22:
2255:
the square root is not real, and one has to choose, for the square root, either non-real square root. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are done in the two terms of the formula, the three choices for the cubic root provide the three branches shown,
2164:
1934:
2490:
2667:
1213:
The informal definition of an algebraic function provides a number of clues about their properties. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual
2949:
307:
1797:
584:
844:
1647:
1145:
774:
2210:
2253:
2030:
987:
1957:
Furthermore, even if one is ultimately interested in real algebraic functions, there may be no means to express the function in terms of addition, multiplication, division and taking
1058:
1390:
2362:
2534:
1860:
388:
929:
204:
2015:
1550:
1446:
1499:
1329:
51:
691:
164:
442:
1280:
3023:, by the common methods of division and extraction of roots, reduce it into an infinite series ascending or descending according to the dimensions of
3176:
2801:
210:
3067:
1238:
1678:
465:
73:
1878:
From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by the
3230:
3205:
3117:
3210:
1879:
779:
2212:
the square root is real and the cubic root is thus well defined, providing the unique real root. On the other hand, for
1569:
34:
3220:
3200:
1070:
1902:) = 0 is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree of
44:
38:
30:
2159:{\displaystyle y=-{\frac {2x}{\sqrt{-108+12{\sqrt {81-12x^{3}}}}}}+{\frac {\sqrt{-108+12{\sqrt {81-12x^{3}}}}}{6}}.}
714:
120:
addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:
3215:
3072:
2772:
2336:
2263:
roots using real numbers only, even though the resulting function is real-valued on the domain of the graph shown.
1883:
865:
2172:
2215:
1159:
942:
313:
55:
2988:
1002:
3092:
2266:
On a more significant theoretical level, using complex numbers allows one to use the powerful techniques of
1340:
708:
2485:{\displaystyle {\frac {1}{2\pi i}}\oint _{\partial \Delta _{i}}{\frac {p_{y}(x_{0},y)}{p(x_{0},y)}}\,dy=1.}
3057:
2662:{\displaystyle f_{i}(x)={\frac {1}{2\pi i}}\oint _{\partial \Delta _{i}}y{\frac {p_{y}(x,y)}{p(x,y)}}\,dy}
1227:
1155:
850:
457:
106:
98:
636:
1832:
330:
3173:
887:
170:
3225:
3062:
3037:
2780:
1971:
1927:
1822:
1215:
449:
117:
113:
3052:
1962:
1826:
1507:
1398:
1245:
642:
606:
453:
445:
109:
3135:
2271:
1458:
1292:
694:
102:
312:
Some algebraic functions, however, cannot be expressed by such finite expressions (this is the
3146:
3087:
3082:
3042:
3004:
2768:
2275:
1863:
1335:
1178:
1174:
660:
616:
321:
126:
615:, with integer coefficients. It can be shown that the same class of functions is obtained if
409:
3184:
3047:
2984:
2267:
1556:
1250:
654:
3180:
2992:
2968:
2525:
1867:
1866:
of branches of the polynomial equation defining our algebraic function is the graph of an
869:
650:
2792:
has at worst algebraic poles and ordinary algebraic branchings over the critical points.
3149:
2776:
1223:
3194:
3008:
2021:
317:
3105:
2976:
2736:
2680:
Note that the foregoing proof of analyticity derived an expression for a system of
872:), a polynomial equation does not implicitly define a single function, but up to
1923:
881:
639:
occur in the coefficients the function is, in general, not algebraic, but it is
90:
3019:
let a quantity denoting the ordinate, be an algebraic function of the abscissa
1933:
3167:
3077:
1887:
877:
776:. A composition of transcendental functions can give an algebraic function:
3154:
2727:
is a point where the number of distinct zeros is smaller than the degree of
2335:
distinct zeros. We shall show that the algebraic function is analytic in a
1965:). For example, consider the algebraic function determined by the equation
3163:
2259:
It may be proven that there is no way to express this function in terms of
2944:{\displaystyle p(x,y)=a_{n}(x)(y-f_{1}(x))(y-f_{2}(x))\cdots (y-f_{n}(x))}
302:{\displaystyle f(x)={\frac {\sqrt {1+x^{3}}}{x^{3/7}-{\sqrt {7}}x^{1/3}}}}
1231:
1219:
3007:. The first discussion of algebraic functions appears to have been in
1559:
of an algebraic function is an algebraic function. For supposing that
3003:
The ideas surrounding algebraic functions go back at least as far as
1158:. The existence of an algebraic function is then guaranteed by the
1792:{\displaystyle b_{m}(y)x^{m}+b_{m-1}(y)x^{m-1}+\cdots +b_{0}(y)=0.}
700:
are considered, and one then talks about "functions algebraic over
1932:
936:
579:{\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0}
2356:
containing each of these zeros. Then by the argument principle
2274:
can be used to show that any algebraic function is in fact an
1237:. This is something of an oversimplification; because of the
15:
3027:, and then find the integral of each of the resulting terms.
