1304:
2614:
4028:
3487:
of the disk of convergence. A power series may diverge at every point on the boundary, or diverge on some points and converge at other points, or converge at all the points on the boundary. Furthermore, even if the series converges everywhere on the boundary (even uniformly), it does not necessarily
90:
to which it converges. In case of multiple singularities of a function (singularities are those values of the argument for which the function is not defined), the radius of convergence is the shortest or minimum of all the respective distances (which are all non-negative numbers) calculated from the
4247:
1293:
562:
The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms. In this second case, extrapolating a plot estimates the radius of
3823:
746:
1516:
increases, these coefficients settle into a regular behavior determined by the nearest radius-limiting singularity. In this case, two main techniques have been developed, based on the fact that the coefficients of a Taylor series are roughly exponential with ratio
492:
2100:
1385:
2461:
2984:
1109:
3096:
960:
4065:
1120:
851:
4387:
3764:
4023:{\displaystyle \sum _{i=1}^{\infty }a_{i}z^{i}{\text{ where }}a_{i}={\frac {(-1)^{n-1}}{2^{n}n}}{\text{ for }}n=\lfloor \log _{2}(i)\rfloor +1{\text{, the unique integer with }}2^{n-1}\leq i<2^{n},}
3316:
3657:
192:
2840:
1697:
1931:
582:
1617:
3114:. It may be cumbersome to try to apply the ratio test to find the radius of convergence of this series. But the theorem of complex analysis stated above quickly solves the problem. At
366:
3565:
2728:
2657:
1928:
The more complicated case is when the signs of the coefficients have a more complex pattern. Mercer and
Roberts proposed the following procedure. Define the associated sequence
1430:
1765:
2563:
2294:
1310:
355:
304:
3166:
2342:
1853:
2337:
4274:
2764:
259:
72:
2537:
1464:
2882:
2223:
2189:
2127:
1891:
1827:
1494:
557:
2496:
2251:
2155:
1919:
1793:
1543:
2271:
1717:
1514:
971:
4302:
So for these particular values the fastest convergence of a power series expansion is at the center, and as one moves away from the center of convergence, the
1553:
The basic case is when the coefficients ultimately share a common sign or alternate in sign. As pointed out earlier in the article, in many cases the limit
4280:. Both the number of terms and the value at which the series is to be evaluated affect the accuracy of the answer. For example, if we want to calculate
3003:
4242:{\displaystyle \sin x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots {\text{ for all }}x}
885:
2253:. This procedure also estimates two other characteristics of the convergence limiting singularity. Suppose the nearest singularity is of degree
1288:{\displaystyle |z-a|<{\frac {1}{\lim _{n\to \infty }{\frac {|c_{n+1}|}{|c_{n}|}}}}=\lim _{n\to \infty }\left|{\frac {c_{n}}{c_{n+1}}}\right|.}
4578:
777:
2861:
4509:
Mercer, G.N.; Roberts, A.J. (1990), "A centre manifold description of contaminant dispersion in channels with varying flow properties",
4660:
4276:
meaning that this series converges for all complex numbers. However, in applications, one is often interested in the precision of a
4603:
4452:
4327:
3704:
4284:
accurate up to five decimal places, we only need the first two terms of the series. However, if we want the same precision for
3210:
3601:
109:
1395:
in the Domb–Sykes plot, plot (b), which intercepts the vertical axis at −2 and has a slope +1. Thus there is a singularity at
741:{\displaystyle C=\limsup _{n\to \infty }{\sqrt{|c_{n}(z-a)^{n}|}}=\limsup _{n\to \infty }\left({\sqrt{|c_{n}|}}\right)|z-a|}
4665:
2772:
2578:
by taking its argument to be a complex variable. The radius of convergence can be characterized by the following theorem:
1622:
857:
4655:
1556:
487:{\displaystyle r=\sup \left\{|z-a|\ \left|\ \sum _{n=0}^{\infty }c_{n}(z-a)^{n}\ {\text{ converges }}\right.\right\}}
4595:
3519:
4640:
2670:, not necessarily on the real line, even if the center and all coefficients are real. For example, the function
2676:
4426:
879:
is usually easier to compute, and when that limit exists, it shows that the radius of convergence is finite.
4535:
4319:
2095:{\displaystyle b_{n}^{2}={\frac {c_{n+1}c_{n-1}-c_{n}^{2}}{c_{n}c_{n-2}-c_{n-1}^{2}}}\quad n=3,4,5,\ldots .}
4540:"O szeregu potęgowym, który jest zbieżny na całem swem kole zbieżności jednostajnie, ale nie bezwzględnie"
2620:
1719:
means the convergence-limiting singularity is on the negative axis. Estimate this limit, by plotting the
3119:
1398:
509:, the behavior of the power series may be complicated, and the series may converge for some values of
4476:
4307:
4048:
2575:
75:
1380:{\displaystyle f(\varepsilon )={\frac {\varepsilon (1+\varepsilon ^{3})}{\sqrt {1+2\varepsilon }}}.}
4303:
4034:
2456:{\textstyle {\frac {1}{2}}\left({\frac {c_{n-1}b_{n}}{c_{n}}}+{\frac {c_{n+1}}{c_{n}b_{n}}}\right)}
1722:
79:
2542:
4492:
4277:
2276:
315:
209:
43:
4467:
Domb, C.; Sykes, M.F. (1957), "On the susceptibility of a ferromagnetic above the Curie point",
4447:, Cambridge Texts in Applied Mathematics, vol. 6, Cambridge University Press, p. 146,
267:
3132:
2845:
The root test shows that its radius of convergence is 1. In accordance with this, the function
1832:
4623:
4618:
4599:
4574:
4448:
3111:
2979:{\displaystyle \arctan(z)=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots .}
2299:
1856:
513:
and diverge for others. The radius of convergence is infinite if the series converges for all
87:
51:
4259:
2736:
244:
57:
4518:
4484:
2989:
It is easy to apply the root test in this case to find that the radius of convergence is 1.
