:
2739:
2738:
2731:
2730:
2720:
2694:
2690:
2684:
2682:
2681:
2676:
2668:
2667:
2649:
2648:
2627:
2626:
2608:
2607:
2595:
2594:
2576:
2575:
2551:
2550:
2532:
2531:
2524:
2523:
2513:
2498:
2477:
2473:
2469:
2465:
2462:of the function
2461:
2416:
2414:
2413:
2408:
2388:
2387:
2367:
2353:
2328:binary relations
2317:
2307:
2301:
2297:
2293:
2276:
2272:
2264:
2252:
2237:
2226:
2215:
2207:postfix notation
2201:
2191:
2173:
2150:
2139:
2120:
2109:
2098:
2094:
2083:
2053:
2047:
2040:
2033:
2020:
2009:
2001:
1997:
1983:
1976:
1969:
1939:
1912:
1908:
1904:
1897:
1886:
1875:
1860:
1854:are defined for
1853:
1846:
1839:inverse function
1836:
1822:
1809:
1800:
1793:
1768:
1739:
1735:
1723:
1674:
1633:
1599:
1593:
1579:
1534:
1512:Cayley's theorem
1491:
1487:
1475:
1463:
1423:
1399:
1386:
1324:
1304:inverse function
1286:
1279:
1277:
1270:
1262:
1240:
1236:
1205:
1203:
1202:
1197:
1195:
1190:
1166:
1152:
1138:
1122:
1112:
1108:
1104:
1096:
1092:
1088:
1075:
1049:
1045:
1041:
1020:
1016:
1000:
989:
985:
974:
966:
924:
883:
869:
855:
836:
830:
809:
798:
791:
763:
761:
760:
755:
736:
730:
724:
722:
721:
716:
704:
702:
701:
696:
684:
682:
681:
676:
652:
646:
636:
630:
628:
627:
622:
550:
548:
547:
542:
530:
528:
527:
522:
506:
504:
503:
498:
470:
463:
456:
268:
267:
261:
253:
252:
246:
238:
237:
233:
223:
222:
216:
208:
207:
201:
193:
192:
188:
178:
177:
171:
163:
162:
158:
148:
147:
141:
133:
132:
126:
118:
117:
113:
80:
57:
21:
5183:
5182:
5178:
5177:
5176:
5174:
5173:
5172:
5148:
5147:
5118:
5115:
5110:
5109:
5100:
5098:
5090:
5089:
5085:
5078:
5059:
5058:
5054:
5047:
5034:
5033:
5029:
5022:
5014:. p. 100.
5003:
5002:
4998:
4991:
4964:
4963:
4952:
4920:
4915:
4914:
4910:
4905:
4901:
4879:
4877:
4873:
4866:
4859:
4858:
4854:
4847:
4829:
4828:
4824:
4817:
4798:
4797:
4793:
4776:
4775:
4771:
4754:
4744:
4743:
4739:
4728:Peano, Giuseppe
4726:
4725:
4721:
4674:
4673:
4666:
4655:for (sin
4585:Benjamin Peirce
4441:
4435:
4425:
4415:
4409:
4399:
4390:in their joint
4372:
4370:
4368:
4348:Cajori, Florian
4346:
4345:
4328:
4324:'s older work.)
4308:
4306:
4289:
4288:
4279:
4272:
4253:
4252:
4248:
4241:
4226:
4225:
4221:
4214:
4199:
4198:
4194:
4187:
4168:
4167:
4163:
4156:
4148:. p. 334.
4137:
4136:
4132:
4123:
4121:
4112:
4111:
4107:
4100:
4081:
4080:
4076:
4067:
4065:
4056:
4055:
4048:
4041:
4033:. p. 232.