2758:
A close analysis of the properties of the function elements
1241:, algebraic functions need not be expressible by radicals.
1813:
However, not every function has an inverse. For example,
1282:
is an algebraic function, since it is simply the solution
2739:
vanish. Hence there are only finitely many such points
2731:, and this occurs only where the highest degree term of
1810:
gives the inverse function, also an algebraic function.
2767:
near the critical points can be used to show that the
1949: + 1 = 0, over the domain 3/2 <
657:
is always an algebraic number. Sometimes, coefficients
394:
In more precise terms, an algebraic function of degree
1660:
is also a solution of this equation for each value of
1461:
2804:
2537:
2365:
2218:
2175:
2033:
1974:
1835:
1681:
1572:
1510:
1401:
1343:
1295:
1253:
1073:
1005:
945:
890:
839:{\displaystyle f(x)=\cos \arcsin x={\sqrt {1-x^{2}}}}
782:
717:
663:
468:
412:
333:
213:
173:
129:
2270:
to discuss algebraic functions. In particular, the
1937:
A graph of three branches of the algebraic function
2293:) be a complex polynomial in the complex variables
939:an overall sign; accordingly, it has two branches:
116:using a finite number of terms, involving only the
2971:acts by permuting the factors, and thus forms the
2943:
2795:Note that, away from the critical points, we have
2661:
2484:
2247:
2204:
2158:
2009:
1930:of an algebraic function can safely be minimized.
1854:
1791:
1641:
1544:
1493:
1440:
1384:
1323:
1274:
1139:
1052:
981:
923:
838:
768:
685:
578:
436:
382:
301:
198:
158:
2991:is related but different notion in the theory of
1642:{\displaystyle a_{n}(x)y^{n}+\cdots +a_{0}(x)=0,}
1961:roots without resorting to complex numbers (see
1140:{\displaystyle p(y,x_{1},x_{2},\dots ,x_{m})=0.}
43:but its sources remain unclear because it lacks
1862:. Another way to understand this, is that the
1501:is an algebraic function, solving the equation
769:{\displaystyle \exp x,\tan x,\ln x,\Gamma (x)}
707:A function which is not algebraic is called a
3013:An Essay on the Principles of Human Knowledge
8:
2205:{\displaystyle x\leq {\frac {3}{\sqrt{4}}},}
2775:over the critical points (and possibly the
2248:{\displaystyle x>{\frac {3}{\sqrt{4}}},}
1829:. The inverse is the algebraic "function"
880:. Consider for example the equation of the
982:{\displaystyle y=\pm {\sqrt {1-x^{2}}}.\,}
316:). This is the case, for example, for the
2923:
2889:
2858:
2830:
2803:
2652:
2608:
2601:
2590:
2582:
2560:
2542:
2536:
2469:
2451:
2424:
2411:
2404:
2396:
2388:
2366:
2364:
2278:, at least in the multiple-valued sense.
2234:
2225:
2217:
2191:
2182:
2174:
2142:
2133:
2118:
2102:
2091:
2082:
2067:
2043:
2032:
2006:
1979:
1973:
1845:
1834:
1768:
1743:
1718:
1705:
1686:
1680:
1615:
1596:
1577:
1571:
1515:
1509:
1484:
1468:
1460:
1400:
1350:
1342:
1320:
1294:
1252:
1122:
1103:
1090:
1072:
1041:
1022:
1004:
978:
967:
955:
944:
920:
908:
895:
889:
828:
816:
781:
716:
668:
662:
619:are accepted for the coefficients of the
555:
530:
505:
492:
473:
467:
411:
347:
332:
286:
282:
271:
258:
254:
242:
229:
212:
189:
172:
148:
128:
74:Learn how and when to remove this message
2963:are by definition the distinct zeros of
649:The value of an algebraic function at a
2495:By continuity, this also holds for all
1926:values. Thus, problems to do with the
1053:{\displaystyle y=f(x_{1},\dots ,x_{m})}
1664:. Indeed, interchanging the roles of
1385:{\displaystyle y={\frac {p(x)}{q(x)}}}
1060:which solves a polynomial equation in
3068:List of special functions and eponyms
7:
1392:is algebraic, being the solution to
1239:fundamental theorem of Galois theory
3187:'s Internet Encyclopedia of Science
1204:Algebraic functions in one variable
1165:Formally, an algebraic function in
999:is similarly defined as a function
711:, as it is for example the case of
3174:Definition of "Algebraic function"
3136:Definition of "Algebraic function"
2587:
2583:
2393:
2389:
754:
14:
1855:{\displaystyle x=\pm {\sqrt {y}}}
876:functions, sometimes also called
646:generated by these coefficients.
383:{\displaystyle f(x)^{5}+f(x)+x=0}
924:{\displaystyle y^{2}+x^{2}=1.\,}
199:{\displaystyle f(x)={\sqrt {x}}}
112:. Algebraic functions are often
20:
2672:which is an analytic function.
