1282:
874:
301:
1546:
617:
1362:. A differential algebra is an algebra with the extra operation of derivation (algebraic version of differentiation). Using the derivation operation new equations can be written and their solutions used in
943:
1468:
467:
517:
403:
567:
1093:
350:
195:
662:
972:
688:
719:
1374:, two special types of transcendental extensions (the logarithm and the exponential) can be added to the field building a tower containing elementary functions.
742:
1046:
1026:
1190:
778:
1998:
1414:′ is used.) The derivation captures the properties of differentiation, so that for any two elements of the base field, the derivation is linear
209:
1482:
1688:
1652:
1335:
132:
in the 1930s. Many textbooks and dictionaries do not give a precise definition of the elementary functions, and mathematicians differ on it.
578:
694:. Additionally, certain classes of functions may be obtained by others using the final two rules. For example, the exponential function
1559:. If the base field is over the rationals, care must be taken when extending the field to add the needed transcendental constants.
975:
1694:
880:
2087:
1706:
1420:
1812:
622:
All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions
1853:
721:
composed with addition, subtraction, and division provides the hyperbolic functions, while initial composition with
99:
1736:
1682:
472:
413:
24:
2148:
2034:
2021:
478:
1993:
2158:
572:
360:
47:
1318:
under arithmetic operations, root extraction and composition. The elementary functions are closed under
528:
1676:
1977:
1961:
1346:
are defined as the elementary functions and, recursively, the integrals of the
Liouvillian functions.
2153:
1051:
51:
2111:
1142:
625:
All functions obtained by root extraction of a polynomial with coefficients in elementary functions
312:
159:
67:
1691: – Says when antiderivatives of elementary functions can be expressed as elementary functions
200:
1700:
643:
83:
1845:
1838:
1670:
1339:
1294:
951:
407:
43:
39:
1315:
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75:
1359:
1343:
1323:
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1149:
1115:, but others allow them. Some have proposed extending the set to include, for example, the
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691:
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114:
91:
87:
8:
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522:
2124:
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2029:
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1590:
1298:
1277:{\displaystyle \mathrm {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt,}
1116:
1031:
1011:
998:
869:{\displaystyle {\frac {e^{\tan x}}{1+x^{2}}}\sin \left({\sqrt {1+(\log x)^{2}}}\right)}
63:
1028:, is also elementary as it can be expressed as the composition of a power and root of
2121:
2083:
1901:
1849:
1742:
1732:
1371:
1302:
1104:
994:
153:
103:
79:
2097:
2075:
2043:
1989:
1973:
1962:"Premier mémoire sur la détermination des intégrales dont la valeur est algébrique"
1957:
1893:
1169:
129:
125:
121:
95:
1978:"Second mémoire sur la détermination des intégrales dont la valeur est algébrique"
1928:
1392:
1363:
1285:
2079:
1314:
It follows directly from the definition that the set of elementary functions is
1331:
1184:
1136:
1108:
1005:
296:{\displaystyle x,\ x^{2},\ {\sqrt {x}}\ (x^{\frac {1}{2}}),\ x^{\frac {2}{3}},}
59:
2142:
1905:
1112:
979:
1746:
1474:
1541:{\displaystyle \partial (u\cdot v)=\partial u\cdot v+u\cdot \partial v\,.}
2015:
2011:
31:
2070:
Davenport, James H. (2007). "What Might "Understand a
Function" Mean?".
1994:"Note sur la détermination des intégrales dont la valeur est algébrique"
1284:
a fact that may not be immediately obvious, but can be proven using the
2055:
1913:
1881:
1319:
990:
770:
71:
20:
2129:
2074:. Lecture Notes in Computer Science. Vol. 4573. pp. 55–65.
1703: – Analytic function that does not satisfy a polynomial equation
1130:
354:
2047:
1897:
1358:, or a function in elementary form, is considered in the context of
612:{\displaystyle \operatorname {arsinh} x,\ \operatorname {arcosh} x,}
986:
55:
1673: – Mathematical formula involving a given set of operations
1731:(3rd ed.). Houston, Tex.: Publish or Perish. p. 359.