4022:
4021:
4014:
4009:
4004:
4003:
3990:
3972:
3954:
3945:
3941:
3931:category theory
3924:
3920:
3915:
3907:Lambda calculus
3845:
3836:
3824:
3820:
3817:
3812:
3811:
3807:
3804:
3776:
3728:
3721:category theory
3624:
3623:
3609:
3467:
3466:
3450:
3434:
3424:
3422:Generalizations
3380:
3358:
3327:
3308:
3274:
3252:
3221:
3202:
3185:
3184:
3130:
3121:
3115:
3108:
3099:
3093:
3049:
3030:
3017:
2995:
2976:
2963:
2941:
2922:
2911:
2910:
2906:
2903:
2894:
2888:
2884:
2881:
2872:
2866:
2865:-ary functions
2862:
2859:
2855:
2851:
2818:
2793:
2768:
2749:
2722:
2715:
2707:
2706:
2692:
2688:
2659:
2634:
2618:
2599:
2586:
2561:
2542:
2515:
2508:
2500:
2499:
2497:
2492:
2479:
2475:
2471:
2467:
2463:
2460:
2452:
2445:
2433:
2427:
2419:operator theory
2379:
2374:
2373:
2366:
2358:
2349:
2346:
2340:
2309:
2303:
2299:
2298:and then apply
2295:
2289:
2274:
2270:
2262:
2255:prefix notation
2239:
2228:
2217:
2210:
2193:
2187:
2180:
2164:
2153:Benjamin Peirce
2141:
2122:
2111:
2100:
2096:
2085:
2081:
2059:iteration count
2049:
2042:
2035:
2025:
2011:
2007:
1999:
1985:
1984:, the equation
1981:
1974:
1955:
1948:when used with
1914:
1910:
1906:
1899:
1892:
1884:
1866:
1855:
1848:
1841:
1824:
1814:
1808:
1802:
1795:
1788:
1787:By convention,
1748:
1737:
1730:
1724:
1677:
1675:
1636:
1634:
1603:
1595:
1581:
1569:
1566:
1560:
1541:symmetric group
1522:
1473:
1451:
1407:
1403:transformations
1387:
1373:
1370:
1364:
1307:
1281:
1272:
1266:
1264:
1246:
1238:
1234:
1221:functions, the
1169:
1168:
1154:
1140:
1124:
1114:
1110:
1106:
1102:
1094:
1090:
1080:
1051:
1047:
1043:
1039:
1028:
1018:
1002:
991:
987:
976:
972:
968:
928:
926:
886:
871:
857:
842:
832:
818:
800:
793:
786:
774:
746:
745:
732:
726:
707:
706:
687:
686:
661:
660:
648:
642:
641:after applying
632:
553:
552:
533:
532:
513:
512:
489:
488:
474:
438:
399:Binary relation
385:
352:
272:
266:
258:
251:
243:
232:
228:
221:
213:
206:
198:
187:
183:
176:
168:
157:
153:
146:
138:
131:
123:
112:
108:
67:
53:
46:
35:
28:
23:
22:
15:
12:
11:
5:
5181:
5179:
5171:
5170:
5165:
5160:
5150:
5149:
5146:
5145:
5134:
5114:
5113:External links
5111:
5108:
5107:
5083:
5076:
5070:. p. 65.
5052:
5045:
5027:
5020:
4996:
4989:
4950:
4931:(8): 677–691.
4908:
4899:
4852:
4845:
4822:
4815:
4791:
4769:
4737:
4719:
4664:
4591:," "log
4571:), sin.
4551:), log.
4437:
4431:
4421:
4411:
4405:
4395:
4366:
4326:
4277:
4270:
4264:. p. 24.