2010:{\displaystyle y^{3}-xy+1=0.\,}
2938:
2935:
2929:
2910:
2904:
2901:
2895:
2876:
2873:
2870:
2864:
2845:
2842:
2836:
2820:
2808:
2646:
2634:
2626:
2614:
2554:
2548:
2463:
2444:
2436:
2417:
1880:fundamental theorem of algebra
1780:
1774:
1736:
1730:
1698:
1692:
1627:
1621:
1589:
1583:
1533:
1527:
1480:
1474:
1429:
1423:
1411:
1405:
1376:
1370:
1362:
1356:
1311:
1305:
1269:
1263:
1128:
1077:
1047:
1015:
792:
786:
763:
757:
680:
674:
567:
561:
523:
517:
485:
479:
428:
422:
365:
359:
344:
337:
223:
217:
183:
177:
139:
133:
1:
1922:to assume complex as well as
1882:, the complex numbers are an
1545:{\displaystyle y^{n}-p(x)=0.}
1441:{\displaystyle q(x)y-p(x)=0.}
2312:is such that the polynomial
1494:{\textstyle y={\sqrt{p(x)}}}
1150:It is normally assumed that
1064: + 1 variables:
849:As a polynomial equation of
653:, and more generally, at an
101:that can be defined as the
2256:in the accompanying image.
1874:The role of complex numbers
1324:{\displaystyle y-p(x)=0.\,}
693:that are polynomial over a
3247:
3073:List of types of functions
1884:algebraically closed field
1455:th root of any polynomial
866:algebraically closed field
3122:Modern Algebra, Volume II
1209:Introduction and overview
1169:variables over the field
1160:implicit function theorem
2989:universal covering space
2973:monodromy representation
2518:) has only one root in Î
686:{\displaystyle a_{i}(x)}
320:, which is the function
159:{\displaystyle f(x)=1/x}
29:This article includes a
3231:Algebraic number theory
3093:Transcendental function
2350:non-overlapping discs Î
709:transcendental function
589:where the coefficients
437:{\displaystyle y=f(x),}
58:more precise citations.
3206:Functions and mappings
3058:Function (mathematics)
2945:
2663:
2486:
2346:. Choose a system of
2249:
2206:
2160:
2011:
1954:
1856:
1793:
1643:
1546:
1495:
1442:
1386:
1325:
1276:
1275:{\displaystyle y=p(x)}
1156:irreducible polynomial
1141:
1054:
993:algebraic function in
983:
925:
840:
770:
687:
637:transcendental numbers
580:
438:
384:
303:
200:
160:
3211:Meromorphic functions
3118:van der Waerden, B.L.
2946:
2664:
2499:in a neighborhood of
2487:
2250:
2207:
2161:
2012:
1936:
1857:
1794:
1672:and gathering terms,
1644:
1547:
1496:
1443:
1387:
1326:
1277:
1244:First, note that any
1173:is an element of the
1142:
1055:
984:
926:
841:
771:
688:
581:
439:
385:
304:
201:
161:
114:algebraic expressions
86:Mathematical function
3150:"Algebraic Function"
3140:Encyclopedia of Math
3063:Generalized function
3038:Algebraic expression
3015:in which he writes:
2802:
2535:
2363:
2216:
2173:
2031:
1972:
1918:, provided we allow
1833:
1823:horizontal line test
1679:
1570:
1508:
1459:
1399:
1341:
1334:More generally, any
1293:
1251:
1216:algebraic operations
1071:
1003:
943:
888:
780:
715:
661:
607:polynomial functions
466:
410:
331:
314:AbelâRuffini theorem
211:
171:
127:
118:algebraic operations
3053:Elementary function
2746:, ...,
1963:casus irreducibilis
1945: −
1246:polynomial function
1191:, ...,
860:roots (and exactly
641:algebraic over the
454:polynomial equation
110:polynomial equation
3221:Types of functions
3201:Analytic functions
3179:2020-10-26 at the
3164:Algebraic Function
3147:Weisstein, Eric W.
