1977:. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees). This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity (e.g., by replacing the latitude/longitude representation with an
3164:
3052:
which are defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function
1828:
is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its
1953:
In real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, not to the original function.
1972:
occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this is the apparent singularity at the 90 degree latitude in
2672:
1058:
291:
3053:
will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, even though it must connect two different branch points (such as
722:
625:
2739:
2357:
2933:
Other than isolated singularities, complex functions of one variable may exhibit other singular behaviour. These are termed nonisolated singularities, of which there are two types:
1270:
1613:
1449:
3312:
3009:
1709:
2125:
3226:
3115:
2290:
3050:
3153:
1945:
2557:
1914:
1878:
1819:
1783:
1389:
1353:
897:
788:
555:
515:
180:
97:
3263:
3083:
748:
651:
3366:
2914:
2888:
2858:
2825:
2795:
2765:
2698:
2586:
2522:
2496:
2470:
2440:
2412:
2383:
2316:
2240:
2214:
2185:
2155:
2091:
2062:
2028:
359:
323:
1738:
1642:
1537:
1480:
1140:
926:
817:
435:
1506:
1107:
996:
206:
123:
3339:
1310:
1290:
1197:
1169:
1081:
966:
946:
857:
837:
475:
455:
3191:– infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are
3426:
may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like
3379:
of bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of the
3419:
366:
1996:, there are several classes of singularities. These include the isolated singularities, the nonisolated singularities, and the branch points.
3655:
47:
3371:
An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of
3458:-axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the
3265:
More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses
390:
3183:
occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in
3702:
2591:
1007:
3188:
222:
38:
is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be
3539:
660:
563:
3529:
2703:
2321:
3572:
2443:
1202:
1110:
209:
1844:
is a term borrowed from complex analysis (see below). This is the case when either one or the other limits
1740:
is defined, and regardless of its value if it is defined (but which does not match that of the two limits).
1560:
1396:
3503:
3268:
1963:
126:
3549:
3524:
3480:
2975:
2955:
2861:
2158:
1974:
1649:
1144:
370:
2981:
1656:
397:(sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities:
2099:
3495:
2188:
2031:
1821:
does not exist (possibly both). This has two subtypes, which are usually not considered separately:
3487:
3391:
spun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using the
3380:
3198:
3168:
3088:
2245:
1830:
54:
3014:
3596:
3519:
3507:
3472:
3427:
3415:
3403:
3399:
3120:
1927:
1834:
1553:
374:
362:
326:
3341:
so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time
1001:
There are some functions for which these limits do not exist at all. For example, the function
3651:
3645:
3620:
3544:
3534:
3229:
3172:
2527:
1883:
1847:
1788:
1752:
1358:
1322:
866:
757:
524:
484:
140:
59:
3383:(for example, the tendency of a chalk to skip when dragged across a blackboard), and how the
3235:
3056:
2771:. The derivative at a non-essential singularity itself has a non-essential singularity, with
3491:
1993:
727:
630:
130:
3344:
2893:
2867:
2837:
2804:
2774:
2744:
2677:
2565:
2501:
2475:
2449:
2419:
2388:
2362:
2295:
2219:
2193:
2164:
2134:
2070:
2041:
2007:
1546:
Type I discontinuities can be further distinguished as being one of the following subtypes:
332:
296:
3476:
1714:
1618:
1513:
1456:
1116:
902:
793:
411:
216:
17:
3392:
1485:
1086:
975:
185:
102:
3321:
27:
Point where a function, a curve or another mathematical object does not behave regularly
3468:
3372:
2959:
2945:
2921:
2560:
2094:
2035:
1391:
exist and are finite, but at least one of the following three conditions also applies:
1295:
1275:
1182:
1154:
1066:
951:
931:
842:
822:
460:
440:
134:
3163:
3696:
3423:
386:
2971:
2941:
39:
3672:
1539:
has a defined value, which, however, does not match the value of the two limits.
31:
3384:
394:
43:
3376:
3192:
3184:
2948:
expansions on each of them, then no such expansion is possible at its limit.
1142:
settles in on. Borrowing from complex analysis, this is sometimes called an
408:
To describe the way these two types of limits are being used, suppose that
1272:, as one expects for a smooth function. All the values must be finite. If
2065:
1978:
405:, which can also be divided into two subtypes (though usually is not).
3406:" (simplistic models yield infinite human population in finite time).
1924:
approach no limit, not even if valid answers are extended to include
972:, regardless of the actual value the function has at the point where
2944:
of isolated singularities. If they are all poles, despite admitting
2954:: any non-isolated set (e.g. a curve) on which functions cannot be
3162:
3388:
2890:
if it is neither a removable singularity nor a pole. The point
2917:
1292:
is not a point of continuity, then a discontinuity occurs at
2958:
around (or outside them if they are closed curves in the
1644:
is defined, and regardless of its value if it is defined.