4246:
4239:
4219:
4212:
4192:
4185:
4161:
4154:
4130:
4105:
4098:
4074:
4046:
4039:
4011:
4010:
4008:
4005:
4002:
4001:
3939:
3917:
3916:
3914:
3911:
3910:
3909:
3904:
3899:
3894:
3889:
3884:
3879:
3873:
3868:
3863:
3857:
3852:
3844:
3841:
3810:is encoded as
3803:
3800:
3782:like functions
3755:, and thus in
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3658:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3562:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3423:
3420:
3401:
3398:
3395:
3390:
3387:
3383:
3379:
3376:
3373:
3368:
3365:
3361:
3357:
3354:
3351:
3348:
3345:
3342:
3337:
3334:
3330:
3326:
3323:
3320:
3315:
3311:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3284:
3281:
3277:
3273:
3270:
3267:
3262:
3259:
3255:
3251:
3248:
3245:
3242:
3239:
3236:
3231:
3228:
3224:
3220:
3217:
3214:
3209:
3205:
3201:
3198:
3195:
3192:
3126:
3119:
3104:
3097:
3067:
3064:
3061:
3056:
3052:
3048:
3045:
3042:
3037:
3033:
3029:
3024:
3020:
3016:
3013:
3010:
3007:
3002:
2998:
2994:
2991:
2988:
2983:
2979:
2975:
2970:
2966:
2962:
2959:
2956:
2953:
2948:
2944:
2940:
2937:
2934:
2929:
2925:
2921:
2918:
2909:-ary function
2899:
2892:
2877:
2870:
2833:
2830:
2825:
2821:
2817:
2814:
2811:
2806:
2803:
2800:
2796:
2792:
2789:
2786:
2781:
2778:
2775:
2771:
2767:
2764:
2761:
2756:
2752:
2748:
2745:
2742:
2737:
2734:
2729:
2725:
2719:
2714:
2674:
2671:
2666:
2662:
2658:
2655:
2652:
2647:
2644:
2641:
2637:
2633:
2630:
2625:
2621:
2617:
2614:
2611:
2606:
2602:
2598:
2593:
2589:
2585:
2582:
2579:
2574:
2571:
2568:
2564:
2560:
2557:
2554:
2549:
2545:
2541:
2538:
2535:
2530:
2527:
2522:
2518:
2512:
2507:
2488:
2484:
2456:
2444:
2441:
2429:Main article:
2426:
2423:
2406:
2403:
2400:
2397:
2394:
2391:
2386:
2382:
2362:
2342:Main article:
2339:
2336:
2259:linear algebra
2179:
2176:
1946:exponentiation
1878:
1877:
1811:
1804:
1728:natural number
1676:
1635:
1602:
1562:Main article:
1559:
1556:
1464:is called the
1366:Main article:
1363:
1360:
1348:multiplication
1233:The functions
1227:cubic function
1223:absolute value
1193:
1188:
1185:
1182:
1179:
1176:
1038:. That is, if
1027:
1024:
1023:
1022:
969:
927:
885:
841:) is given by
811:
773:
770:
753:
714:
694:
674:
671:
668:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
560:
540:
520:
496:
476:
475:
473:
472:
465:
458:
450:
447:
446:
440:
439:
437:
436:
431:
426:
421:
416:
411:
406:
401:
395:
392:
391:
387:
386:
384:
383:
378:
373:
368:
362:
359:
358:
354:
353:
351:
350:
345:
340:
335:
330:
325:
320:
315:
310:
305:
300:
295:
290:
284:
281:
280:
274:
273:
271:
270:
264:
255:
249:
240:
230:
225:
219:
210:
204:
195:
185:
180:
174:
165:
155:
150:
144:
135:
129:
120:
110:
104:
101:
100:
89:
88:
82:
81:
64:
63:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5180:
5169:
5166:
5164:
5161:
5159:
5156:
5155:
5153:
5143:
5139:
5135:
5131:
5127:
5126:
5121:
5117:
5116:
5112:
5097:
5096:Maths History
5093:
5087:
5084:
5079:
5073:
5069:
5065:
5064:
5056:
5053:
5048:
5046:0-8218-0627-0
5042:
5038:
5031:
5028:
5023:
5017:
5013:
5009:
5008:
5000:
4997:
4992:
4986:
4982:
4978:
4974:
4970:
4969:
4961:
4959:
4957:
4955:
4951:
4946:
4942:
4938:
4934:
4930:
4926:
4919:
4912:
4909:
4903:
4900:
4896:
4892:
4888:
4887:Prentice Hall
4876:on 2016-03-04
4872:
4865:
4864:
4856:
4853:
4848:
4842:
4838:
4837:
4832:
4831:Gallier, Jean
4826:
4823:
4818:
4812:
4808:
4804:
4803:
4795:
4792:
4787:
4783:
4779:
4773:
4770:
4765:
4759:
4751:
4747:
4741:
4738:
4733:
4729:
4723:
4720:
4715:
4711:
4707:
4703:
4699:
4695:
4691:
4687:
4683:
4682:
4677:
4671:
4669:
4665:
4660:
4658:
4654:
4650:
4646:
4642:
4638:
4634:
4630:
4626:
4622:
4618:
4614:
4610:
4607:), sin
4606:
4602:
4598:
4594:
4590:
4586:
4582:
4578:
4574:
4570:
4566:
4562:
4558:
4554:
4550:
4546:
4542:
4538:
4534:
4531:, log.
