45:
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are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well known since numerical implementations are widely available.
571:
The fundamental problem of symbolic integration is thus, given an elementary function specified by a closed-form expression, to decide whether its antiderivative is an elementary function, and, if it is, to find a closed-form expression for this antiderivative.
2108:
981:
Closed-form expressions are an important sub-class of analytic expressions, which contain a finite number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not include
968:
If an analytic expression involves only the algebraic operations (addition, subtraction, multiplication, division, and exponentiation to a rational exponent) and rational constants then it is more specifically referred to as an
2314:
For purposes of numeric computations, being in closed form is not in general necessary, as many limits and integrals can be efficiently computed. Some equations have no closed form solution, such as those that represent the
920:
constructed using well-known operations that lend themselves readily to calculation. Similar to closed-form expressions, the set of well-known functions allowed can vary according to context but always includes the
593:
450:
allows showing that if a solution of a polynomial equation has a closed form involving exponentials, logarithms or trigonometric functions, then it has also a closed form that does not involve these functions.
2294:, and they include some but not all transcendental numbers. In contrast, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers. Closed-form numbers can be studied via
1884:
320:
2259:, but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in (
2021:
860:
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381:
1010:
can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.
1079:
512:
2008:
1929:
583:; antiderivatives are not always rational fractions, but are always elementary functions that may involve logarithms and polynomial roots. This is usually proved with
2207:
have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the
Liouvillian numbers (not to be confused with
815:
791:
771:
1963:
2743:
469:
states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms. A simple example is the equation
1819:
62:
2407:
937:
However, the class of expressions considered to be analytic expressions tends to be wider than that for closed-form expressions. In particular,
1072:
743:{\displaystyle \int {\frac {f(x)}{g(x)}}\,dx=\sum _{\alpha \in \operatorname {Roots} (g(x))}{\frac {f(\alpha )}{g'(\alpha )}}\ln(x-\alpha ),}
536:
of functions that are specified by closed-form expressions. In this context, the basic functions used for defining closed forms are commonly
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2283:
algebraic, exponential, and logarithmic operations. "EL" stands both for "exponential–logarithmic" and as an abbreviation for "elementary".
2419:
869:
Changing the definition of "well known" to include additional functions can change the set of equations with closed-form solutions. Many
1950:
The integral of a closed-form expression may or may not itself be expressible as a closed-form expression. This study is referred to as
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if and only if at least one solution can be expressed as an analytic expression. There is a subtle distinction between a "closed-form
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925:(addition, subtraction, multiplication, and division), exponentiation to a real exponent (which includes extraction of the
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2154:. In particular, as the section is written, it seems that Liouvillian numbers and elementary numbers are exactly the same.
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closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve
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is not in closed form because the summation entails an infinite number of elementary operations. However, by summing a
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1951:
1939:
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27:"Closed formula" redirects here. For "closed formula" in the sense of a logic formula with no free variables, see
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466:
1114:
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1893:
462:(degree 4). The size of these expressions increases significantly with the degree, limiting their usefulness.
1970:
A standard example of an elementary function whose antiderivative does not have a closed-form expression is:
1446:
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203:
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Jonathan M. Borwein and
Richard E. Crandall (January 2013), "Closed Forms: What They Are and Why We Care",
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2504:"Siewert solutions of transcendental equations, generalized Lambert functions and physical applications"
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There is software that attempts to find closed-form expressions for numerical values, including RIES,
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closed under exponentiation and logarithm—this need not be algebraically closed, and corresponds to
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of this object, that is, an expression of this object in terms of previous ways of specifying it.
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Equations or systems too complex for closed-form or analytic solutions can often be analysed by
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2103:{\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt.}
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229:: given an object specified with such tools, a natural problem is to find, if possible, a
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for deciding whether a particular polynomial equation can be solved in radicals.
17:
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2323:. Therefore, the future states of these systems must be computed numerically.
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587:. The need for logarithms and polynomial roots is illustrated by the formula
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are also allowed, since they can be expressed in terms of the preceding ones.
393:; that is, a closed-form expression for which the allowed functions are only
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Whether a number is a closed-form number is related to whether a number is
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to be well known. It is possible to solve the quintic equation if general
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548:. Functions that have a closed form for these basic functions are called
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can be expressed as a closed-form expression; and it is said to have an
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2717:
2757:
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1052:. A closed-form or analytic solution is sometimes referred to as an
2520:
2290:. Formally, Liouvillian numbers and elementary numbers contain the
2398: – Elementary functions and their finitely iterated integrals
2368: – Process of mathematical modelling, performed on a computer
2340:
2011:
1044:" in the discussion of a "closed-form solution", discussed in (
2701:
Chow, Timothy Y. (May 1999), "What is a Closed-Form Number?",
2132:
206:. However, the set of basic functions depends on the context.