1109:. The limits in this case are not infinite, but rather
2667:{\displaystyle \ f(z)={\frac {g(z)}{\ (z-a)^{n}\ }}\ }
3347:
3324:
3271:
3238:
3201:
3123:
3091:
3059:
3017:
2984:
2896:
2870:
2840:
2807:
2777:
2747:
2706:
2680:
2594:
2568:
2530:
2504:
2478:
2452:
2422:
2391:
2365:
2324:
2298:
2248:
2222:
2196:
2167:
2137:
2102:
2073:
2044:
2010:
1930:
1886:
1850:
1791:
1755:
1717:
1659:
1621:
1563:
1516:
1488:
1459:
1399:
1361:
1325:
1298:
1278:
1205:
1185:
1157:
1119:
1089:
1069:
1053:{\displaystyle g(x)=\sin \left({\frac {1}{x}}\right)}
1010:
978:
954:
934:
905:
869:
845:
825:
796:
760:
730:
663:
633:
566:
527:
487:
463:
443:
414:
335:
299:
225:
188:
143:
105:
62:
3644:
Berresford, Geoffrey C.; Rockett, Andrew M. (2015).
3483:
which is lower than at other points of the variety.
3360:
3333:
3306:
3257:
3220:
3147:
3109:
3077:
3044:
3003:
2908:
2882:
2852:
2819:
2789:
2759:
2733:
2692:
2666:
2580:
2551:
2516:
2490:
2464:
2434:
2406:
2377:
2351:
2310:
2284:
2234:
2208:
2179:
2149:
2119:
2085:
2056:
2022:
1939:
1908:
1872:
1813:
1777:
1732:
1703:
1636:
1607:
1531:
1500:
1474:
1443:
1383:
1347:
1304:
1284:
1264:
1191:
1163:
1134:
1101:
1075:
1052:
990:
960:
940:
920:
891:
851:
831:
811:
782:
742:
716:
645:
619:
549:
509:
469:
449:
429:
353:
317:
286:{\displaystyle \left\{(x,y):y^{3}-x^{2}=0\right\}}
285:
200:
174:
117:
91:
687:
590:
3441:defines a curve that has a cusp at the origin
2924:has infinitely many powers of negative degree.
325:coordinate system has a singularity (called a
3475:, the singularities are the points where the
2385:is a continuous replacement for the function
8:
2722:
2716:
2340:
2334:
717:{\displaystyle f(c^{+})=\lim _{x\to c}f(x)}
620:{\displaystyle f(c^{-})=\lim _{x\to c}f(x)}
42:in some particular way, such as by lacking
3410:Algebraic geometry and commutative algebra
3352:
3346:
3323:
3295:
3279:
3270:
3243:
3237:
3206:
3200:
3122:
3090:
3058:
3016:
2988:
2983:
2895:
2869:
2839:
2806:
2776:
2746:
2734:{\displaystyle \ U\smallsetminus \{a\}~.}
2705:
2679:
2649:
2613:
2593:
2567:
2529:
2503:
2477:
2451:
2421:
2390:
2364:
2352:{\displaystyle \ U\smallsetminus \{a\}~.}
2323:
2297:
2247:
2221:
2195:
2166:
2136:
2107:
2106:
2101:
2072:
2043:
2009:
1929:
1916:does not exist, but not because it is an
1897:
1885:
1861:
1849:
1802:
1790:
1766:
1754:
1716:
1692:
1670:
1658:
1620:
1596:
1574:
1562:
1515:
1487:
1458:
1432:
1410:
1398:
1372:
1360:
1336:
1324:
1297:
1277:
1253:
1216:
1204:
1184:
1156:
1118:
1088:
1068:
1036:
1009:
977:
953:
933:
904:
880:
868:
844:
824:
795:
771:
759:
729:
690:
674:
662:
632:
593:
577:
565:
538:
526:
498:
486:
462:
457:, and for any value of its argument, say
442:
413:
334:
298:
266:
253:
224:
187:
167:
159:
142:
104:
81:
61:
3561:
2472:if there exists a holomorphic function
3232:, where the exponent is (negative) 1:
2831:so that the singularity is removable).
1265:{\displaystyle f(c^{-})=f(c)=f(c^{+})}
367:singular point of an algebraic variety
3486:An equivalent definition in terms of
1608:{\displaystyle f(c^{-})\neq f(c^{+})}
1444:{\displaystyle f(c^{-})\neq f(c^{+})}
7:
3567:
3565:
3422:is a point of the variety where the
3307:{\displaystyle (t_{0}-t)^{-\alpha }}
1151:The possible cases at a given value
3420:singularity of an algebraic variety
3195:for various exponents of the form
3101:
1934:
1063:does not tend towards anything as
25:
3650:. Cengage Learning. p. 151.