1435:
2501:
2194:
2160:
2105:
1862:
1798:
1472:
535:
1104:{\displaystyle \lim _{n\to \infty }{\frac {|c_{n+1}(z-a)^{n+1}|}{|c_{n}(z-a)^{n}|}}<1.}
869:
4306:
slows down until you reach the boundary (if it exists) and cross over, in which case the
3423:
The singularities nearest 0, which is the center of the power series expansion, are at ±2
2466:
2228:
2132:
1896:
1770:
1520:
4480:
4670:
4587:
2256:
1702:
1499:
752:
514:
205:
1303:
4649:
4496:
3775:
is the function represented by this series on the unit disk, then the derivative of
2667:
1389:
83:
3769:
has radius of convergence 1 and converges everywhere on the boundary absolutely. If
763: > 1. It follows that the power series converges if the distance from
2873:
91:
center of the disk of convergence to the respective singularities of the function.
47:
39:
3091:{\displaystyle {\frac {z}{e^{z}-1}}=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}z^{n}}
82:
inside the open disk of radius equal to the radius of convergence, and it is the
4570:
3811:
2613:
31:
17:
3681:
876:
4628:
1392:
955:{\displaystyle r=\lim _{n\to \infty }\left|{\frac {c_{n}}{c_{n+1}}}\right|.}
573:
4488:
1469:
Usually, in scientific applications, only a finite number of coefficients
4406:
3570:
has radius of convergence 1 and diverges at every point on the boundary.
2574:
A power series with a positive radius of convergence can be made into a
856:
and diverges if the distance exceeds that number; this statement is the
559:
then you take certain limits and find the precise radius of convergence.
965:
This is shown as follows. The ratio test says the series converges if
846:{\displaystyle r={\frac {1}{\limsup _{n\to \infty }{\sqrt{|c_{n}|}}}}}
3480:
3122:. The only non-removable singularities are therefore located at the
4522:
864: = 1/0 is interpreted as an infinite radius, meaning that
360:
Some may prefer an alternative definition, as existence is obvious:
2617:
Radius of convergence (white) and Taylor approximations (blue) for
2296:
to the real axis. Then the slope of the linear fit given above is
4396:
is greater than a particular number depending on the coefficients
4291:
we must evaluate and sum the first five terms of the series. For
4256:= 0, we find out that the radius of convergence of this series is
1302:
532:
The first case is theoretical: when you know all the coefficients
3669:
but converges for all other points on the boundary. The function
4539:
1299:
Practical estimation of radius in the case of real coefficients
3430:. The distance from the center to either of those points is 2
3405:. Consequently the singular points of this function occur at
2606:
is strictly less than the radius of convergence is called the
4382:{\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}
3759:{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}z^{n}}
476:
3311:{\displaystyle e^{z}=e^{x}e^{iy}=e^{x}(\cos(y)+i\sin(y)),}
576:
to the terms of the series. The root test uses the number
4295:, one requires the first 18 terms of the series, and for
3652:{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}z^{n},}
2766:
has no real roots. Its Taylor series about 0 is given by
187:{\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n},}
2225:
estimates the reciprocal of the radius of convergence,
1893:
estimates the reciprocal of the radius of convergence,
572:
The radius of convergence can be found by applying the
2504:
2469:
2345:
1625:
1559:
4330:
4262:
4068:
3826:
3707:
3604:
3522:
3213:
3135:
3006:
2885:
2775:
2739:
2679:
2623:
2598:
cannot be defined in a way that makes it holomorphic.
2545:
2302:
2279:
2259:
2231:
2197:
2163:
2135:
2108:
1934:
1899:
1865:
1835:
1801:
1773:
1725:
1705:
1523:
1502:
1475:
1438:
1401:
1313:
1123:
974:
888:
780:
755:. The root test states that the series converges if
585:
538:
369:
318:
270:
247:
112:
60:
3349:
is necessarily 1. Therefore, the absolute value of
2835:{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}z^{2n}.}
1692:{\textstyle 1/r=\lim _{n\to \infty }{c_{n}/c_{n-1}}}
4381:
4268:
4241:
4022:
3758:
3651:
3559:
3310:
3160:
3090:
2978:
2834:
2758:
2722:
2651:
2557:
2531:
2490:
2455:
2331:
2288:
2265:
2245:
2217:
2183:
2149:
2121:
2094:
1913:
1885:
1847:
1821:
1787:
1759:
1711:
1691:
1611:
1537:
1508:
1488:
1458:
1424:
1379:
1287:
1103:
954:
845:
740:
551:
486:
349:
298:
253:
186:
66:
3446:If the power series is expanded around the point
3126:points where the denominator is zero. We solve
1641:
1612:{\textstyle \lim _{n\to \infty }{c_{n}/c_{n-1}}}
1561:
1229:
1153:
976:
896:
794:
665:
593:
376:
3662:has radius of convergence 1, and diverges for
54:. It is either a non-negative real number or
4320:abscissa of convergence of a Dirichlet series
4314:Abscissa of convergence of a Dirichlet series
3491:Example 1: The power series for the function
3118:= 0, there is in effect no singularity since
2733:has no singularities on the real line, since
8:
4431:. Krishna Prakashan Media. 16 November 2010.
4392:Such a series converges if the real part of
3968:
3943:
2582:The radius of convergence of a power series
3560:{\displaystyle \sum _{n=0}^{\infty }z^{n},}
4033:has radius of convergence 1 and converges
4368:
4358:
4352:
4346:
4335:
4329:
4299:we need to evaluate the first 141 terms.
4261:
4231:
4209:
4203:
4184:
4178:
4154:
4118:
4102:
4096:
4085:
4067:
4011:
3986:
3977:
3950:
3932:
3920:
3902:
3886:
3877:
3868:
3862:
3852:
3842:
3831:
3825:
3750:
3738:
3729:
3723:
3712:
3706:
3640:
3626:
3620:
3609:
3603:
3548:
3538:
3527:
3521:
3257:
3241:
3231:
3218:
3212:
3140:
3134:
3082:
3062:
3056:
3050:
3039:
3017:
3007:
3005:
2956:
2950:
2936:
2930:
2916:
2910:
2884:
2857:, which are at a distance 1 from 0.