1882:"Algebraic Properties of the Elementary Functions of Analysis"
1679: – Study of Galois symmetry groups of differential fields
2119:
1709: – Formula that visually represents itself when graphed
636:
Certain elementary functions of a single complex variable
632:
a finite number of any of the previously listed functions
1869:
Weisstein, Eric W. "Elementary
Function." From MathWorld
938:{\displaystyle -i\log \left(x+i{\sqrt {1-x^{2}}}\right)}
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363:
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212:
162:
1463:{\displaystyle \partial (u+v)=\partial u+\partial v}
1811:Subbotin, Igor Ya.; Bilotskii, N. N. (March 2008).
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189:
1813:"Algorithms and Fundamental Concepts of Calculus"
1810:
128:treatment of elementary functions was started by
113:All elementary functions are continuous on their
2140:
1726:
1999:Journal für die reine und angewandte Mathematik
1929:"A new elementary function for our curricula?"
744:instead provides the trigonometric functions.
124:in a series of papers from 1833 to 1841. An
1685: – System of arithmetic in proof theory
1398:for example) together with a derivation map
1326:. Importantly, the elementary functions are
1098:
2072:Towards Mechanized Mathematical Assistants
2028:
1988:
1972:
1956:
1820:Journal of Research in Innovative Teaching
1783:
1771:
1759:
1689:Liouville's theorem (differential algebra)
1410:is a new function. Sometimes the notation
752:Examples of elementary functions include:
145:Elementary functions of a single variable
2069:
1534:
1264:
462:{\displaystyle \sin x,\ \cos x,\ \tan x,}
120:Elementary functions were introduced by
2032:(1972). "Integration in finite terms".
1926:
1349:
1111:or discontinuous functions such as the
2141:
1366:of the algebra. By starting with the
512:{\displaystyle \arcsin x,\ \arccos x,}
2120:
1936:Australian Senior Mathematics Journal
1879:
747:
2010:
1806:
1804:
1795:
1695:Tarski's high school algebra problem
1122:Some examples of functions that are
398:{\displaystyle \log x,\ \log _{a}x}
13:
2063:
1528:
1507:
1486:
1454:
1445:
1424:
1354:The mathematical definition of an
1201:
1198:
1195:
562:{\displaystyle \sinh x,\ \cosh x,}
14:
2170:
2105:
1801:
1707:Tupper's self-referential formula
140:
1982:Journal de l'École Polytechnique
1966:Journal de l'École Polytechnique
1103:Many mathematicians exclude non-
1088:{\textstyle |x|={\sqrt {x^{2}}}}
2115:at Encyclopaedia of Mathematics
1920:
1886:American Journal of Mathematics
1840:Ordinary Differential Equations
473:Inverse trigonometric functions
1873:
1862:
1830:
1789:
1777:
1765:
1753:
1720:
1683:Elementary function arithmetic
1501:
1489:
1439:
1427:
1211:
1205:
1064:
1056:
948:The last function is equal to
851:
838:
266:
248:
25:Elementary function arithmetic
23:. For the logical system, see
19:For the complexity class, see
1:
2035:American Mathematical Monthly
1950:
1667: – Mathematical function
1391:(rational functions over the
345:{\displaystyle e^{x},\ a^{x}}
190:{\displaystyle 2,\ \pi ,\ e,}
1697: – Mathematical problem
1566:of a differential extension
1322:. They are not closed under
573:Inverse hyperbolic functions
54:) that is defined as taking
7:
2080:10.1007/978-3-540-73086-6_5
1658:
657:{\displaystyle {\sqrt {z}}}
135:
10:
2175:
1677:Differential Galois theory
1309:
628:All functions obtained by
18:
1880:Risch, Robert H. (1979).
1727:Spivak, Michael. (1994).
967:{\displaystyle \arccos x}
1713:
1570:of a differential field
1324:limits and infinite sums
1099:Non-elementary functions
1844:. Dover. 1985. p.
1701:Transcendental function
1295:nonelementary integrals
1109:absolute value function
1006:absolute value function
763:Multiplication, e.g. (2
408:Trigonometric functions
1927:Stewart, Seán (2005).
1671:Closed-form expression
1542:
1464:
1340:nonelementary integral
1278:
1089:
1042:
1022:
968:
939:
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738:
715:
684:
683:{\displaystyle \log z}
658:
613:
563:
513:
463:
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346:
297:
191:
2125:"Elementary function"
1543:
1465:
1344:Liouvillian functions
1279:
1143:Liouvillian functions
1090:
1043:
1023:
969:
940:
871:
739:
716:
714:{\displaystyle e^{z}}
685:
659:
614:
564:
514:
464:
400:
347:
307:Exponential functions
298:
192:
94:functions, and their
16:Mathematical function
2149:Differential algebra
2113:Elementary functions
2017:Differential Algebra
1984:. tome XIV: 149–193.