3597:"Singularity | complex functions"
3573:"Singularities, Zeros, and Poles"
3318:for time, reversing direction to
3004:{\displaystyle \ {\sqrt {z\ }}\ }
1749:discontinuity occurs when either
1704:{\displaystyle f(c^{-})=f(c^{+})}
1171:for the argument are as follows.
437:is a function of a real argument
2446:or non-essential singularity of
2120:{\displaystyle \ \mathbb {C} ~.}
3490:may be given, which extends to
1482:is not defined for the case of
1319:discontinuity occurs when both
899:is the value that the function
790:is the value that the function
129:is not defined, as involving a
3398:Hypothetical examples include
3292:
3272:
3189:Partial Differential Equations
3139:
3133:
3033:
3027:
2974:are generally the result of a
2646:
2633:
2625:
2619:
2607:
2601:
2543:
2537:
2276:
2270:
2261:
2255:
1903:
1890:
1867:
1854:
1808:
1795:
1772:
1759:
1727:
1721:
1698:
1685:
1676:
1663:
1631:
1625:
1602:
1589:
1580:
1567:
1526:
1520:
1469:
1463:
1438:
1425:
1416:
1403:
1378:
1365:
1342:
1329:
1259:
1246:
1237:
1231:
1222:
1209:
1129:
1123:
1020:
1014:
915:
909:
886:
873:
806:
800:
777:
764:
711:
705:
694:
680:
667:
614:
608:
597:
583:
570:
544:
531:
504:
491:
424:
418:
401:, which has two subtypes, and
348:
336:
312:
300:
243:
231:
168:
160:
153:
147:
72:
66:
1:
3464:-axis is a "double tangent."
3221:{\displaystyle x^{-\alpha },}
3110:{\displaystyle \ z=\infty \ }
2285:{\displaystyle \ f(z)=g(z)\ }
1711:, also regardless of whether
3430:. For example, the equation
3375:is lost on each bounce, the
3155:) which are fixed in place.
3045:{\displaystyle \ \log(z)\ ,}
2916:is an essential singularity
393:, or discontinuities of the
3621:"Singularity (mathematics)"
3148:{\displaystyle \ \log(z)\ }
1940:{\displaystyle \pm \infty }
928:tends towards as the value
819:tends towards as the value
389:, singularities are either
3719:
3540:Pathological (mathematics)
1961:
182:also has a singularity at
18:Mathematical singularities
3577:mathfaculty.fullerton.edu
3228:of which the simplest is
2929:Nonisolated singularities
1113:: there is no value that
125:, where the value of the
3530:Degeneracy (mathematics)
3504:local ring at this point
2552:{\displaystyle \ g(a)\ }
1958:Coordinate singularities
1909:{\displaystyle f(c^{+})}
1873:{\displaystyle f(c^{-})}
1814:{\displaystyle f(c^{+})}
1778:{\displaystyle f(c^{-})}
1615:, regardless of whether
1384:{\displaystyle f(c^{+})}
1348:{\displaystyle f(c^{-})}
892:{\displaystyle f(c^{+})}
783:{\displaystyle f(c^{-})}
550:{\displaystyle f(c^{+})}
510:{\displaystyle f(c^{-})}
175:{\displaystyle g(x)=|x|}
92:{\displaystyle f(x)=1/x}
3601:Encyclopedia Britannica
3452:. One could define the
3258:{\displaystyle x^{-1}.}
3181:finite-time singularity
3159:Finite-time singularity
3078:{\displaystyle \ z=0\ }
1922:Essential singularities
1650:removable discontinuity
369:. For singularities in
361:. For singularities in
3362:
3335:
3308:
3259:
3222:
3176:
3149:
3111:
3079:
3046:
3005:
2956:analytically continued
2910:
2884:
2854:
2821:
2791:
2761:
2741:The least such number
2735:
2694:
2668:
2582:
2553:
2518:
2492:
2466:
2436:
2408:
2379:
2353:
2312:
2286:
2236:
2210:
2181:
2151:
2121:
2087:
2058:
2032:complex differentiable
2030:is a function that is
2024:
2000:Isolated singularities
1970:coordinate singularity
1964:Coordinate singularity
1941:
1918:infinite discontinuity
1910:
1874:
1826:infinite discontinuity
1815:
1779:
1734:
1705:
1638:
1609:
1533:
1502:
1476:
1445:
1385:
1349:
1306:
1286:
1266:
1193:
1165:
1136:
1103:
1077:
1054:
992:
962:
942:
922:
893:
853:
833:
813:
784:
744:
743:{\displaystyle x>c}
718:
647:
646:{\displaystyle x<c}
621:
551:
511:
471:
451:
431:
355:
319:
287:
202:
176:
119:
93:
3703:Mathematical analysis
3677:mathworld.wolfram.com
3625:TheFreeDictionary.com
3550:Removable singularity
3525:Defined and undefined
3363:
3361:{\displaystyle t_{0}}
3336:
3309:
3260:
3223:
3166:
3150:
3112:
3080:
3047:
3006:
2976:multi-valued function
2911:
2909:{\displaystyle \ a\ }
2885:
2883:{\displaystyle \ f\ }
2862:essential singularity
2855:
2853:{\displaystyle \ a\ }
2822:
2820:{\displaystyle \ n\ }
2792:
2790:{\displaystyle \ n\ }
2762:
2760:{\displaystyle \ n\ }
2736:
2695:
2693:{\displaystyle \ z\ }
2669:
2583:
2581:{\displaystyle \ n\ }
2554:
2519:
2517:{\displaystyle \ U\ }
2493:
2491:{\displaystyle \ g\ }
2467:
2465:{\displaystyle \ f\ }
2437:
2435:{\displaystyle \ a\ }
2409:
2407:{\displaystyle \ f~.}
2380:
2378:{\displaystyle \ g\ }
2354:
2313:
2311:{\displaystyle \ z\ }
2287:
2237:
2235:{\displaystyle \ U\ }
2211:
2209:{\displaystyle \ g\ }
2182:
2180:{\displaystyle \ f\ }
2159:removable singularity
2152:
2150:{\displaystyle \ a\ }
2122:
2088:
2086:{\displaystyle \ U\ }
2059:
2057:{\displaystyle \ a\ }
2025:
2023:{\displaystyle \ f\ }
1975:spherical coordinates
1942:
1911:
1875:
1842:essential singularity
1816:
1780:
1735:
1706:
1639:
1610:
1534:
1503:
1477:
1446:
1386:
1350:
1307:
1287:
1267:
1194:
1166:
1145:essential singularity
1137:
1104:
1078:
1055:
993:
963:
943:
923:
894:
854:
834:
814:
785:
745:
719:
648:
622:
552:
512:
472:
452:
432:
371:differential geometry
356:
354:{\displaystyle (0,0)}
320:
318:{\displaystyle (x,y)}
288:
203:
177:
120:
99:has a singularity at
94:
3473:projective varieties
3345:
3322:
3269:
3236:
3199:
3121:
3089:
3057:
3015:
2982:
2894:
2868:
2838:
2805:
2775:
2745:
2704:
2678:
2592:
2566:
2528:
2502:
2476:
2450:
2420:
2389:
2363:
2322:
2296:
2246:
2220:
2194:
2189:holomorphic function
2165:
2135:
2100:
2071:
2042:
2008:
1928:
1884:
1848:
1789:
1753:
1733:{\displaystyle f(c)}
1715:
1657:
1637:{\displaystyle f(c)}
1619:
1561:
1532:{\displaystyle f(c)}
1514:
1486:
1475:{\displaystyle f(x)}
1457:
1397:
1359:
1323:
1296:
1276:
1203:
1183:
1155:
1135:{\displaystyle g(x)}
1117:
1087:
1067:
1008:
976:
952:
932:
921:{\displaystyle f(x)}
903:
867:
843:
823:
812:{\displaystyle f(x)}
794:
758:
728:
661:
631:
564:
525:
485:
461:
441:
430:{\displaystyle f(x)}
412:
333:
297:
223:
186:
141:
103:
60:
3671:Weisstein, Eric W.