2820:
2810:
2791:
2780:
2774:
2750:
2738:
2723:{\displaystyle f(z)={\frac {1}{1+z^{2}}}}
2711:
2695:
2678:
2640:
2624:
2622:
2570:Radius of convergence in complex analysis
2544:
2517:
2508:
2503:
2482:
2473:
2468:
2439:
2429:
2412:
2406:
2395:
2384:
2368:
2361:
2346:
2344:
2321:
2301:
2278:
2258:
2235:
2230:
2201:
2196:
2167:
2162:
2139:
2134:
2113:
2107:
2052:
2041:
2022:
2012:
2000:
1995:
1976:
1960:
1953:
1944:
1939:
1933:
1903:
1898:
1869:
1864:
1834:
1805:
1800:
1777:
1772:
1745:
1736:
1730:
1724:
1704:
1676:
1667:
1661:
1656:
1644:
1629:
1624:
1596:
1587:
1581:
1576:
1564:
1558:
1527:
1522:
1501:
1480:
1474:
1448:
1437:
1414:
1400:
1348:
1329:
1312:
1264:
1254:
1248:
1232:
1214:
1208:
1199:
1192:
1180:
1171:
1168:
1156:
1146:
1138:
1124:
1122:
1087:
1081:
1059:
1050:
1043:
1031:
1003:
994:
991:
979:
973:
931:
921:
915:
899:
887:
833:
827:
821:
812:
809:
797:
787:
779:
733:
719:
708:
702:
696:
687:
684:
668:
654:
648:
642:
620:
611:
608:
596:
584:
543:
537:
470:
461:
439:
429:
418:
398:
384:
368:
333:
319:
317:
285:
271:
269:
246:
175:
153:
143:
132:
111:
74:. When it is positive, the power series
59:
3434:, so the radius of convergence is 2
2612:
2602:The set of all points whose distance to
4418:
759: < 1 and diverges if
4565:Brown, James; Churchill, Ruel (1989),
3412:= a nonzero integer multiple of 2
2191:via a linear fit. The intercept with
27:Domain of convergence of power series
7:
2862:analyticity of holomorphic functions
2652:{\displaystyle {\frac {1}{1+z^{2}}}}
2498:, then a linear fit extrapolated to
1432:and so the radius of convergence is
2876:can be expanded in a power series:
4567:Complex variables and applications
4347:
4263:
4097:
3843:
3724:
3621:
3539:
3389:is real, that happens only if cos(
3051:
2792:
1842:
1651:
1571:
1239:
1163:
986:
906:
804:
675:
603:
430:
261:such that the series converges if
248:
144:
61:
25:
3450:and the radius of convergence is
2860:For a proof of this theorem, see
2157:, and graphically extrapolate to
1795:, and graphically extrapolate to
1425:{\displaystyle \varepsilon =-1/2}
524:Finding the radius of convergence
497:On the boundary, that is, where |
3799:of Example 2. It turns out that
3573:Example 2: The power series for
3401:is an integer multiple of 2
241:is a nonnegative real number or
3979:, the unique integer with
3333:is real, the absolute value of
2666:means the nearest point in the
2061:
4641:What is radius of convergence?
4141:
4126:
4115:
4105:
3965:
3959:
3899:
3889:
3361:is real, that happens only if
3302:
3299:
3293:
3278:
3272:
3263:
2898:
2892:
2807:
2797:
2689:
2683:
2590:is equal to the distance from
2318:
2306:
1648:
1568:
1549:is the radius of convergence.
1354:
1335:
1323:
1317:
1236:
1215:
1200:
1193:
1172:
1160:
1139:
1125:
1088:
1078:
1065:
1051:
1044:
1028:
1015:
995:
983:
903:
828:
813:
801:
734:
720:
703:
688:
672:
649:
639:
626:
612:
600:
458:
445:
399:
385:
334:
320:
286:
272:
172:
159:
122:
116:
1:
3456:, then the set of all points
2853:) has singularities at ±
2102:Plot the finitely many known
1760:{\displaystyle c_{n}/c_{n-1}}
42:is the radius of the largest
4318:An analogous concept is the
3817:Example 4: The power series
3698:Example 3: The power series
3120:the singularity is removable
2997:Consider this power series:
2558:{\displaystyle \cos \theta }
1388:The solid green line is the
227:-th complex coefficient, and
208:constant, the center of the
4544:Prace Matematyczno-Fizyczne
3442:Convergence on the boundary
3101:where the rational numbers
2872:The arctangent function of
2594:to the nearest point where
2289:{\displaystyle \pm \theta }
751:"lim sup" denotes the
350:{\displaystyle |z-a|>r.}
4687:
4596:Princeton University Press
4590:; Shakarchi, Rami (2003),
4059:If we expand the function
2993:A more complicated example
1496:are known. Typically, as
875:The limit involved in the
299:{\displaystyle |z-a|<r}
237:The radius of convergence
4661:Convergence (mathematics)
4594:, Princeton, New Jersey:
3161:{\displaystyle e^{z}-1=0}
1921:. This plot is called a
1848:{\displaystyle n=\infty }
1619:exists, and in this case
80:uniformly on compact sets
4428:Mathematical Analysis-II
3369:is purely imaginary and
2332:{\displaystyle -(p+1)/r}
4269:{\displaystyle \infty }
4037:on the entire boundary
2759:{\displaystyle 1+z^{2}}
858:Cauchy–Hadamard theorem
254:{\displaystyle \infty }
67:{\displaystyle \infty }
4489:10.1098/rspa.1957.0078
4383:
4351:
4270:
4243:
4101:
4024:
3847:
3760:
3728:
3653:
3625:
3561:
3543:
3312:
3162:
3092:
3055:
2980:
2836:
2796:
2760:
2724:
2660:
2653:
2559:
2533:
2532:{\textstyle 1/n^{2}=0}
2492:
2457:
2333:
2290:
2267:
2247:
2219:
2185:
2151:
2123:
2096:
1915:
1887:
1859:. The intercept with
1849:
1823:
1789:
1761:
1713:
1693:
1613:
1539:
1510:
1490:
1466:
1460:
1459:{\displaystyle r=1/2.}
1426:
1381:
1307:Plots of the function
1289:
1114:That is equivalent to
1105:
956:
847:
742:
553:
488:
434:
351:
300:
255:
233:is a complex variable.
188:
148:
68:
4469:Proc. R. Soc. Lond. A
4384:
4331:
4271:
4244:
4081:
4025:
3827:
3761:
3708:
3654:
3605:
3562:
3523:
3488:converge absolutely.