38:
2422: – Formula that visually represents itself when graphed
1879:{\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {x}{2^{n}}}}
1957:
The basic theorem of differential Galois theory is due to
315:{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}
2389: – Study of algorithms using numerical approximation
873:
cannot be expressed in closed form, unless one considers
2245:
polynomials (roots of polynomials); this is defined in (
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in the sense of rational approximation), EL numbers and
36:
Mathematical formula involving a given set of operations
2555:"RIES - Find Algebraic Equations, Given Their Solution"
2416: – Components of a mathematical or logical formula
2391:
Pages displaying short descriptions of redirect targets
2382:
Pages displaying short descriptions of redirect targets
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Pages displaying short descriptions of redirect targets
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consists essentially of the search of closed forms for
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There are expressions in radicals for all solutions of
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exponentiation and logarithms, but allow explicit and
2359: – Solution in radicals of a polynomial equation
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this expression can be expressed in the closed form:
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213:arises when new ways are introduced for specifying
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1961:in the 1830s and 1840s and hence referred to as
2380: – Addition, multiplication, division, ...
2113:Mathematical modelling and computer simulation
1154:Addition, subtraction, and multiplication only
977:Comparison of different classes of expressions
1576:Gamma function and factorial of a non-integer
1073:
8:
2744:Notices of the American Mathematical Society
2160:. There might be a discussion about this on
934:), logarithms, and trigonometric functions.
2791:Closed-form continuous-time neural networks
1954:, by analogy with algebraic Galois theory.
1812:Transformation into closed-form expressions
2473:"Numerical Solution, Closed-Form Solution"
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2180:Learn how and when to remove this message
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129:Learn how and when to remove this message
1807:Dealing with non-closed-form expressions
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2432:
1049:
1038:
965:in particular, are typically excluded.
949:are usually allowed, and often so are
187:. Commonly, the allowed functions are
7:
2408:Tarski's high school algebra problem
2260:
2246:
1045:
173:connected by arithmetic operations (
67:adding citations to reliable sources
2404: – Type of regression analysis
389:, a closed form of a solution is a
2125:(for an example in physics, see).
1854:
385:More generally, in the context of
25:
2420:Tupper's self-referential formula
2298:, in which a major result is the
871:cumulative distribution functions
855:{\displaystyle \deg f<\deg g.}
2137:
579:; that is, for fractions of two
439:{\displaystyle (+,-,\times ,/).}
329:of the solutions to the general
43:
2444:inverse trigonometric functions
2327:Conversion from numerical forms
2302:, and a major open question is
2014:a multiplicative constant) the
1147:Elementary arithmetic operation
558:inverse trigonometric functions
54:needs additional citations for
2343:, Plouffe's Inverter, and the
2273:, is the smallest subfield of
2255:was originally referred to as
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376:{\displaystyle ax^{2}+bx+c=0.}
1:
2704:American Mathematical Monthly
2661:"Inverse Symbolic Calculator"
2374: – Mathematical function
2010:whose one antiderivative is (
237:Example: roots of polynomials
29:Sentence (mathematical logic)
2448:inverse hyperbolic functions
2410: – Mathematical problem
2296:transcendental number theory
2194:Transcendental number theory
566:inverse hyperbolic functions
507:{\displaystyle x^{5}-x-1=0.}
2695:Integration in finite terms
2345:Inverse Symbolic Calculator
2003:{\displaystyle e^{-x^{2}},}
1706:Infinite continued fraction
1522:Inverse hyperbolic function
923:basic arithmetic operations
910:expression in analytic form
2827:
2514:(1). De Gruyter: 232–242.
2191:
1952:differential Galois theory
1943:
1940:Differential Galois theory
1937:
1934:Differential Galois theory
897:
26:
2300:Gelfond–Schneider theorem
1222:Finite continued fraction
1000:Stone–Weierstrass theorem
2263:, pp. 441–442), denoted
1924:{\displaystyle f(x)=2x.}
1115:Mathematical expressions
887:hypergeometric functions
78:"Closed-form expression"
2606:"Number identification"
2502:Barsan, Victor (2018).