4530:
4526:
4522:
4518:
4514:
4510:
4506:
4502:
4498:
4494:
4490:
4486:
4482:
4478:
4474:
4470:
4466:
4465:
4460:
4456:
4452:
4450:
4449:John Herschel
4445:
4440:
4434:
4429:
4424:
4419:
4414:
4408:
4403:
4398:
4394:article: "log
4393:
4389:
4385:
4381:
4369:
4363:
4359:
4355:
4354:
4349:
4343:
4341:
4339:
4337:
4335:
4333:
4331:
4327:
4323:
4320:and mentions
4319:
4315:
4304:
4300:
4296:
4292:
4286:
4284:
4282:
4278:
4273:
4267:
4263:
4259:
4258:
4250:
4247:
4242:
4236:
4232:
4231:
4223:
4220:
4215:
4209:
4205:
4204:
4196:
4193:
4188:
4182:
4179:. p. 2.
4178:
4174:
4173:
4165:
4162:
4157:
4151:
4147:
4143:
4142:
4134:
4131:
4119:
4115:
4109:
4106:
4101:
4095:
4091:
4087:
4086:
4078:
4075:
4064:
4060:
4059:"Composition"
4053:
4051:
4047:
4042:
4036:
4032:
4028:
4027:
4019:
4017:
4013:
4006:
3998:
3993:
3988:
3984:
3980:
3975:
3971:
3967:
3961:
3957:
3952:
3948:
3943:
3940:
3936:
3932:
3928:
3922:
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3912:
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3905:
3903:
3900:
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3877:
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3869:
3867:
3864:
3861:
3858:
3856:
3853:
3850:
3847:
3846:
3842:
3840:
3834:
3830:
3829:Degree symbol
3821:RING OPERATOR
3801:
3798:
3797:
3793:
3788:
3787:
3783:
3779:
3774:
3773:
3771:
3764:
3762:
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3170:
3166:
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3158:
3154:
3150:
3146:
3142:
3137:
3135:
3129:
3125:
3118:
3113:
3107:
3103:
3096:
3091:
3087:
3086:superposition
3083:
3078:
3065:
3054:
3050:
3046:
3043:
3040:
3035:
3031:
3022:
3018:
3014:
3011:
3008:
3000:
2996:
2992:
2989:
2986:
2981:
2977:
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2935:
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2898:
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2880:
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2743:
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2723:
2712:
2704:
2702:
2698:
2685:
2672:
2664:
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2656:
2653:
2650:
2645:
2642:
2639:
2635:
2631:
2623:
2619:
2615:
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2609:
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2600:
2596:
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2572:
2569:
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2558:
2555:
2552:
2547:
2543:
2536:
2533:
2528:
2525:
2520:
2516:
2505:
2496:
2491:
2487:
2482:
2459:
2455:
2450:
2442:
2440:
2438:
2432:
2424:
2422:
2420:
2404:
2401:
2398:
2395:
2392:
2389:
2384:
2380:
2371:
2365:
2361:
2357:
2352:
2345:
2337:
2335:
2333:
2329:
2325:
2321:
2316:
2312:
2306:
2292:
2286:
2284:
2280:
2268:
2260:
2256:
2250:
2246:
2242:
2236:
2232:
2224:
2220:
2213:
2208:
2203:
2200:
2196:
2190:
2185:
2177:
2175:
2171:
2167:
2162:
2158:
2154:
2148:
2144:
2137:
2133:
2129:
2125:
2118:
2114:
2107:
2103:
2092:
2088:
2078:
2076:
2072:
2068:
2064:
2060:
2055:
2052:
2045:
2038:
2032:
2028:
2022:
2018:
2014:
2005:
1996:
1992:
1988:
1978:
1971:
1967:
1963:
1959:
1953:
1951:
1947:
1943:
1937:
1933:
1929:
1925:
1921:
1917:
1902:
1895:
1890:
1882:
1873:
1869:
1864:
1858:
1851:
1844:
1840:
1835:
1831:
1827:
1821:
1817:
1812:
1807:
1798:
1791:
1786:
1785:
1784:
1782:
1781:
1776:
1772:
1767:
1763:
1759:
1755:
1751:
1746:
1745:
1733:
1729:
1721:
1717:
1713:
1709:
1705:
1701:
1697:
1693:
1689:
1685:
1681:
1672:
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1656:
1652:
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1644:
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1631:
1627:
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1615:
1611:
1607:
1601:
1598:
1592:
1588:
1584:
1578:
1575:
1572:
1565:
1557:
1555:
1553:
1548:
1546:
1542:
1538:
1533:
1529:
1525:
1519:
1517:
1513:
1509:
1505:
1501:
1486:
1485:shear mapping
1481:
1477:
1471:
1467:
1462:
1458:
1454:
1449:
1446:. The set of
1445:
1444:de Rham curve
1441:
1437:
1436:
1431:
1427:
1422:
1418:
1414:
1410:
1405:
1404:
1398:
1394:
1390:
1384:
1380:
1376:
1369:
1361:
1359:
1357:
1353:
1349:
1344:
1342:
1338:
1334:
1330:
1326:
1323:
1319:
1315:
1311:
1305:
1301:
1297:
1293:
1288:
1284:
1275:
1269:
1261:
1257:
1253:
1249:
1244:
1228:
1224:
1220:
1215:
1211:
1209:
1191:
1186:
1180:
1174:
1165:
1161:
1157:
1151:
1147:
1143:
1137:
1133:
1132:
1127:
1121:
1117:
1100:
1087:
1083:
1077:
1074:
1070:
1066:
1062:
1058:
1054:
1037:
1033:
1025:
1014:
1010:
1006:
998:
994:
983:
979:
970:
964:
960:
956:
952:
948:
944:
940:
936:
932:
922:
918:
914:
910:
906:
902:
898:
894:
890:
882:
878:
874:
868:
864:
860:
853:
849:
845:
840:
835:
829:
825:
821:
816:
812:
807:
803:
796:
789:
784:
783:
778:
771:
769:
767:
766:associativity
751:
743:
738:
735:
729:
712:
692:
672:
666:
658:
654:
651:
645:
640:
635:
612:
606:
600:
597:
591:
582:
579:
576:
570:
564:
558:
538:
518:
510:
494:
487:
483:
471:
466:
464:
459:
457:
452:
451:
449:
448:
445:
441:
435:
432:
430:
427:
425:
422:
420:
417:
415:
412:
410:
407:
405:
402:
400:
397:
396:
394:
393:
388:
382:
379:
377:
374:
372:
369:
367:
364:
363:
361:
360:
357:Constructions
355:
349:
346:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
285:
283:
282:
279:
275:
269:
256:
254:
241:
239:
226:
224:
211:
209:
196:
194:
181:
179:
166:
164:
151:
149:
136:
134:
121:
119:
106:
105:
103:
102:
99:
95:
90:
87:
83:
78:
74:
70:
66:
65:
62:
58:
55:
51:
44:
40:
39:operator ring
33:
19:
5123:
5099:. Retrieved
5095:
5086:
5062:
5055:
5036:
5030:
5006:
4999:
4967:
4928:
4924:
4911:
4902:
4878:. Retrieved