1968:. tome XIV: 124–148.
1483:
1475:Leibniz product rule
1421:
1360:differential algebra
1350:Differential algebra
1191:
1158:logarithmic integral
1052:
1032:
1012:
952:
881:
779:
725:
698:
668:
644:
579:
529:
523:Hyperbolic functions
479:
414:
361:
313:
210:
160:
2030:Rosenlicht, Maxwell
1653:Liouville's theorem
1639: / a for
1576:elementary function
1356:elementary function
1336:Liouville's theorem
1243:
999:algebraic functions
201:Rational powers of
36:elementary function
2159:Types of functions
2122:Weisstein, Eric W.
1665:Algebraic function
1538:
1473:and satisfies the
1460:
1379:differential field
1372:rational functions
1299:Dirichlet integral
1274:
1229:
1117:Lambert W function
1105:analytic functions
1085:
1038:
1018:
995:rational functions
964:
935:
866:
748:Composite examples
737:{\displaystyle iz}
734:
711:
680:
654:
609:
559:
509:
459:
395:
342:
293:
187:
154:Constant functions
2089:978-3-540-73083-5
1990:Liouville, Joseph
1974:Liouville, Joseph
1958:Liouville, Joseph
1555:is a constant if
1303:elliptic integral
1227:
1226:
1083:
1041:{\displaystyle x}
1021:{\displaystyle x}
1001:are elementary.
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2154:Computer algebra
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1582:if the function
1557:∂h = 0
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1297:, including the
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756:Addition, e.g. (
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130:Joseph Fels Ritt
122:Joseph Liouville
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2064:Further reading
2048:10.2307/2318066
1953:
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1898:10.2307/2373917
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1784:Liouville 1833c
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1772:Liouville 1833b
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1760:Liouville 1833a
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1320:differentiation
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1286:Risch algorithm
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1141:non-elementary
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2106:External links
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2042:(9): 963–972.
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1137:gamma function
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976:inverse cosine
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1113:step function
1110:
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1078:
1074:
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1035:
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1007:
1002:
1000:
996:
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980:complex plane
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84:trigonometric
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2016:
2012:Ritt, Joseph
2003:
1997:
1981:
1965:
1939:
1935:
1922:
1889:
1885:
1875:
1864:
1839:
1832:
1823:
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1791:
1779:
1767:
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1631:, that is, ∂
1628:
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1180:) integrals.
1177:
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1121:
1107:such as the
1102:
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947:
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635:
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119:
112:
107:
68:compositions
42:of a single
35:
29:
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1602:exponential
1562:A function
1551:An element
1384:is a field
1332:integration
1150:exponential
1008:, for real
991:polynomials
692:multivalued
92:exponential
46:(typically
32:mathematics
2143:Categories
2006:: 347–359.
1951:References
1942:(2): 8–26.
1738:0914098896
1651:(see also
1406:. (Here ∂
1364:extensions
771:Polynomial
640:, such as
355:Logarithms
88:hyperbolic
76:polynomial
21:ELEMENTARY
2130:MathWorld
1992:(1833c).
1976:(1833b).
1960:(1833a).
1906:0002-9327
1796:Ritt 1950
1629:logarithm
1591:algebraic
1529:∂
1526:⋅
1514:⋅
1508:∂
1496:⋅
1487:∂
1455:∂
1446:∂
1425:∂
1393:rationals
1250:−
1231:∫
1224:π
1131:tetration
987:monomials
959:
916:−
894:
885:−
845:
824:
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773:functions
690:, may be
675:
630:composing
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173:π
149:include:
126:algebraic
2014:(1950).
1747:31441929
1729:Calculus
1659:See also
136:Examples
96:inverses
80:rational
72:finitely
60:products
44:variable
40:function
2098:8049737
2056:2318066
1914:2373917
1310:Closure
1170:Fresnel
115:domains
98:(e.g.,
52:complex
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2052:JSTOR
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1910:JSTOR
1816:(PDF)
1714:Notes
1627:is a
1593:over
1578:over
1368:field
106:, or
74:many
64:roots
38:is a
34:, an
2084:ISBN
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1743:OCLC
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1624:, or
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664:and
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2076:doi
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790:tan
760:+1)
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