2198:Three subfields of the
1105:Closed-form expressions
918:mathematical expression
865:Alternative definitions
554:trigonometric functions
465:In higher degrees, the
204:trigonometric functions
2777:"Closed-Form Solution"
2539:10.1515/phys-2018-0034
2310:Numerical computations
2119:mathematical modelling
2104:
2004:
1946:Nonelementary integral
1925:
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1447:Trigonometric function
1095:Polynomial expressions
1090:Arithmetic expressions
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231:closed-form expression
2304:Schanuel's conjecture
2269:, and referred to as
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2005:
1926:
1881:
1838:
1651:Infinite sum (series)
1100:Algebraic expressions
957:. On the other hand,
857:
812:
788:
768:
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509:
441:
378:
317:
157:if it is formed with
2635:"Plouffe's Inverter"
2440:Hyperbolic functions
2396:Liouvillian function
2321:Hodgkin–Huxley model
2228:algebraically closed
2225:, form the smallest
2150:confusing or unclear
2022:
1974:
1894:
1820:
1547:Root of a polynomial
1397:Exponential function
1110:Analytic expressions
1023:closed-form solution
971:algebraic expression
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581:polynomial functions
562:hyperbolic functions
550:elementary functions
542:exponential function
530:Symbolic integration
525:Symbolic integration
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467:Abel–Ruffini theorem
401:
391:solution in radicals
387:polynomial equations
336:
251:
215:mathematical objects
196:exponential function
185:function composition
63:improve this article
2610:SymPy documentation
2530:2018OPhy...16...34B
2402:Symbolic regression
2372:Elementary function
2366:Computer simulation
2217:Liouvillian numbers
2158:clarify the section
2123:computer simulation
2069:
1964:Liouville's theorem
1497:Hyperbolic function
1372:Irrational exponent
1019:system of equations
1004:continuous function
990:; neither includes
988:continued fractions
955:continued fractions
906:analytic expression
894:Analytic expression
795:coprime polynomials
211:closed-form problem
2774:Weisstein, Eric W.
2483:on 4 February 2012
2387:Numerical solution
2378:Finitary operation
2357:Algebraic solution
2317:Three-body problem
2257:elementary numbers
2213:elementary numbers
2129:Closed-form number
2100:
2055:
2000:
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1551:algebraic solution
1021:is said to have a
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753:which is valid if
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577:rational functions
519:algorithmic method
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331:quadratic equation
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2811:Special functions
2584:Maple Online Help
2292:algebraic numbers
2209:Liouville numbers
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1347:Integer factorial
1322:Rational exponent
1054:explicit solution
1031:analytic solution
998:. Indeed, by the
939:special functions
900:Analytic function
875:special functions
810:{\displaystyle g}
786:{\displaystyle g}
766:{\displaystyle f}
714:
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460:quartic equations
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243:quadratic formula
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18:Analytic solution
16:(Redirected from
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2667:on 29 March 2012
2663:. Archived from
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2641:on 19 April 2012
2637:. Archived from
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2612:. Archived from
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2553:Munafo, Robert.
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2479:. Archived from
2477:riskglossary.com
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1816:The expression:
1681:Infinite product