2941:
2659:
2506:. In particular,
2482:
2272:argument principle
2245:
2202:
2156:
2007:
1955:
1852:
1789:
1652:for each value of
1639:
1555:Surprisingly, the
1542:
1491:
1438:
1382:
1321:
1272:
1179:rational functions
1137:
1050:
979:
921:
836:
766:
683:
576:
434:
380:
299:
196:
156:
95:algebraic function
31:list of references
3216:Special functions
3088:Special functions
3083:Rational function
3043:Analytic function
2783:extension of the
2777:point at infinity
2703:), provided that
2686:function elements
2650:
2576:
2467:
2382:
2276:analytic function
2240:
2239:
2197:
2196:
2151:
2147:
2139:
2097:
2096:
2088:
1850:
1825:: it fails to be
1806:as a function of
1563:is a solution to
1489:
1380:
1336:rational function
1175:algebraic closure
973:
834:
617:algebraic numbers
297:
276:
248:
194:
84:
83:
76:
3238:
3185:David J. Darling
3160:
3159:
3125:
3113:
3110:Complex Analysis
3048:Complex function
2993:Riemann surfaces
2985:monodromy action
2950:
2948:
2947:
2942:
2928:
2927:
2894:
2893:
2863:
2862:
2835:
2834:
2697:
2668:
2666:
2665:
2660:
2651:
2649:
2629:
2613:
2612:
2602:
2597:
2596:
2595:
2594:
2577:
2575:
2561:
2547:
2546:
2491:
2489:
2488:
2483:
2468:
2466:
2456:
2455:
2439:
2429:
2428:
2416:
2415:
2405:
2403:
2402:
2401:
2400:
2383:
2381:
2367:
2301:. Suppose that
2268:complex analysis
2254:
2252:
2251:
2246:
2241:
2238:
2230:
2226:
2211:
2209:
2208:
2203:
2198:
2195:
2187:
2183:
2165:
2163:
2162:
2157:
2152:
2146:
2141:
2140:
2138:
2137:
2119:
2104:
2103:
2098:
2095:
2090:
2089:
2087:
2086:
2068:
2053:
2052:
2044:
2016:
2014:
2013:
2008:
1984:
1983:
1861:
1859:
1858:
1853:
1851:
1846:
1798:
1796:
1795:
1790:
1773:
1772:
1754:
1753:
1729:
1728:
1710:
1709:
1691:
1690:
1648:
1646:
1645:
1640:
1620:
1619:
1601:
1600:
1582:
1581:
1557:inverse function
1551:
1549:
1548:
1543:
1520:
1519:
1500:
1498:
1497:
1492:
1490:
1488:
1483:
1469:
1447:
1445:
1444:
1439:
1391:
1389:
1388:
1383:
1381:
1379:
1365:
1351:
1330:
1328:
1327:
1322:
1286:to the equation
1281:
1279:
1278:
1273:
1230:, and taking an
1177:of the field of
1146:
1144:
1143:
1138:
1127:
1126:
1108:
1107:
1095:
1094:
1059:
1057:
1056:
1051:
1046:
1045:
1027:
1026:
988:
986:
985:
980:
974:
972:
971:
956:
931:This determines
930:
928:
927:
922:
913:
912:
900:
899:
845:
843:
842:
837:
835:
833:
832:
817:
775:
773:
772:
767:
703:
699:
692:
690:
689:
684:
673:
672:
655:algebraic number
634:
614:
604:
585:
583:
582:
577:
560:
559:
541:
540:
516:
515:
497:
496:
478:
477:
452:and satisfies a
443:
441:
440:
435:
405:
400:in one variable
399:
389:
387:
386:
381:
352:
351:
308:
306:
305:
300:
298:
296:
295:
294:
290:
277:
272:
267:
266:
262:
247:
246:
231:
230:
205:
203:
202:
197:
195:
190:
165:
163:
162:
157:
152:
79:
72:
68:
65:
59:
54:this article by
45:inline citations
24:
23:
16:
3246:
3245:
3241:
3240:
3239:
3237:
3236:
3235:
3191:
3190:
3181:Wayback Machine
3145:
3144:
3132:
3116:
3104:
3101:
3034:
3001:
2969:monodromy group
2962:
2919:
2885:
2854:
2826:
2800:
2799:
2791:
2769:monodromy cover
2766:
2754:
2745:
2698:
2695:
2678:
2630:
2604:
2603:
2586:
2578:
2565:
2538:
2533:
2532:
2526:residue theorem
2524:, given by the
2523:
2505:
2447:
2440:
2420:
2407:
2406:
2392:
2384:
2371:
2361:
2360:
2355:
2345:
2322:
2308: â
2307:
2214:
2213:
2171:
2170:
2129:
2105:
2078:
2054:
2045:
2029:
2028:
1975:
1970:
1969:
1876:
1868:algebraic curve
1831:
1830:
1817: =
1764:
1739:
1714:
1701:
1682:
1677:
1676:
1611:
1592:
1573:
1568:
1567:
1511:
1506:
1505:
1470:
1457:
1456:
1397:
1396:
1366:
1352:
1339:
1338:
1291:
1290:
1249:
1248:
1211:
1206:
1199:
1190:
1118:
1099:
1086:
1069:
1068:
1037:
1018:
1001:
1000:
963:
941:
940:
904:
891:
886:
885:
870:complex numbers
824:
778:
777:
713:
712:
701:
697:
664:
659:
658:
651:rational number
628:
620:
610:
598:
590:
551:
526:
501:
488:
469:
464:
463:
408:
407:
401:
395:
343:
329:
328:
278:
250:
249:
238:
209:
208:
169:
168:
125:
124:
87:
80:
69:
63:
60:
49:
35:related reading
25:
21:
12:
11:
5:
3244:
3242:
3234:
3233:
3228:
3223:
3218:
3213:
3208:
3203:
3193:
3192:
3189:
3188:
3171:
3161:
3142:
3131:
3130:External links
3128:
3127:
3126:
3114:
3112:. McGraw Hill.