3488:commutative algebra
3404:Doomsday's equation
3169:reciprocal function
1501:{\displaystyle x=c}
1177:point of continuity
1102:{\displaystyle c=0}
991:{\displaystyle x=c}
201:{\displaystyle x=0}
118:{\displaystyle x=0}
55:reciprocal function
3520:Catastrophe theory
3508:regular local ring
3492:abstract varieties
3416:algebraic geometry
3400:Heinz von Foerster
3358:
3334:{\displaystyle -t}
3331:
3304:
3255:
3218:
3177:
3145:
3107:
3075:
3042:
3001:
2952:Natural boundaries
2906:
2880:
2850:
2817:
2787:
2757:
2731:
2690:
2664:
2578:
2549:
2514:
2488:
2462:
2432:
2404:
2375:
2349:
2308:
2282:
2232:
2216:defined on all of
2206:
2187:if there exists a
2177:
2147:
2117:
2083:
2054:
2020:
1937:
1906:
1870:
1835:vertical asymptote
1811:
1775:
1730:
1701:
1634:
1605:
1554:jump discontinuity
1529:
1498:
1472:
1441:
1381:
1345:
1302:
1282:
1262:
1189:
1161:
1132:
1099:
1073:
1050:
988:
958:
938:
918:
889:
849:
829:
809:
780:
740:
714:
701:
643:
617:
604:
557:, are defined by:
547:
519:right-handed limit
507:
467:
447:
427:
375:singularity theory
363:algebraic geometry
351:
315:
283:
208:, since it is not
198:
172:
115:
89:
3657:978-1-305-46505-3
3545:Singular solution
3535:Hyperbolic growth
3230:hyperbolic growth
3173:hyperbolic growth
3144:
3126:
3106:
3094:
3074:
3062:
3038:
3020:
3000:
2996:
2995:
2987:
2905:
2899:
2879:
2873:
2849:
2843:
2816:
2810:
2786:
2780:
2769:order of the pole
2756:
2750:
2727:
2709:
2689:
2683:
2663:
2659:
2657:
2632:
2597:
2577:
2571:
2548:
2533:
2513:
2507:
2487:
2481:
2461:
2455:
2431:
2425:
2400:
2394:
2374:
2368:
2345:
2327:
2307:
2301:
2281:
2251:
2231:
2225:
2205:
2199:
2176:
2170:
2146:
2140:
2113:
2105:
2082:
2076:
2053:
2047:
2019:
2013:
1984:representation).
1305:{\displaystyle c}
1285:{\displaystyle c}
1192:{\displaystyle c}
1164:{\displaystyle c}
1076:{\displaystyle x}
1044:
961:{\displaystyle c}
941:{\displaystyle x}
852:{\displaystyle c}
832:{\displaystyle x}
724:, constrained by
686:
627:, constrained by
589:
479:left-handed limit
470:{\displaystyle c}
450:{\displaystyle x}
53:For example, the
44:differentiability
16:(Redirected from
3710:
3687:
3686:
3684:
3683:
3668:
3662:
3661:
3647:Applied Calculus
3641:
3635:
3634:
3632:
3631:
3617:
3611:
3610:
3608:
3607:
3593:
3587:
3586:
3584:
3583:
3569:
3463:
3457:
3451:
3440:
3381:Painlevé paradox
3367:
3365:
3364:
3359:
3357:
3356:
3340:
3338:
3337:
3332:
3313:
3311:
3310:
3305:
3303:
3302:
3284:
3283:
3264:
3262:
3261:
3256:
3251:
3250:
3227:
3225:
3224:
3219:
3214:
3213:
3154:
3152:
3151:
3146:
3142:
3124:
3116:
3114:
3113:
3108:
3104:
3092:
3084:
3082:
3081:
3076:
3072:
3060:
3051:
3049:
3048:
3043:
3036:
3018:
3010:
3008:
3007:
3002:
2998:
2997:
2993:
2989:
2985:
2915:
2913:
2912:
2907:
2903:
2897:
2889:
2887:
2886:
2881:
2877:
2871:
2859:
2857:
2856:
2851:
2847:
2841:
2830:
2826:
2824:
2823:
2818:
2814:
2808:
2800:
2796:
2794:
2793:
2788:
2784:
2778:
2766:
2764:
2763:
2758:
2754:
2748:
2740:
2738:
2737:
2732:
2725:
2707:
2699:
2697:
2696:
2691:
2687:
2681:
2673:
2671:
2670:
2665:
2661:
2660:
2658:
2655:
2654:
2653:
2630:
2628:
2614:
2595:
2587:
2585:
2584:
2579:
2575:
2569:
2558:
2556:
2555:
2550:
2546:
2531:
2523:
2521:
2520:
2515:
2511:
2505:
2497:
2495:
2494:
2489:
2485:
2479:
2471:
2469:
2468:
2463:
2459:
2453:
2441:
2439:
2438:
2433:
2429:
2423:
2413:
2411:
2410:
2405:
2398:
2392:
2384:
2382:
2381:
2376:
2372:
2366:
2358:
2356:
2355:
2350:
2343:
2325:
2317:
2315:
2314:
2309:
2305:
2299:
2291:
2289:
2288:
2283:
2279:
2249:
2241:
2239:
2238:
2233:
2229:
2223:
2215:
2213:
2212:
2207:
2203:
2197:
2186:
2184:
2183:
2178:
2174:
2168:
2156:
2154:
2153:
2148:
2144:
2138:
2126:
2124:
2123:
2118:
2111:
2110:
2103:
2092:
2090:
2089:
2084:
2080:
2074:
2063:
2061:
2060:
2055:
2051:
2045:
2029:
2027:
2026:
2021:
2017:
2011:
1994:complex analysis
1988:Complex analysis
1981:
1946:
1944:
1943:
1938:
1915:
1913:
1912:
1907:
1902:
1901:
1879:
1877:
1876:
1871:
1866:
1865:
1820:
1818:
1817:
1812:
1807:
1806:
1784:
1782:
1781:
1776:
1771:
1770:
1739:
1737:
1736:
1731:
1710:
1708:
1707:
1702:
1697:
1696:
1675:
1674:
1643:
1641:
1640:
1635:
1614:
1612:
1611:
1606:
1601:
1600:
1579:
1578:
1538:
1536:
1535:
1530:
1507:
1505:
1504:
1499:
1481:
1479:
1478:
1473:
1450:
1448:
1447:
1442:
1437:
1436:
1415:
1414:
1390:
1388:
1387:
1382:
1377:
1376:
1354:
1352:
1351:
1346:
1341:
1340:
1311:
1309:
1308:
1303:
1291:
1289:
1288:
1283:
1271:
1269:
1268:
1263:
1258:
1257:
1221:
1220:
1198:
1196:
1195:
1190:
1170:
1168:
1167:
1162:
1141:
1139:
1138:
1133:
1108:
1106:
1105:
1100:
1082:
1080:
1079:
1074:
1059:
1057:
1056:
1051:
1049:
1045:
1037:
997:
995:
994:
989:
967:
965:
964:
959:
947:
945:
944:
939:
927:
925:
924:
919:
898:
896:
895:
890:
885:
884:
863:, and the value
858:
856:
855:
850:
838:
836:
835:
830:
818:
816:
815:
810:
789:
787:
786:
781:
776:
775:
749:
747:
746:
741:
723:
721:
720:
715:
700:
679:
678:
652:
650:
649:
644:
626:
624:
623:
618:
603:
582:
581:
556:
554:
553:
548:
543:
542:
516:
514:
513:
508:
503:
502:
476:
474:
473:
468:
456:
454:
453:
448:
436:
434:
433:
428:
360:
358:
357:
352:
324:
322:
321:
316:
292:
290:
289:
284:
282:
278:
271:
270:
258:
257:
207:
205:
204:
199:
181:
179:
178:
173:
171:
163:
131:division by zero
124:
122:
121:
116:
98:
96:
95:
90:
85:
21:
3718:
3717:
3713:
3712:
3711:
3709:
3708:
3707:
3693:
3692:
3691:
3690:
3681:
3679:
3670:
3669:
3665:
3658:
3643:
3642:
3638:
3629:
3627:
3619:
3618:
3614:
3605:
3603:
3595:
3594:
3590:
3581:
3579:
3571:
3570:
3563:
3558:
3516:
3477:Jacobian matrix
3459:
3453:
3442:
3431:
3412:
3348:
3343:
3342:
3320:
3319:
3291:
3275:
3267:
3266:
3239:
3234:
3233:
3202:
3197:
3196:
3161:
3119:
3118:
3087:
3086:
3055:
3054:
3013:
3012:
2980:
2979:
2969:
2931:
2892:
2891:
2866:
2865:
2836:
2835:
2828:
2803:
2802:
2798:
2773:
2772:
2743:
2742:
2702:
2701:
2676:
2675:
2645:
2629:
2615:
2590:
2589:
2564:
2563:
2559:nonzero, and a
2526:
2525:
2500:
2499:
2474:
2473:
2448:
2447:
2418:
2417:
2387:
2386:
2361:
2360:
2320:
2319:
2294:
2293:
2244:
2243:
2218:
2217:
2192:
2191:
2163:
2162:
2133:
2132:
2098:
2097:
2095:complex numbers
2069:
2068:
2040:
2039:
2006:
2005:
2002:
1990:
1979:
1966:
1960:
1926:
1925:
1893:
1882:
1881:
1857:
1846:
1845:
1798:
1787:
1786:
1762:
1751:
1750:
1713:
1712:
1688:
1666:
1655:
1654:
1617:
1616:
1592:
1570:
1559:
1558:
1512:
1511:
1484:
1483:
1455:
1454:
1428:
1406:
1395:
1394:
1368:
1357:
1356:
1332:
1321:
1320:
1294:
1293:
1274:
1273:
1249:
1212:
1201:
1200:
1181:
1180:
1153:
1152:
1115:
1114:
1085:
1084:
1065:
1064:
1032:
1006:
1005:
974:
973:
950:
949:
930:
929:
901:
900:
876:
865:
864:
841:
840:
821:
820:
792:
791:
767:
756:
755:
726:
725:
670:
659:
658:
629:
628:
573:
562:
561:
534:
523:
522:
494:
483:
482:
459:
458:
439:
438:
410:
409:
391:discontinuities
383:
331:
330:
295:
294:
262:
249:
230:
226:
221:
220:
217:algebraic curve
184:
183:
139:
138:
101:
100:
58:
57:
28:
23:
22:
15:
12:
11:
5:
3716:
3714:
3706:
3705:
3695:
3694:
3689:
3688:
3663:
3656:
3636:
3612:
3588:
3560:
3559:
3557:
3554:
3553:
3552:
3547:
3542:
3537:
3532:
3527:
3522:
3515:
3512:
3411:
3408:
3402:'s facetious "
3373:kinetic energy
3355:
3351:
3330:
3327:
3301:
3298:
3294:
3290:
3287:
3282:
3278:
3274:
3254:
3249:
3246:
3242:
3217:
3212:
3209:
3205:
3160:
3157:
3141:
3138:
3135:
3132:
3129:
3103:
3100:
3097:
3071:
3068:
3065:
3041:
3035:
3032:
3029:
3026:
3023:
2992:
2968:
2965:
2964:
2963:
2960:Riemann sphere
2949:
2946:Laurent series
2938:Cluster points
2930:
2927:
2926:
2925:
2922:Laurent series
2918:if and only if
2902:
2876:
2846:
2832:
2813:
2783:
2767:is called the
2753:
2730:
2724:
2721:
2718:
2715:
2712:
2686:
2652:
2648:
2644:
2641:
2638:
2635:
2627:
2624:
2621:
2618:
2612:
2609:
2606:
2603:
2600:
2574:
2561:natural number
2545:
2542:
2539:
2536:
2510:
2484:
2458:
2428:
2414:
2403:
2397:
2371:
2348:
2342:
2339:
2336:
2333:
2330:
2304:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2228:
2202:
2173:
2143:
2116:
2109:
2079:
2050:
2016:
2001:
1998:
1989:
1986:
1962:Main article:
1959:
1956:
1951:
1950:
1949:
1948:
1936:
1933:
1905:
1900:
1896:
1892:
1889:
1869:
1864:
1860:
1856:
1853:
1838:
1810:
1805:
1801:
1797:
1794:
1774:
1769:
1765:
1761:
1758:
1743:
1742:
1741:
1729:
1726:
1723:
1720:
1700:
1695:
1691:
1687:
1684:
1681:
1678:
1673:
1669:
1665:
1662:
1645:
1633:
1630:
1627:
1624:
1604:
1599:
1595:
1591:
1588:
1585:
1582:
1577:
1573:
1569:
1566:
1548:
1547:
1544:
1541:
1540:
1528:
1525:
1522:
1519:
1509:
1497:
1494:
1491:
1471:
1468:
1465:
1462:
1452:
1440:
1435:
1431:
1427:
1424:
1421:
1418:
1413:
1409:
1405:
1402:
1380:
1375:
1371:
1367:
1364:
1344:
1339:
1335:
1331:
1328:
1313:
1301:
1281:
1261:
1256:
1252:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1219:
1215:
1211:
1208:
1188:
1179:is a value of
1160:
1131:
1128:
1125:
1122:
1098:
1095:
1092:
1072:
1061:
1060:
1048:
1043:
1040:
1035:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
987:
984:
981:
957:
937:
917:
914:
911:
908:
888:
883:
879:
875:
872:
848:
828:
808:
805:
802:
799:
779:
774:
770:
766:
763:
752:
751:
739:
736:
733:
713:
710:
707:
704:
699:
696:
693:
689:
685:
682:
677:
673:
669:
666:
655:
654:
642:
639:
636:
616:
613:
610:
607:
602:
599:
596:
592:
588:
585:
580:
576:
572:
569:
546:
541:
537:
533:
530:
506:
501:
497:
493:
490:
466:
446:
426:
423:
420:
417:
382:
379:
350:
347:
344:
341:
338:
314:
311:
308:
305:
302:
281:
277:
274:
269:
265:
261:
256:
252:
248:
245:
242:
239:
236:
233:
229:
210:differentiable
197:
194:
191:
170:
166:
162:
158:
155:
152:
149:
146:
135:absolute value
114:
111:
108:
88:
84:
80:
77:
74:
71:
68:
65:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3715:
3704:
3701:
3700:
3698:
3678:
3674:
3673:"Singularity"
3667:
3664:
3659:
3653:
3649:
3648:
3640:
3637:
3626:
3622:
3616:
3613:
3602:
3598:
3592:
3589:
3578:
3574:
3568:
3566:
3562:
3555:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3517:
3513:
3511:
3509:
3505:
3501:
3498:: A point is
3497:
3493:
3489:
3484:
3482:
3478:
3474:
3470:
3465:
3462:
3456:
3449:
3445:
3438:
3434:
3429:
3425:
3424:tangent space
3421:
3417:
3409:
3407:
3405:
3401:
3396:
3394:
3390:
3386:
3382:
3378:
3374:
3369:
3353:
3349:
3328:
3325:
3317:
3299:
3296:
3288:
3285:
3280:
3276:
3252:
3247:
3244:
3240:
3231:
3215:
3210:
3207:
3203:
3194:
3190:
3186:
3182:
3174:
3171:, exhibiting
3170:
3165:
3158:
3156:
3136:
3130:
3127:
3098:
3095:
3069:
3066:
3063:
3039:
3030:
3024:
3021:
2990:
2977:
2973:
2972:Branch points
2967:Branch points
2966:
2961:
2957:
2953:
2950:
2947:
2943:
2939:
2936:
2935:
2934:
2928:
2923:
2919:
2900:
2874:
2863:
2844:
2833:
2811:
2797:increased by
2781:
2770:
2751:
2728:
2719:
2713:
2710:
2684:
2650:
2642:
2639:
2636:
2622:
2616:
2610:
2604:
2598:
2572:
2562:
2540:
2534:
2508:
2482:
2456:
2445:
2426:
2415:
2401:
2395:
2369:
2359:The function
2346:
2337:
2331:
2328:
2302:
2273:
2267:
2264:
2258:
2252:
2226:
2200:
2190:
2171:
2160:
2141:
2130:
2129:
2128:
2114:
2096:
2077:
2067:
2048:
2037:
2033:
2014:
2004:Suppose that
1999:
1997:
1995:
1987:
1985:
1983:
1976:
1971:
1965:
1957:
1955:
1931:
1923:
1919:
1898:
1894:
1887:
1862:
1858:
1851:
1843:
1839:
1836:
1832:
1827:
1823:
1822:
1803:
1799:
1792:
1767:
1763:
1756:
1748:
1744:
1724:
1718:
1693:
1689:
1682:
1679:
1671:
1667:
1660:
1652:
1651:
1646:
1628:
1622:
1597:
1593:
1586:
1583:
1575:
1571:
1564:
1556:
1555:
1550:
1549:
1545:
1543:
1542:
1523:
1517:
1510:
1495:
1492:
1489:
1466:
1460:
1453:
1433:
1429:
1422:
1419:
1411:
1407:
1400:
1393:
1392:
1373:
1369:
1362:
1337:
1333:
1326:
1318:
1314:
1299:
1279:
1254:
1250:
1243:
1240:
1234:
1228:
1225:
1217:
1213:
1206:
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495:
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421:
415:
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400:
396:
392:
388:
387:real analysis
381:Real analysis
380:
378:
376:
372:
368:
364:
345:
342:
339:
328:
309:
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279:
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3676:
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3628:. Retrieved
3624:
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3604:. Retrieved
3600:
3591:
3580:. Retrieved
3576:
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3436:
3432:
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3397:
3393:Euler's Disk
3370:
3315:
3180:
3178:
2970:
2951:
2942:limit points
2937:
2932:
2768:
2003:
1991:
1969:
1967:
1952:
1921:
1917:
1841:
1825:
1747:type II
1746:
1653:occurs when
1648:
1557:occurs when
1552:
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969:
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403:type II
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40:well-behaved
35:
29:
2801:(except if
2498:defined on
2066:open subset
2038:of a point
1317:type I
1083:approaches
948:approaches
839:approaches
477:, then the
399:type I
219:defined by
48:analyticity
36:singularity
32:mathematics
3682:2019-12-12
3630:2019-12-12
3606:2019-12-12
3582:2019-12-12
3556:References
3387:rate of a
3385:precession
3193:power laws
3185:kinematics
2978:, such as
2834:The point
2588:such that
2416:The point
2242:such that
2131:The point
2036:complement
1199:for which
754:The value
517:, and the
395:derivative
3506:is not a
3377:frequency
3326:−
3300:α
3297:−
3286:−
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1935:∞
1932:±
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1768:−
1672:−
1584:≠
1576:−
1420:≠
1412:−
1338:−
1218:−
1111:undefined
1030:
773:−
695:→
598:→
579:−
500:−
260:−
137:function
3697:Category
3514:See also
3500:singular
2674:for all
2292:for all
998: .
127:function
3502:if the
3496:schemes
3314:(using
2093:of the
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1982:-vector
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373:, see
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3494:and
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3471:and
3467:For
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3187:and
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