1626:Special function
1297:Integer nth root
1272:Integer exponent
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961:in general, and
943:Bessel functions
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914:analytic formula
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1199:
1197:Finite product
1193:
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1077:
1070:
1062:
978:
975:
947:gamma function
895:
892:
883:gamma function
879:error function
866:
863:
851:
848:
845:
842:
839:
836:
833:
830:
806:
782:
762:
751:
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712:
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664:
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628:
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619:
614:
611:
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435:
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311:
305:
302:
295:
292:
289:
286:
281:
277:
271:
268:
265:
259:
256:
238:
235:
181:integer powers
137:
136:
51:
49:
42:
35:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2823:
2812:
2809:
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2792:
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2696:
2692:
2688:
2687:
2683:
2666:
2662:
2656:
2653:
2640:
2636:
2630:
2627:
2616:on 2018-07-06
2615:
2611:
2607:
2601:
2598:
2585:
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2549:
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2205:
2201:
2195:
2184:
2181:
2173:
2163:
2162:the talk page
2159:
2153:
2151:
2146:This section
2144:
2135:
2134:
2128:
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2120:
2112:
2110:
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2017:
2013:
1997:
1990:
1986:
1982:
1978:
1968:
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1234:
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1051:
1047:
1043:
1042:
1036:
1032:
1028:
1024:
1020:
1016:
1011:
1009:
1008:unit interval
1005:
1001:
997:
993:
989:
985:
976:
974:
972:
966:
964:
960:
956:
952:
948:
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567:
563:
559:
555:
551:
547:
543:
539:
535:
531:
524:
522:
520:
516:
515:Galois theory
501:
498:
495:
492:
489:
486:
481:
477:
468:
463:
461:
457:
452:
449:
433:
426:
422:
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416:
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207:
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186:
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156:
152:
148:
144:
133:
130:
122:
111:
108:
104:
101:
97:
94:
90:
87:
83:
80: –
79:
75:
74:Find sources:
68:
64:
58:
57:
52:This article
50:
46:
41:
40:
34:
30:
19:
2780:
2751:(1): 50–65,
2748:
2742:
2718:math/9805045
2708:
2702:
2694:
2669:. Retrieved
2665:the original
2655:
2643:. Retrieved
2639:the original
2629:
2618:. Retrieved
2614:the original
2609:
2600:
2588:. Retrieved
2583:
2574:
2562:. Retrieved
2558:
2548:
2511:
2508:Open Physics
2507:
2497:
2485:. Retrieved
2481:the original
2476:
2466:
2435:
2414:Term (logic)
2330:
2313:
2285:
2280:
2275:
2270:
2265:
2256:
2251:
2242:
2238:
2233:
2231:subfield of
2226:
2221:
2216:
2203:
2197:
2176:
2170:October 2020
2167:
2156:Please help
2147:
2116:
1969:
1962:
1956:
1949:
1815:
1655:power series
1104:
1053:
1040:
1039:closed-form
1034:
1030:
1022:
1012:
980:
967:
941:such as the
936:
928:
913:
909:
905:
903:
877:such as the
868:
752:
574:
570:
552:and include
528:
517:provides an
464:
453:
448:field theory
384:
326:
324:
240:
230:
210:
208:
189:
154:
140:
125:
116:
106:
99:
92:
85:
73:
61:Please help
56:verification
53:
33:
2691:Ritt, J. F.
2586:. Maplesoft
2487:31 December
1653:(including
819:square free
327:closed form
155:closed form
143:mathematics
2800:Categories
2620:2016-12-01
2580:"identify"
2521:1703.10052
2458:References
2271:EL numbers
2249:, p. 60).
2219:, denoted
2192:See also:
2152:to readers
1944:See also:
1756:Derivative
1172:Finite sum
898:See also:
797:such that
538:logarithms
217:, such as
176:+, −, ×, /
167:finite set