4871:the original
4862:
4855:
4835:
4825:
4801:
4794:
4785:
4772:
4749:
4740:
4731:
4722:
4685:
4679:
4656:
4652:
4648:
4644:
4640:
4636:
4632:
4628:
4624:
4620:
4616:
4612:
4611:, sin
4608:
4604:
4600:
4596:
4592:
4588:
4580:
4572:
4568:
4564:
4560:
4556:
4552:
4548:
4544:
4540:
4536:
4532:
4528:
4524:
4520:
4516:
4515:, ΔΔΔ
4512:
4508:
4504:
4500:
4496:
4492:
4488:
4484:
4480:
4476:
4472:
4462:
4458:
4457:, tan
4454:
4447:
4443:
4438:
4432:
4427:
4422:
4417:
4412:
4406:
4401:
4396:
4392:Encyclopédie
4391:
4379:
4377:
4371:. Retrieved
4352:
4307:. Retrieved
4298:
4256:
4249:
4229:
4222:
4202:
4195:
4171:
4164:
4140:
4133:
4122:. Retrieved
4120:. 2020-01-16
4117:
4108:
4084:
4077:
4066:. Retrieved
4062:
4025:
3991:
3973:
3959:
3955:
3942:
3926:
3921:
3805:
3789:
3785:
3781:
3777:
3775:
3767:
3766:
3757:group theory
3747:applies for
3742:
3738:
3734:
3730:
3706:
3695:
3614:
3610:
3603:
3465:
3459:
3455:
3451:
3445:
3439:
3435:
3425:
3413:
3180:
3177:homomorphism
3172:
3168:
3164:
3160:
3156:
3147:is called a
3144:
3138:
3127:
3123:
3116:
3105:
3101:
3094:
3089:
3085:
3081:
3079:
2900:
2896:
2889:
2878:
2874:
2867:
2845:
2705:
2700:
2686:
2494:
2489:
2485:
2480:
2457:
2453:
2446:
2434:
2363:
2359:
2355:
2350:
2347:
2314:
2310:
2304:
2290:
2287:
2248:
2244:
2240:
2234:
2230:
2222:
2218:
2211:
2204:
2198:
2194:
2188:
2184:group theory
2181:
2169:
2165:
2151:was used by
2146:
2142:
2135:
2131:
2127:
2123:
2116:
2112:
2105:
2101:
2090:
2086:
2079:
2056:
2050:
2043:
2036:
2030:
2026:
2023:
2016:
2012:
1994:
1990:
1986:
1979:
1972:
1965:
1961:
1957:
1954:
1942:trigonometry
1935:
1931:
1927:
1923:
1919:
1915:
1900:
1893:
1880:
1879:
1871:
1867:
1856:
1849:
1842:
1833:
1829:
1825:
1819:
1815:
1805:
1796:
1789:
1778:
1765:
1761:
1757:
1753:
1749:
1741:
1731:
1725:
1719:
1715:
1711:
1707:
1703:
1699:
1695:
1691:
1687:
1683:
1679:
1670:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1629:
1625:
1621:
1617:
1613:
1609:
1605:
1596:
1590:
1586:
1582:
1576:
1570:
1567:
1549:
1544:
1537:permutations
1531:
1527:
1523:
1520:
1497:
1469:
1460:
1456:
1452:
1447:
1439:
1433:
1420:
1416:
1412:
1408:
1401:
1396:
1392:
1388:
1382:
1378:
1374:
1371:
1345:
1327:
1321:
1317:
1313:
1309:
1289:
1282:
1273:
1267:
1259:
1255:
1251:
1247:
1241:are said to
1232:
1163:
1155:
1149:
1145:
1141:
1130:
1125:
1119:
1115:
1105:, such that
1085:
1081:
1078:
1072:
1068:
1064:
1060:
1056:
1052:
1029:
1012:
1008:
1004:
996:
992:
981:
977:
962:
958:
954:
950:
946:
942:
938:
934:
930:
920:
916:
912:
908:
904:
900:
896:
892:
888:
880:
876:
872:
870:is given by
866:
862:
858:
851:
847:
843:
839:real numbers
833:
827:
823:
819:
815:infinite set
805:
801:
794:
787:
739:
733:
727:
656:
655:
649:
643:
633:
485:
479:
424:Higher-order
370:
76:
72:
68:
54:
4975:. pp.