3313:
3171:by recalling that if
3163:
3093:
3035:
2981:
2837:
2776:
2761:
2725:
2654:
2616:
2560:
2534:
2493:
2458:
2334:
2291:
2268:
2248:
2220:
2218:{\displaystyle 1/n=0}
2186:
2184:{\displaystyle 1/n=0}
2152:
2124:
2122:{\displaystyle b_{n}}
2097:
1916:
1888:
1886:{\displaystyle 1/n=0}
1850:
1824:
1822:{\displaystyle 1/n=0}
1790:
1762:
1714:
1694:
1614:
1540:
1511:
1491:
1489:{\displaystyle c_{n}}
1461:
1427:
1382:
1306:
1290:
1106:
957:
848:
743:
554:
552:{\displaystyle c_{n}}
489:
472: converges
414:
352:
301:
256:
189:
128:
69:
36:radius of convergence
4666:Mathematical physics
4445:Perturbation Methods
4443:Hinch, E.J. (1991),
4328:
4260:
4066:
3824:
3705:
3680:of Example 1 is the
3602:
3520:
3211:
3133:
3004:
2883:
2773:
2737:
2677:
2621:
2586:centered on a point
2576:holomorphic function
2543:
2502:
2491:{\textstyle 1/n^{2}}
2467:
2343:
2300:
2277:
2257:
2229:
2195:
2161:
2133:
2106:
1932:
1897:
1863:
1833:
1799:
1771:
1723:
1703:
1623:
1557:
1521:
1500:
1473:
1436:
1399:
1311:
1121:
972:
886:
778:
583:
536:
367:
316:
268:
245:
110:
76:converges absolutely
58:
50:in which the series
48:center of the series
4511:SIAM J. Appl. Math.
4481:1957RSPSA.240..214D
4441:See Figure 8.1 in:
4304:rate of convergence
4233: for all
4055:Rate of convergence
4049:converge absolutely
3329:to be real. Since
2608:disk of convergence
2246:{\displaystyle 1/r}
2150:{\displaystyle 1/n}
2057:
2005:
1949:
1914:{\displaystyle 1/r}
1788:{\displaystyle 1/n}
1538:{\displaystyle 1/r}
99:For a power series
4656:Analytic functions
4379:
4266:
4239:
4020:
3756:
3649:
3588:, expanded around
3557:
3513:, which is simply
3506:, expanded around
3308:
3158:
3088:
2976:
2832:
2756:
2720:
2661:
2649:
2555:
2529:
2488:
2453:
2329:
2286:
2263:
2243:
2215:
2181:
2147:
2119:
2092:
2037:
1991:
1935:
1911:
1883:
1845:
1819:
1785:
1757:
1709:
1689:
1655:
1609:
1575:
1535:
1506:
1486:
1467:
1456:
1422:
1377:
1285:
1243:
1167:
1101:
990:
952:
910:
843:
808:
738:
679:
607:
568:Theoretical radius
549:
484:
347:
296:
251:
184:
64:
4624:Convergence tests
4580:978-0-07-010905-6
4475:(1221): 214–228,
4374:
4252:around the point
4234:
4223:
4198:
4148:
4051:on the boundary.
3980:
3935:
3930:
3871:
3870: where
3744:
3634:
3353:can be 1 only if
3112:Bernoulli numbers
3076:
3030:
2965:
2945:
2925:
2718:
2664:The nearest point
2647:
2539:has intercept at
2446:
2401:
2354:
2339:. Further, plot
2266:{\displaystyle p}
2059:
1712:{\displaystyle r}
1640:
1560:
1509:{\displaystyle n}
1372:
1371:
1276:
1228:
1223:
1220:
1152:
1093:
975:
943:
895:
841:
838:
793:
713:
664:
659:
592:
528:Two cases arise:
473:
469:
413:
405:
88:analytic function
16:(Redirected from
4678:
4608:
4592:Complex Analysis
4583:
4552:
4551:
4532:
4526:
4525:
4517:(6): 1547–1565,
4506:
4500:
4499:
4464:
4458:
4457:
4439:
4433:
4432:
4423:
4409:of convergence.
4388:
4386:
4385:
4380:
4375:
4373:
4372:
4363:
4362:
4353:
4350:
4345:
4298:
4294:
4290:
4283:
4278:numerical answer
4275:
4273:
4272:
4267:
4248:
4246:
4245:
4240:
4235:
4232:
4224:
4222:
4214:
4213:
4204:
4199:
4197:
4189:
4188:
4179:
4168:
4167:
4149:
4147:
4124:
4123:
4122:
4103:
4100:
4095:
4046:
4044:
4029:
4027:
4026:
4021:
4016:
4015:
3997:
3996:
3981:
3978:
3955:
3954:
3936:
3933:
3931:
3929:
3925:
3924:
3914:
3913:
3912:
3887:
3882:
3881:
3872:
3869:
3867:
3866:
3857:
3856:
3846:
3841:
3809:
3774:
3765:
3763:
3762:
3757:
3755:
3754:
3745:
3743:
3742:
3730:
3727:
3722:
3694:
3679:
3668:
3658:
3656:
3655:
3650:
3645:
3644:
3635:
3627:
3624:
3619:
3594:
3587:
3566:
3564:
3563:
3558:
3553:
3552:
3542:
3537:
3512:
3505:
3478:
3473:
3461:
3455:
3437:
3433:
3426:
3415:
3404:
3384:
3365:= 0. Therefore
3348:
3317:
3315:
3314:
3309:
3262:
3261:
3249:
3248:
3236:
3235:
3223:
3222:
3203:
3184:
3167:
3165:
3164:
3159:
3145:
3144:
3097:
3095:
3094:
3089:
3087:
3086:
3077:
3075:
3067:
3066:
3057:
3054:
3049:
3031:
3029:
3022:
3021:
3008:
2985:
2983:
2982:
2977:
2966:
2961:
2960:
2951:
2946:
2941:
2940:
2931:
2926:
2921:
2920:
2911:
2868:A simple example
2841:
2839:
2838:
2833:
2828:
2827:
2815:
2814:
2795:
2790:
2765:
2763:
2762:
2757:
2755:
2754:
2729:
2727:
2726:
2721:
2719:
2717:
2716:
2715:
2696:
2658:
2656:
2655:
2650:
2648:
2646:
2645:
2644:
2625:
2564:
2562:
2561:
2556:
2538:
2536:
2535:
2530:
2522:
2521:
2512:
2497:
2495:
2494:
2489:
2487:
2486:
2477:
2462:
2460:
2459:
2454:
2452:
2448:
2447:
2445:
2444:
2443:
2434:
2433:
2423:
2422:
2407:
2402:
2400:
2399:
2390:
2389:
2388:
2379:
2378:
2362:
2355:
2347:
2338:
2336:
2335:
2330:
2325:
2295:
2293:
2292:
2287:
2272:
2270:
2269:
2264:
2252:
2250:
2249:
2244:
2239:
2224:
2222:
2221:
2216:
2205:
2190:
2188:
2187:
2182:
2171:
2156:
2154:
2153:
2148:
2143:
2128:
2126:
2125:
2120:
2118:
2117:
2101:
2099:
2098:
2093:
2060:
2058:
2056:
2051:
2033:
2032:
2017:
2016:
2006:
2004:
1999:
1987:
1986:
1971:
1970:
1954:
1948:
1943:
1920:
1918:
1917:
1912:
1907:
1892:
1890:
1889:
1884:
1873:
1854:
1852:
1851:
1846:
1828:
1826:
1825:
1820:
1809:
1794:
1792:
1791:
1786:
1781:
1766:
1764:
1763:
1758:
1756:
1755:
1740:
1735:
1734:
1718:
1716:
1715:
1710:
1698:
1696:
1695:
1690:
1688:
1687:
1686:
1671:
1666:
1665:
1654:
1633:
1618:
1616:
1615:
1610:
1608:
1607:
1606:
1591:
1586:
1585:
1574:
1544:
1542:
1541:
1536:
1531:
1515:
1513:
1512:
1507:
1495:
1493:
1492:
1487:
1485:
1484:
1465:
1463:
1462:
1457:
1452:
1431:
1429:
1428:
1423:
1418:
1386:
1384:
1383:
1378:
1373:
1358:
1357:
1353:
1352:
1330:
1294:
1292:
1291:
1286:
1281:
1277:
1275:
1274:
1259:
1258:
1249:
1242:
1224:
1222:
1221:
1219:
1218:
1213:
1212:
1203:
1197:
1196:
1191:
1190:
1175:
1169:
1166:
1147:
1142:
1128:
1110:
1108:
1107:
1102:
1094:
1092:
1091:
1086:
1085:
1064:
1063:
1054:
1048:
1047:
1042:
1041:
1014:
1013:
998:
992:
989:
961:
959:
958:
953:
948:
944:
942:
941:
926:
925:
916:
909:
852:
850:
849:
844:
842:
840:
839:
837:
832:
831:
826:
825:
816:
810:
807:
788:
747:
745:
744:
739:
737:
723:
718:
714:
712:
707:
706:
701:
700:
691:
685:
678:
660:
658:
653:
652:
647:
646:
625:
624:
615:
609:
606:
558:
556:
555:
550:
548:
547:
493:
491:
490:
485:
483:
479:
478:
475:
474:
471:
467:
466:
465:
444:
443:
433:
428:
411:
403:
402:
388:
356:
354:
353:
348:
337:
323:
309:and diverges if
305:
303:
302:
297:
289:
275:
260:
258:
257:
252:
193:
191:
190:
185:
180:
179:
158:
157:
147:
142:
73:
71:
70:
65:
21:
4686:
4685:
4681:
4680:
4679:
4677:
4676:
4675:
4646:
4645:
4637:
4615:
4606:
4586:
4581:
4564:
4561:
4556:
4555:
4534:
4533:
4529:
4523:10.1137/0150091
4508:
4507:
4503:
4466:
4465:
4461:
4455:
4442:
4440:
4436:
4425:
4424:
4420:
4415:
4404:
4364:
4354:
4326:
4325:
4316:
4296:
4292:
4285:
4281:
4258:
4257:
4215:
4205:
4190:
4180:
4150:
4125:
4114:
4104:
4064:
4063:
4057:
4047:, but does not
4040:
4038:
4007:
3982:
3946:
3934: for
3916:
3915:
3898:
3888:
3873:
3858:
3848:
3822:
3821:
3800:
3770:
3746:
3734:
3703:
3702:
3685:
3670:
3663:
3636:
3600:
3599:
3589:
3574:
3544:
3518:
3517:
3507:
3492:
3465:
3463:
3457:
3451:
3444:
3435:
3431:
3424:
3413:
3402:
3397:) = 0, so that
3370:
3334:
3253:
3237:
3227:
3214:
3209:
3208:
3186:
3172:
3136:
3131:
3130:
3109:
3078:
3068:
3058:
3013:
3012:
3002:
3001:
2995:
2952:
2932:
2912:
2881:
2880:
2870:
2816:
2806:
2771:
2770:
2746:
2735:
2734:
2707:
2700:
2675:
2674:
2636:
2629:
2619:
2618:
2572:
2541:
2540:
2513:
2500:
2499:
2478:
2465:
2464:
2435:
2425:
2424:
2408:
2391:
2380:
2364:
2363:
2360:
2356:
2341:
2340:
2298:
2297:
2275:
2274:
2255:
2254:
2227:
2226:
2193:
2192:
2159:
2158:
2131:
2130:
2109:
2104:
2103:
2018:
2008:
2007:
1972:
1956:
1955:
1930:
1929:
1923:Domb–Sykes plot
1895:
1894:
1861:
1860:
1831:
1830:
1797:
1796:
1769:
1768:
1741:
1726:
1721:
1720:
1701:
1700:
1672:
1657:
1621:
1620:
1592:
1577:
1555:
1554:
1519:
1518:
1498:
1497:
1476:
1471:
1470:
1434:
1433:
1397:
1396:
1387:
1344:
1331:
1309:
1308:
1301:
1260:
1250:
1244:
1204:
1198:
1176:
1170:
1151:
1119:
1118:
1077:
1055:
1049:
1027:
999:
993:
970:
969:
927:
917:
911:
884:
883:
870:entire function
817:
811:
792:
776:
775:
692:
686:
680:
638:
616:
610:
581:
580:
570:
539:
534:
533:
526:
515:complex numbers
457:
435:
410:
406:
383:
379:
365:
364:
314:
313:
266:
265:
243:
242:
222:
212:of convergence,
171:
149:
108:
107:
97:
56:
55:
28:
23:
22:
18:Domb–Sykes plot
15:
12:
11:
5:
4684:
4682:
4674:
4673:
4668:
4663:
4658:
4648:
4647:
4644:
4643:
4636:
4635:External links
4633:
4632:
4631:
4626:
4621:
4619:Abel's theorem
4614:
4611:
4610:
4609:
4604:
4584:
4579:
4560:
4557:
4554:
4553:
4536:Sierpiński, W.