3100:
3097:
3096:
3095:
3090:
3085:
3080:
3075:
3070:
3065:
3060:
3055:
3050:
3045:
3040:
3033:
3030:
3029:
3028:
3005:René Descartes
3000:
2997:
2958:
2952:
2951:
2940:
2937:
2934:
2931:
2926:
2922:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2892:
2888:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2861:
2857:
2853:
2850:
2847:
2844:
2841:
2838:
2833:
2829:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2787:
2762:
2750:
2743:
2725:critical point
2709:critical point
2691:
2677:
2674:
2670:
2669:
2658:
2655:
2648:
2645:
2642:
2639:
2636:
2633:
2628:
2625:
2622:
2619:
2616:
2611:
2607:
2600:
2593:
2589:
2585:
2581:
2574:
2571:
2568:
2564:
2559:
2556:
2553:
2550:
2545:
2541:
2519:
2503:
2493:
2492:
2481:
2478:
2475:
2472:
2465:
2462:
2459:
2454:
2450:
2446:
2443:
2438:
2435:
2432:
2427:
2423:
2419:
2414:
2410:
2399:
2395:
2391:
2387:
2380:
2377:
2374:
2370:
2351:
2343:
2320:
2305:
2281:Formally, let
2244:
2237:
2233:
2229:
2224:
2221:
2201:
2194:
2190:
2186:
2181:
2178:
2167:
2166:
2155:
2150:
2145:
2136:
2132:
2128:
2125:
2122:
2117:
2114:
2111:
2108:
2101:
2094:
2085:
2081:
2077:
2074:
2071:
2066:
2063:
2060:
2057:
2051:
2048:
2042:
2039:
2036:
2018:
2017:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1982:
1978:
1914:at each point
1875:
1872:
1849:
1844:
1841:
1838:
1800:
1799:
1788:
1785:
1782:
1779:
1776:
1771:
1767:
1763:
1760:
1757:
1752:
1749:
1746:
1742:
1738:
1735:
1732:
1727:
1724:
1721:
1717:
1713:
1708:
1704:
1700:
1697:
1694:
1689:
1685:
1650:
1649:
1638:
1635:
1632:
1629:
1626:
1623:
1618:
1614:
1610:
1607:
1604:
1599:
1595:
1591:
1588:
1585:
1580:
1576:
1553:
1552:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1518:
1514:
1487:
1482:
1479:
1476:
1473:
1467:
1464:
1451:Moreover, the
1449:
1448:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1378:
1375:
1372:
1369:
1364:
1361:
1358:
1355:
1349:
1346:
1332:
1331:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1271:
1268:
1265:
1262:
1259:
1256:
1224:multiplication
1210:
1207:
1205:
1202:
1195:
1188:
1148:
1147:
1136:
1133:
1130:
1125:
1121:
1117:
1114:
1111:
1106:
1102:
1098:
1093:
1089:
1085:
1082:
1079:
1076:
1049:
1044:
1040:
1036:
1033:
1030:
1025:
1021:
1017:
1014:
1011:
1008:
977:
970:
966:
962:
959:
954:
951:
948:
935:, except only
919:
916:
911:
907:
903:
898:
894:
868:, such as the
864:roots over an
831:
827:
823:
820:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
785:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
682:
679:
676:
671:
667:
624:
594:
587:
586:
575:
572:
569:
566:
563:
558:
554:
550:
547:
544:
539:
536:
533:
529:
525:
522:
519:
514:
511:
508:
504:
500:
495:
491:
487:
484:
481:
476:
472:
433:
430:
427:
424:
421:
418:
415:
406:is a function
392:
391:
379:
376:
373:
370:
367:
364:
361:
358:
355:
350:
346:
342:
339:
336:
310:
309:
293:
289:
285:
281:
275:
270:
265:
261:
257:
253:
245:
241:
237:
234:
228:
225:
222:
219:
216:
206:
193:
188:
185:
182:
179:
176:
166:
155:
151:
147:
144:
141:
138:
135:
132:
85:
82:
81:
39:external links
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
3243:
3232:
3229:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3204:
3202:
3199:
3198:
3196:
3186:
3182:
3178:
3175:
3172:
3169:
3165:
3162:
3157:
3156:
3151:
3148:
3143:
3141:
3137:
3134:
3133:
3129:
3123:
3119:
3115:
3111:
3107:
3106:Ahlfors, Lars
3103:
3102:
3098:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3035:
3031:
3026:
3022:
3018:
3017:
3016:
3014:
3010:
3009:Edward Waring
3006:
2998:
2996:
2994:
2990:
2986:
2982:
2978:
2974:
2970:
2966:
2961:
2957:
2932:
2924:
2920:
2916:
2913:
2907:
2898:
2890:
2886:
2882:
2879:
2867:
2859:
2855:
2851:
2848:
2839:
2831:
2827:
2823:
2817:
2814:
2811:
2805:
2798:
2797:
2796:
2793:
2790:
2786:
2782:
2779:). Thus the
2778:
2774:
2770:
2765:
2761:
2756:
2753:
2749:
2742:
2738:
2734:
2730:
2726:
2722:
2718:
2714:
2710:
2706:
2702:
2694:
2690:
2687:
2683:
2675:
2673:
2656:
2653:
2643:
2640:
2637:
2631:
2623:
2620:
2617:
2609:
2605:
2598:
2591:
2579:
2572:
2569:
2566:
2562:
2557:
2551:
2543:
2539:
2531:
2530:
2529:
2527:
2522:
2517:
2513:
2509:
2502:
2498:
2479:
2476:
2473:
2470:
2460:
2457:
2452:
2448:
2441:
2433:
2430:
2425:
2421:
2412:
2408:
2397:
2385:
2378:
2375:
2372:
2368:
2359:
2358:
2357:
2354:
2349:
2342:
2338:
2334:
2330:
2326:
2319:
2315:
2311:
2304:
2300:
2296:
2292:
2288:
2284:
2279:
2277:
2273:
2269:
2264:
2262:
2257:
2242:
2235:
2231:
2227:
2222:
2219:
2199:
2192:
2188:
2184:
2179:
2176:
2153:
2148:
2143:
2134:
2130:
2126:
2123:
2120:
2115:
2112:
2109:
2106:
2099:
2092:
2083:
2079:
2075:
2072:
2069:
2064:
2061:
2058:
2055:
2049:
2046:
2040:
2037:
2034:
2027:
2026:
2025:
2023:
2022:cubic formula
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1980:
1976:
1968:
1967:
1966:
1964:
1960:
1952:
1948:
1944:
1940:
1935:
1931:
1929:
1925:
1921:
1917:
1913:
1909:
1905:
1901:
1897:
1893:
1889:
1886:. Hence any
1885:
1881:
1873:
1871:
1869:
1865:
1847:
1842:
1839:
1836:
1828:
1824:
1820:
1816:
1811:
1809:
1805:
1786:
1783:
1777:
1769:
1765:
1761:
1758:
1755:
1750:
1747:
1744:
1740:
1733:
1725:
1722:
1719:
1715:
1711:
1706:
1702:
1695:
1687:
1683:
1675:
1674:
1673:
1671:
1667:
1663:
1659:
1655:
1636:
1633:
1630:
1624:
1616:
1612:
1608:
1605:
1602:
1597:
1593:
1586:
1578:
1574:
1566:
1565:
1564:
1562:
1558:
1539:
1536:
1530:
1524:
1521:
1516:
1512:
1504:
1503:
1502:
1485:
1477:
1471:
1465:
1462:
1454:
1435:
1432:
1426:
1420:
1417:
1414:
1408:
1402:
1395:
1394:
1393:
1373:
1367:
1359:
1353:
1347:
1344:
1337:
1317:
1314:
1308:
1302:
1299:
1296:
1289:
1288:
1287:
1285:
1266:
1260:
1257:
1254:
1247:
1242:
1240:
1236:
1234:
1229:
1225:
1221:
1217:
1208:
1203:
1201:
1198:
1194:
1187:
1183:
1180:
1176:
1172:
1168:
1163:
1161:
1157:
1154:should be an
1153:
1134:
1131:
1123:
1119:
1115:
1112:
1109:
1104:
1100:
1096:
1091:
1087:
1083:
1080:
1074:
1067:
1066:
1065:
1063:
1042:
1038:
1034:
1031:
1028:
1023:
1019:
1012:
1009:
1006:
998:
996:
989:
975:
968:
964:
960:
957:
952:
949:
946:
938:
934:
917:
914:
909:
905:
901:
896:
892:
883:
879:
875:
871:
867:
863:
859:
855:
852:
847:
829:
825:
821:
818:
813:
810:
807:
804:
801:
798:
795:
789:
783:
760:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
710:
705:
696:
677:
669:
665:
656:
652:
647:
645:
644:
638:
632:
627:
623:
618:
613:
608:
602:
597:
593:
573:
570:
564:
556:
552:
548:
545:
542:
537:
534:
531:
527:
520:
512:
509:
506:
502:
498:
493:
489:
482:
474:
470:
462:
461:
460:
459:
455:
451:
447:
431:
425:
419:
416:
413:
404:
398:
377:
374:
371:
368:
362:
356:
353:
348:
340:
334:
327:
326:
325:
323:
319:
318:Bring radical
315:
291:
287:
283:
279:
273:
268:
263:
259:
255:
251:
243:
239:
235:
232:
226:
220:
214:
207:
191:
186:
180:
174:
167:
153:
149:
145:
142:
136:
130:
123:
122:
121:
119:
115:
111:
108:
104:
100:
96:
92:
78:
75:
67:
57:
53:
47:
46:
40:
36:
32:
27:
18:
17:
3153:
3139:
3121:
3109:
3024:
3020:
3012:
3002:
2980:
2977:Galois group
2972:
2964:
2959:
2955:
2953:
2794:
2788:
2784:
2763:
2759:
2757:
2751:
2747:
2740:
2737:discriminant
2732:
2728:
2724:
2720:
2716:
2712:
2708:
2704:
2700:
2692:
2688:
2685:
2681:
2679:
2671:
2520:
2515:
2511:
2507:
2500:
2496:
2494:
2352:
2347:
2340:
2337:neighborhood
2332:
2328:
2324:
2317:
2313:
2309:
2302:
2298:
2294:
2290:
2286:
2282:
2280:
2265:
2260:
2258:
2168:
2019:
1958:
1956:
1950:
1946:
1942:
1938:
1919:
1915:
1911:
1907:
1903:
1899:
1895:
1891:
1877:
1818:
1814:
1812:
1807:
1803:
1801:
1669:
1665:
1661:
1657:
1653:
1651:
1560:
1554:
1452:
1450:
1333:
1283:
1243:
1232:
1212:
1196:
1192:
1185:
1181:
1170:
1166:
1164:
1151:
1149:
1061:
994:
992:
990:
932:
873:
861:
857:
853:
848:
706:
648:
640:
630:
625:
621:
611:
600:
595:
591:
588:
456:of positive
402:
396:
393:
311:
94:
88:
70:
61:
50:Please help
42:
3226:Polynomials
3124:. Springer.
2781:holomorphic
882:unit circle
324:defined by
107:irreducible
91:mathematics
56:introducing
3195:Categories
3168:PlanetMath
3099:References
3078:Polynomial
2954:since the
2684:different
2020:Using the
1888:polynomial
1827:one-to-one
1821:fails the
856:has up to
446:continuous
322:implicitly
3155:MathWorld
2917:−
2908:⋯
2883:−
2852:−
2707:is not a
2676:Monodromy
2588:Δ
2584:∂
2580:∮
2570:π
2394:Δ
2390:∂
2386:∮
2376:π
2180:≤
2124:−
2107:−
2073:−
2056:−
2041:−
2024:, we get
1986:−
1890:relation
1843:±
1759:⋯
1748:−
1723:−
1606:⋯
1522:−
1418:−
1300:−
1113:…
1032:…
997:variables
961:−
953:±
822:−
808:
802:
755:Γ
746:
734:
722:
546:⋯
535:−
510:−
269:−
64:June 2023
3177:Archived
3120:(1931).
3108:(1979).
3032:See also
3011:'s 1794
2983:. (The
2773:ramified
2719:,
2514:,
2323:,
2289:,
1953:< 50.
1941:, where
1898:,
1802:Writing
1228:division
1220:addition
878:branches
444:that is
99:function
3138:in the
2999:History
2987:on the
2975:of the
2967:. The
2735:or the
2696:
1656:, then
1235:th root
635:'s. If
448:in its
52:improve
2723:). A
1928:domain
1910:) for
851:degree
805:arcsin
458:degree
450:domain
105:of an
2327:) of
937:up to
643:field
97:is a
93:, an
37:, or
2331:has
2297:and
2223:>
2169:For
1924:real
1668:and
695:ring
605:are
103:root
3183:in
3166:at
2995:.)
2979:of
2771:is
2711:of
2339:of
2261:nth
2110:108
2059:108
1959:nth
1906:in
1864:set
1200:).
991:An
799:cos
731:tan
719:exp
704:".
609:of
89:In
3197::
3152:.
2755:.
2528::
2480:1.
2127:12
2121:81
2116:12
2076:12
2070:81
2065:12
2004:0.
1947:xy
1870:.
1787:0.
1540:0.
1436:0.
1318:0.
1226:,
1222:,
1218::
1162:.
1135:0.
918:1.
884::
846:.
743:ln
41:,
33:,
3170:.
3158:.