147:expression
89:newspapers
2782:MathWorld
2261:Chow 1999
2247:Ritt 1948
2076:−
2057:∫
2050:π
2029:
1983:−
1855:∞
1840:∑
1422:Logarithm
1046:Chow 1999
1037:" and a "
992:integrals
963:integrals
844:
832:
732:α
729:−
720:
708:α
689:α
657:
651:∈
648:α
644:∑
598:∫
493:−
487:−
446:In fact,
420:×
414:−
285:−
270:±
264:−
227:integrals
200:logarithm
171:functions
169:of basic
163:variables
159:constants
119:June 2014
2693:(1948),
2671:30 April
2645:30 April
2590:30 April
2564:30 April
2351:See also
2333:identify
2281:explicit
2243:implicit
2239:explicit
1781:Integral
1247:Variable
1122:Constant
1035:function
1027:solution
1015:equation
945:and the
701:′
151:equation
2806:Algebra
2735:2589148
2526:Bibcode
2319:or the
2148:may be
1006:on the
932:th root
916:) is a
192:th root
103:scholar
2733:
2215:. The
1048:) and
1041:number
1002:, any
996:limits
959:limits
564:, and
223:series
219:limits
202:, and
183:) and
179:, and
165:and a
153:is in
105:
98:
91:
84:
76:
2731:JSTOR
2713:arXiv
2516:arXiv
2427:Notes
2341:SymPy
2337:Maple
2012:up to
1731:Limit
1050:below
654:Roots
325:is a
145:, an
110:JSTOR
96:books
2673:2012
2647:2012
2592:2012
2566:2012
2559:MROB
2489:2012
2446:and
2339:and
2121:and
1800:Yes
1775:Yes
1750:Yes
1725:Yes
1700:Yes
1675:Yes
1645:Yes
1620:Yes
1595:Yes
1570:Yes
1541:Yes
1516:Yes
1491:Yes
1466:Yes
1441:Yes
1416:Yes
1391:Yes
1366:Yes
1341:Yes
1316:Yes
1291:Yes
1266:Yes
1241:Yes
1216:Yes
1191:Yes
1166:Yes
1141:Yes
953:and
838:<
821:and
793:are
773:and
575:For
544:and
241:The
225:and
209:The
82:news
2753:doi
2723:doi
2709:106
2534:doi
2335:in
2026:erf
1642:Yes
1617:Yes
1592:Yes
1567:Yes
1538:Yes
1535:Yes
1513:Yes
1510:Yes
1488:Yes
1485:Yes
1463:Yes
1460:Yes
1438:Yes
1435:Yes
1413:Yes
1410:Yes
1388:Yes
1385:Yes
1363:Yes
1360:Yes
1357:Yes
1338:Yes
1335:Yes
1332:Yes
1313:Yes
1310:Yes
1307:Yes
1288:Yes
1285:Yes
1282:Yes
1279:Yes
1263:Yes
1260:Yes
1257:Yes
1254:Yes
1238:Yes
1235:Yes
1232:Yes
1226:Yes
1213:Yes
1210:Yes
1207:Yes
1204:Yes
1201:Yes
1188:Yes
1185:Yes
1182:Yes
1179:Yes
1176:Yes
1163:Yes
1160:Yes
1157:Yes
1151:Yes
1138:Yes
1135:Yes
1132:Yes
1129:Yes
1126:Yes
1017:or
994:or
986:or
912:or
904:An
881:or
841:deg
829:deg
817:is
149:or
141:In
65:by
2802::
2779:.
2749:60
2747:,
2729:,
2721:,
2707:,
2608:.
2582:.
2557:.
2532:.
2524:.
2512:16
2510:.
2506:.
2475:.
2442:,
2347:.
2306:.
2018::
1967:.
1797:No
1794:No
1791:No
1788:No
1785:No
1772:No
1769:No
1766:No
1763:No
1760:No
1747:No
1744:No
1741:No
1738:No
1735:No
1719:No
1716:No
1713:No
1710:No
1694:No
1691:No
1688:No
1685:No
1669:No
1666:No
1663:No
1660:No
1657:)
1639:No
1636:No
1633:No
1630:No
1614:No
1611:No
1608:No
1605:No
1589:No
1586:No
1583:No
1580:No
1564:No
1561:No
1558:No
1555:No
1532:No
1529:No
1526:No
1507:No
1504:No
1501:No
1482:No
1479:No
1476:No
1457:No
1454:No
1451:No
1432:No
1429:No
1426:No
1407:No
1404:No
1401:No
1382:No
1379:No
1376:No
1354:No
1351:No
1329:No
1326:No
1304:No
1301:No
1276:No
1251:No
1229:No
1056:.
973:.
717:ln
568:.
560:,
556:,
540:,
502:0.
371:0.
221:,
198:,
194:,
161:,
2785:.
2755::
2725::
2715::
2675:.
2649:.
2623:.
2594:.
2568:.
2542:.
2536::
2528::
2518::
2491:.
2276:C
2266:E
2252:L
2234:C
2222:L
2204:C
2183:)
2177:(
2172:)
2168:(
2164:.
2098:.
2095:t
2092:d
2084:2
2080:t
2072:e
2066:x
2061:0
2046:2
2041:=
2038:)
2035:x
2032:(
1998:,
1991:2
1987:x
1979:e
1919:.
1916:x
1913:2
1910:=
1907:)
1904:x
1901:(
1898:f
1870:n
1866:2
1862:x
1850:0
1847:=
1844:n
1836:=
1833:)
1830:x
1827:(
1824:f
1081:e
1074:t
1067:v
929:n
850:.
847:g
835:f
805:g
781:g
761:f
738:,
735:)
726:x
723:(
711:)
705:(
698:g
692:)
686:(
683:f
675:)
672:)
669:x
666:(
663:g
660:(
640:=
637:x
634:d
627:)
624:x
621:(
618:g
613:)
610:x
607:(
604:f
499:=
496:1
490:x
482:5
478:x
434:.
431:)
427:/
423:,
417:,
411:,
408:+
405:(
395:n
368:=
365:c
362:+
359:x
356:b
353:+
348:2
344:x
340:a
310:.
304:a
301:2
294:c
291:a
288:4
280:2
276:b
267:b
258:=
255:x
190:n
132:)
126:(
121:)
117:(
107:·
100:·
93:·
86:·
59:.
31:.
20:)
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