4527:
4501:
4459:
4453:
4434:
4417:
4416:
4414:
4411:
4400:
4390:
4389:
4378:
4371:
4367:
4361:
4357:
4349:
4344:
4341:
4338:
4334:
4315:
4312:
4310:will diverge.
4265:
4250:
4249:
4238:
4230:
4227:
4221:
4218:
4212:
4208:
4202:
4196:
4193:
4187:
4183:
4177:
4174:
4171:
4166:
4163:
4160:
4157:
4153:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4121:
4117:
4113:
4110:
4107:
4099:
4094:
4091:
4088:
4084:
4080:
4077:
4074:
4071:
4056:
4053:
4031:
4030:
4019:
4014:
4010:
4006:
4003:
4000:
3995:
3992:
3989:
3985:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3953:
3949:
3945:
3942:
3939:
3928:
3923:
3919:
3911:
3908:
3905:
3901:
3897:
3894:
3891:
3885:
3880:
3876:
3865:
3861:
3855:
3851:
3845:
3840:
3837:
3834:
3830:
3783:) is equal to
3767:
3766:
3753:
3749:
3741:
3737:
3733:
3726:
3721:
3718:
3715:
3711:
3660:
3659:
3648:
3643:
3639:
3633:
3630:
3623:
3618:
3615:
3612:
3608:
3568:
3567:
3556:
3551:
3547:
3541:
3536:
3533:
3530:
3526:
3443:
3440:
3421:
3420:
3393:) = 1 and sin(
3321:and then take
3319:
3318:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3260:
3256:
3252:
3247:
3244:
3240:
3234:
3230:
3226:
3221:
3217:
3169:
3168:
3157:
3154:
3151:
3148:
3143:
3139:
3105:
3099:
3098:
3085:
3081:
3074:
3071:
3065:
3061:
3053:
3048:
3045:
3042:
3038:
3034:
3028:
3025:
3020:
3016:
3011:
2994:
2991:
2987:
2986:
2975:
2972:
2969:
2964:
2959:
2955:
2949:
2944:
2939:
2935:
2929:
2924:
2919:
2915:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2869:
2866:
2843:
2842:
2831:
2826:
2823:
2819:
2813:
2809:
2805:
2802:
2799:
2794:
2789:
2786:
2783:
2779:
2753:
2749:
2745:
2742:
2731:
2730:
2714:
2710:
2706:
2703:
2699:
2694:
2691:
2688:
2685:
2682:
2643:
2639:
2635:
2632:
2628:
2600:
2599:
2571:
2568:
2567:
2566:
2554:
2551:
2548:
2528:
2525:
2520:
2516:
2511:
2507:
2485:
2481:
2476:
2472:
2451:
2442:
2438:
2432:
2428:
2421:
2418:
2415:
2411:
2405:
2398:
2394:
2387:
2383:
2377:
2374:
2371:
2367:
2359:
2353:
2350:
2328:
2324:
2320:
2317:
2314:
2311:
2308:
2305:
2285:
2282:
2273:and has angle
2262:
2242:
2238:
2234:
2214:
2211:
2208:
2204:
2200:
2180:
2177:
2174:
2170:
2166:
2146:
2142:
2138:
2116:
2112:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2055:
2050:
2047:
2044:
2040:
2036:
2031:
2028:
2025:
2021:
2015:
2011:
2003:
1998:
1994:
1990:
1985:
1982:
1979:
1975:
1969:
1966:
1963:
1959:
1952:
1947:
1942:
1938:
1926:
1910:
1906:
1902:
1882:
1879:
1876:
1872:
1868:
1844:
1841:
1838:
1818:
1815:
1812:
1808:
1804:
1784:
1780:
1776:
1754:
1751:
1748:
1744:
1739:
1733:
1729:
1708:
1685:
1682:
1679:
1675:
1670:
1664:
1660:
1653:
1650:
1647:
1643:
1639:
1636:
1632:
1628:
1605:
1602:
1599:
1595:
1590:
1584:
1580:
1573:
1570:
1567:
1563:
1534:
1530:
1526:
1505:
1483:
1479:
1455:
1451:
1447:
1444:
1441:
1421:
1417:
1413:
1410:
1407:
1404:
1376:
1370:
1367:
1364:
1361:
1356:
1351:
1347:
1343:
1340:
1337:
1334:
1328:
1325:
1322:
1319:
1316:
1300:
1297:
1296:
1295:
1284:
1280:
1273:
1270:
1267:
1263:
1257:
1253:
1247:
1241:
1238:
1235:
1231:
1227:
1217:
1211:
1207:
1202:
1195:
1189:
1186:
1183:
1179:
1174:
1165:
1162:
1159:
1155:
1150:
1145:
1141:
1137:
1134:
1131:
1127:
1112:
1111:
1100:
1097:
1090:
1084:
1080:
1076:
1073:
1070:
1067:
1062:
1058:
1053:
1046:
1040:
1037:
1034:
1030:
1026:
1023:
1020:
1017:
1012:
1009:
1006:
1002:
997:
988:
985:
982:
978:
963:
962:
951:
947:
940:
937:
934:
930:
924:
920:
914:
908:
905:
902:
898:
894:
891:
854:
853:
836:
830:
824:
820:
815:
806:
803:
800:
796:
795:lim sup
791:
786:
783:
767:to the center
753:limit superior
749:
748:
736:
732:
729:
726:
722:
717:
711:
705:
699:
695:
690:
683:
677:
674:
671:
667:
666:lim sup
663:
657:
651:
645:
641:
637:
634:
631:
628:
623:
619:
614:
605:
602:
599:
595:
594:lim sup
591:
588:
569:
566:
565:
564:
560:
546:
542:
525:
522:
495:
494:
482:
477:
464:
460:
456:
453:
450:
447:
442:
438:
432:
427:
424:
421:
417:
409:
401:
397:
394:
391:
387:
382:
378:
375:
372:
358:
357:
346:
343:
340:
336:
332:
329:
326:
322:
307:
306:
295:
292:
288:
284:
281:
278:
274:
250:
235:
234:
228:
218:
213:
195:
194:
183:
178:
174:
170:
167:
164:
161:
156:
152:
146:
141:
138:
135:
131:
127:
124:
121:
118:
115:
96:
93:
63:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4683:
4672:
4669:
4667:
4664:
4662:
4659:
4657:
4654:
4653:
4651:
4642:
4639:
4638:
4634:
4630:
4627:
4625:
4622:
4620:
4617:
4616:
4612:
4607:
4605:0-691-11385-8
4601:
4597:
4593:
4589:
4585:
4582:
4576:
4572:
4568:
4563:
4562:
4558:
4550:(1): 263–266.