4889:in 1990 as
4782:Molk, Jules
4519:, ΣΣ
4446:)." §533.
3981:(1947) for
3849:Cobweb plot
3827:); see the
3427:Composition
3179:preserving
2697:restriction
1870: = (
1801:'s domain,
1742:functional
1600:. That is:
1516:isomorphism
1432:, called a
1356:commutative
1329:Derivatives
1167:defined by
1139:defined by
1032:associative
482:mathematics
409:Multivalued
371:Composition
366:Restriction
5152:Categories
5101:2024-02-13
4880:2014-08-23
4595:." §537.
4503:, Σ
4499:, Δ
4469:p. 10
4453:sin
4384:Pringsheim
4373:2016-01-18
4309:2020-08-04
4124:2020-08-28
4068:2020-08-28
4007:References
3989:'s (1995)
3985:, or with
3968:'s (1982)
3951:Jules Molk
3802:Typography
3141:operations
2320:Z notation
2267:row vector
2209:, writing
2163:suggested
2161:Jules Molk
1837:admits an
1764: ∘
1760: =
1752: =
1450:functions
1333:chain rule
1300:bijections
1292:one-to-one
1280:only when
1026:Properties
705:first and
507:takes two
343:Surjective
333:Measurable
328:Continuous
303:Polynomial
5130:EMS Press
4973:CRC Press
4758:cite book
4714:118124706
4420:), …, log
4318:1813 work
4177:CRC Press
3983:tetration
3713:morphisms
3636:∘
3580:∈
3561:∧
3554:∈
3527:∈
3521:∃
3509:×
3503:∈
3476:∘
3375:…
3347:…
3322:…
3269:…
3241:…
3216:…
3167:of arity
3159:of arity
3044:…
3012:…
2990:…
2936:…
2905:, is the
2813:…
2777:−
2763:…
2701:co-factor
2654:…
2613:…
2570:−
2556:…
2399:∘
2174:instead.
1508:generated
1500:bijective
1352:pointwise
961:+ 4) = (2
752:∘
670:↦
580:∘
509:functions
495:∘
348:Bijective
338:Injective
313:Algebraic
92:Types by
4945:10385726
4833:(2011).
4784:(1907).
4748:(1852).
4730:(1903).
4303:Archived
4293:(1820).
3970:notation
3843:See also
3818:∘
3725:morphism
3717:category
2850:. Given
2370:operator
2313: ;
2279:matrices
2155:whereas
2121:meaning
2095:for the
2071:fractals
1964:) · sin(
1960:) = sin(
1918: (
1874: )
1718: (
1669: (
1628: (
1535:(called
1472:on
1208:interval
1148:) = 9 −
1136:(−∞,+9]
884:, then:
772:Examples
429:Morphism
414:Implicit
318:Analytic
308:Rational
293:Identity
288:Constant
98:codomain
75: (
61:Function
5144:, 2007.
5132:, 2001
4577:Burmann
4511:
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1492:(green)
1243:commute
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831:(where
799:, then
639:applied
434:Functor
404:Partial
381:Inverse
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792:, and
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323:Smooth
298:Linear
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4979:–80,
4941:S2CID
4921:(PDF)
4874:(PDF)
4867:(PDF)
4710:S2CID
4702:JSTOR
4507:mean
4430:= log
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3997:roots
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3913:Notes
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3092:with
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