3025:x
3021:x
2981:p
2965:p
2960:i
2956:f
2939:)
2936:)
2933:x
2930:(
2925:n
2921:f
2914:y
2911:(
2905:)
2902:)
2899:x
2896:(
2891:2
2887:f
2880:y
2877:(
2874:)
2871:)
2868:x
2865:(
2860:1
2856:f
2849:y
2846:(
2843:)
2840:x
2837:(
2832:n
2828:a
2824:=
2821:)
2818:y
2815:,
2812:x
2809:(
2806:p
2789:i
2785:f
2764:i
2760:f
2752:m
2748:c
2744:1
2741:c
2733:p
2729:p
2721:y
2717:x
2715:(
2713:p
2705:x
2701:x
2699:(
2693:i
2689:f
2682:n
2657:y
2654:d
2647:)
2644:y
2641:,
2638:x
2635:(
2632:p
2627:)
2624:y
2621:,
2618:x
2615:(
2610:y
2606:p
2599:y
2592:i
2573:i
2567:2
2563:1
2558:=
2555:)
2552:x
2549:(
2544:i
2540:f
2521:i
2516:y
2512:x
2510:(
2508:p
2504:0
2501:x
2497:x
2477:=
2474:y
2471:d
2464:)
2461:y
2458:,
2453:0
2449:x
2445:(
2442:p
2437:)
2434:y
2431:,
2426:0
2422:x
2418:(
2413:y
2409:p
2398:i
2379:i
2373:2
2369:1
2353:i
2348:n
2344:0
2341:x
2333:n
2329:y
2325:y
2321:0
2318:x
2316:(
2314:p
2310:C
2306:0
2303:x
2299:y
2295:x
2291:y
2287:x
2285:(
2283:p
2243:,
2236:3
2232:4
2228:3
2220:x
2200:,
2193:3
2189:4
2185:3
2177:x
2154:.
2149:6
2144:3
2135:3
2131:x
2113:+
2100:+
2093:3
2084:3
2080:x
2062:+
2050:x
2047:2
2038:=
2035:y
2001:=
1998:1
1995:+
1992:y
1989:x
1981:3
1977:y
1951:x
1943:y
1939:y
1920:y
1916:x
1912:y
1908:y
1904:p
1900:x
1896:y
1894:(
1892:p
1848:y
1840:=
1837:x
1819:x
1815:y
1808:y
1804:x
1784:=
1781:)
1778:y
1775:(
1770:0
1766:b
1762:+
1756:+
1751:1
1745:m
1741:x
1737:)
1734:y
1731:(
1726:1
1720:m
1716:b
1712:+
1707:m
1703:x
1699:)
1696:y
1693:(
1688:m
1684:b
1670:y
1666:x
1662:y
1658:x
1654:x
1637:,
1634:0
1631:=
1628:)
1625:x
1622:(
1617:0
1613:a
1609:+
1603:+
1598:n
1594:y
1590:)
1587:x
1584:(
1579:n
1575:a
1561:y
1537:=
1534:)
1531:x
1528:(
1525:p
1517:n
1513:y
1486:n
1481:)
1478:x
1475:(
1472:p
1466:=
1463:y
1453:n
1433:=
1430:)
1427:x
1424:(
1421:p
1415:y
1412:)
1409:x
1406:(
1403:q
1377:)
1374:x
1371:(
1368:q
1363:)
1360:x
1357:(
1354:p
1348:=
1345:y
1315:=
1312:)
1309:x
1306:(
1303:p
1297:y
1284:y
1270:)
1267:x
1264:(
1261:p
1258:=
1255:y
1233:n
1197:m
1193:x
1189:1
1186:x
1184:(
1182:K
1171:K
1167:m
1152:p
1132:=
1129:)
1124:m
1120:x
1116:,
1110:,
1105:2
1101:x
1097:,
1092:1
1088:x
1084:,
1081:y
1078:(
1075:p
1062:m
1048:)
1043:m
1039:x
1035:,
1029:,
1024:1
1020:x
1016:(
1013:f
1010:=
1007:y
995:m
976:.
969:2
965:x
958:1
950:=
947:y
933:y
915:=
910:2
906:x
902:+
897:2
893:y
874:n
862:n
858:n
854:n
830:2
826:x
819:1
814:=
811:x
796:=
793:)
790:x
787:(
784:f
764:)
761:x
758:(
752:,
749:x
740:,
737:x
728:,
725:x
702:R
698:R
681:)
678:x
675:(
670:i
666:a
633:)
631:x
629:(
626:i
622:a
612:x
603:)
601:x
599:(
596:i
592:a
574:0
571:=
568:)
565:x
562:(
557:0
553:a
549:+
543:+
538:1
532:n
528:y
524:)
521:x
518:(
513:1
507:n
503:a
499:+
494:n
490:y
486:)
483:x
480:(
475:n
471:a
432:,
429:)
426:x
423:(
420:f
417:=
414:y
403:x
397:n
390:.
378:0
375:=
372:x
369:+
366:)
363:x
360:(
357:f
354:+
349:5
345:)
341:x
338:(
335:f
292:3
288:/
284:1
280:x
274:7
264:7
260:/
256:3
252:x
244:3
240:x
236:+
233:1
227:=
224:)
221:x
218:(
215:f
192:x
187:=
184:)
181:x
178:(
175:f
154:x
150:/
146:1
143:=
140:)
137:x
134:(
131:f
77:)
71:(
66:)
62:(
48:.
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