4549:
4545:
4541:
4537:
4531:
4528:
4524:
4520:
4516:
4512:
4505:
4502:
4498:
4494:
4490:
4486:
4482:
4478:
4474:
4470:
4463:
4460:
4456:
4454:0-521-37897-4
4450:
4446:
4438:
4435:
4430:
4429:
4422:
4419:
4412:
4410:
4408:
4403:
4399:
4395:
4376:
4369:
4365:
4359:
4355:
4342:
4339:
4336:
4332:
4324:
4323:
4322:
4321:
4313:
4311:
4309:
4305:
4300:
4288:
4279:
4255:
4236:
4228:
4225:
4219:
4216:
4210:
4206:
4200:
4194:
4191:
4185:
4181:
4175:
4172:
4169:
4164:
4161:
4158:
4155:
4151:
4144:
4138:
4135:
4132:
4129:
4119:
4111:
4108:
4092:
4089:
4086:
4082:
4078:
4075:
4072:
4069:
4062:
4061:
4060:
4054:
4052:
4050:
4043:
4036:
4017:
4012:
4008:
4004:
4001:
3998:
3993:
3990:
3987:
3983:
3974:
3971:
3962:
3956:
3951:
3947:
3940:
3937:
3926:
3921:
3917:
3909:
3906:
3903:
3895:
3892:
3883:
3878:
3874:
3863:
3859:
3853:
3849:
3838:
3835:
3832:
3828:
3820:
3819:
3818:
3815:
3813:
3807:
3803:
3798:
3794:
3790:
3786:
3782:
3778:
3773:
3751:
3747:
3739:
3735:
3731:
3719:
3716:
3713:
3709:
3701:
3700:
3699:
3696:
3692:
3688:
3683:
3677:
3673:
3666:
3646:
3641:
3637:
3631:
3628:
3616:
3613:
3610:
3606:
3598:
3597:
3596:
3592:
3585:
3581:
3577:
3571:
3554:
3549:
3545:
3534:
3531:
3528:
3524:
3516:
3515:
3514:
3510:
3503:
3499:
3495:
3489:
3486:
3482:
3477:
3472:
3468:
3460:
3454:
3449:
3441:
3439:
3429:
3418:
3411:
3408:
3407:
3406:
3400:
3396:
3392:
3388:
3382:
3378:
3374:
3368:
3364:
3360:
3356:
3352:
3346:
3342:
3338:
3332:
3328:
3324:
3305:
3296:
3290:
3287:
3284:
3281:
3275:
3269:
3266:
3258:
3254:
3250:
3245:
3242:
3238:
3232:
3228:
3224:
3219:
3215:
3207:
3206:
3205:
3201:
3197:
3193:
3189:
3183:
3179:
3175:
3155:
3152:
3149:
3146:
3141:
3137:
3129:
3128:
3127:
3125:
3121:
3117:
3113:
3108:
3104:
3083:
3079:
3072:
3069:
3063:
3059:
3046:
3043:
3040:
3036:
3032:
3026:
3023:
3018:
3014:
3009:
3000:
2999:
2998:
2992:
2990:
2973:
2970:
2967:
2962:
2957:
2953:
2947:
2942:
2937:
2933:
2927:
2922:
2917:
2913:
2907:
2904:
2901:
2895:
2889:
2886:
2879:
2878:
2877:
2875:
2867:
2865:
2863:
2858:
2856:
2852:
2848:
2829:
2824:
2821:
2817:
2811:
2803:
2800:
2787:
2784:
2781:
2777:
2769:
2768:
2767:
2751:
2747:
2743:
2740:
2712:
2708:
2704:
2701:
2697:
2692:
2686:
2680:
2673:
2672:
2671:
2669:
2668:complex plane
2665:
2641:
2637:
2633:
2630:
2626:
2615:
2611:
2609:
2605:
2597:
2593:
2589:
2585:
2581:
2580:
2579:
2577:
2569:
2552:
2549:
2546:
2526:
2523:
2518:
2514:
2509:
2505:
2483:
2479:
2474:
2470:
2449:
2440:
2436:
2430:
2426:
2419:
2416:
2413:
2409:
2403:
2396:
2392:
2385:
2381:
2375:
2372:
2369:
2365:
2357:
2351:
2348:
2326:
2322:
2315:
2312:
2309:
2303:
2283:
2280:
2260:
2240:
2236:
2232:
2212:
2209:
2206:
2202:
2198:
2178:
2175:
2172:
2168:
2164:
2144:
2140:
2136:
2114:
2110:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2053:
2048:
2045:
2042:
2038:
2034:
2029:
2026:
2023:
2019:
2013:
2009:
2001:
1996:
1992:
1988:
1983:
1980:
1977:
1973:
1967:
1964:
1961:
1957:
1950:
1945:
1940:
1936:
1927:
1924:
1908:
1904:
1900:
1880:
1877:
1874:
1870:
1866:
1858:
1839:
1836:
1829:(effectively
1816:
1813:
1810:
1806:
1802:
1782:
1778:
1774:
1752:
1749:
1746:
1742:
1737:
1731:
1727:
1706:
1683:
1680:
1677:
1673:
1668:
1662:
1658:
1645:
1637:
1634:
1630:
1626:
1603:
1600:
1597:
1593:
1588:
1582:
1578:
1565:
1552:
1551:
1550:
1548:
1532:
1528:
1524:
1503:
1481:
1477:
1453:
1449:
1445:
1442:
1439:
1419:
1415:
1411:
1408:
1405:
1402:
1394:
1391:
1390:straight-line
1374:
1368:
1365:
1362:
1359:
1349:
1345:
1341:
1338:
1332:
1326:
1320:
1314:
1305:
1298:
1282:
1278:
1271:
1268:
1265:
1261:
1255:
1251:
1245:
1233:
1225:
1209:
1205:
1187:
1184:
1181:
1177:
1157:
1148:
1143:
1135:
1132:
1129:
1117:
1116:
1115:
1098:
1095:
1082:
1074:
1071:
1068:
1060:
1056:
1038:
1035:
1032:
1024:
1021:
1018:
1010:
1007:
1004:
1000:
980:
968:
967:
966:
949:
945:
938:
935:
932:
928:
922:
918:
912:
900:
892:
889:
882:
881:
880:
878:
873:
871:
867:
863:
860:. Note that
859:
834:
822:
818:
798:
789:
784:
781:
774:
773:
772:
771:is less than
770:
766:
762:
758:
754:
730:
727:
724:
715:
709:
697:
693:
681:
669:
661:
655:
643:
635:
632:
629:
621:
617:
597:
589:
586:
579:
578:
577:
575:
567:
561:
544:
540:
531:
530:
529:
523:
521:
519:
516:
512:
508:
504:
501: −
500:
480:
462:
454:
451:
448:
440:
436:
425:
422:
419:
415:
407:
395:
392:
389:
380:
373:
370:
363:
362:
361:
344:
341:
338:
330:
327:
324:
312:
311:
310:
293:
290:
282:
279:
276:
264:
263:
262:
240:
232:
229:
226:
221:
217:
214:
211:
207:
203:
200:
199:
198:
181:
176:
168:
165:
162:
154:
150:
139:
136:
133:
129:
125:
119:
113:
106:
105:
104:
102:
94:
92:
89:
85:
84:Taylor series
81:
77:
53:
49:
45:
41:
37:
33:
19:
4591:
4588:Stein, Elias
4569:, New York:
4566:
4547:
4543:
4530:
4514:
4510:
4504:
4472:
4468:
4462:
4444:
4437:
4427:
4421:
4401:
4397:
4393:
4391:
4317:
4301:
4286:
4253:
4251:
4058:
4041:
4032:
3816:
3805:
3801:
3796:
3792:
3788:
3784:
3780:
3776:
3771:
3768:
3697:
3690:
3686:
3675:
3671:
3664:
3661:
3590:
3583:
3582:) = −ln(1 −
3579:
3575:
3572:
3569:
3508:
3501:
3497:
3493:
3490:
3484:
3475:
3470:
3466:
3458:
3452:
3447:
3445:
3427:
3422:
3416:
3409:
3398:
3394:
3390:
3386:
3380:
3376:
3372:
3366:
3362:
3358:
3357:is 1; since
3354:
3350:
3344:
3340:
3336:
3330:
3326:
3322:
3320:
3199:
3195:
3191:
3187:
3181:
3177:
3173:
3170:
3123:
3115:
3106:
3102:
3100:
2996:
2988:
2874:trigonometry
2871:
2859:
2854:
2850:
2846:
2844:
2732:
2663:
2662:
2607:
2603:
2601:
2595:
2591:
2587:
2583:
2573:
1922:
1546:
1468:
1113:
964:
874:
865:
861:
855:
768:
764:
760:
756:
750:
571:
563:convergence.
527:
517:
510:
506:
502:
498:
496:
359:
308:
238:
236:
230:
224:
219:
215:
201:
196:
103:defined as:
100:
98:
40:power series
35:
29:
4571:McGraw-Hill
3812:dilogarithm
3595:, which is
3500:) = 1/(1 −
3483:called the
1699:. Negative
32:mathematics
4650:Categories
4559:References
4045:| = 1
3814:function.
3682:derivative
3462:such that
1857:linear fit
877:ratio test
95:Definition
4629:Root test
4497:119974403
4348:∞
4333:∑
4264:∞
4229:⋯
4226:−
4176:−
4109:−
4098:∞
4083:∑
4073:
4035:uniformly
3999:≤
3991:−
3969:⌋
3957:
3944:⌊
3907:−
3893:−
3844:∞
3829:∑
3725:∞
3710:∑
3622:∞
3607:∑
3540:∞
3525:∑
3474:| =
3385:. Since
3291:
3270:
3147:−
3052:∞
3037:∑
3024:−
2971:⋯
2948:−
2908:−
2890:
2801:−
2793:∞
2778:∑
2553:θ
2550:
2373:−
2304:−
2284:θ
2281:±
2087:…
2046:−
2035:−
2027:−
1989:−
1981:−
1843:∞
1750:−
1681:−
1652:∞
1649:→
1601:−
1572:∞
1569:→
1409:−
1403:ε
1393:asymptote
1369:ε
1346:ε
1333:ε
1321:ε
1240:∞
1237:→
1164:∞
1161:→
1133:−
1072:−
1022:−
987:∞
984:→
907:∞
904:→
805:∞
802:→
728:−
676:∞
673:→
633:−
604:∞
601:→
574:root test
452:−
431:∞
416:∑
393:−
328:−
280:−
249:∞
166:−
145:∞
130:∑
62:∞
52:converges
4613:See also
4538:(1918).
4407:abscissa
4297:sin(100)
4282:sin(0.1)
3485:boundary
3110:are the
1855:) via a
4477:Bibcode
4293:sin(10)
3810:is the
2463:versus
2129:versus
1767:versus
223:is the
206:complex
86:of the
46:at the
4602:
4577:
4495:
4451:
4405:: the
4308:series
4039:|
3481:circle
3464:|
3190:= cos(
2887:arctan
1545:where
868:is an
468:
412:
404:
204:is a
197:where
34:, the
4671:Radii
4493:S2CID
4413:Notes
3795:with
3479:is a
3383:) = 1
3204:then
3124:other
38:of a
4600:ISBN
4575:ISBN
4449:ISBN
4005:<
3379:sin(
3375:) +
3371:cos(
3343:sin(
3339:) +
3335:cos(
3325:and
3198:sin(
3194:) +
3185:and
1144:<
1096:<
505:| =
339:>
291:<
210:disk
78:and
44:disk
4519:doi
4485:doi
4473:240
4289:= 1
4070:sin
3948:log
3684:of
3667:= 1
3593:= 0
3511:= 0
3288:sin
3267:cos
2547:cos
1642:lim
1562:lim
1230:lim
1154:lim
977:lim
897:lim
377:sup
30:In
4652::
4598:,
4573:,
4548:29
4546:.
4542:.
4515:50
4513:,
4491:,
4483:,
4471:,
3791:)/
3695:.
3469:−
3438:.
3182:iy
3180:+
3176:=
2864:.
2610:.
1454:2.
1099:1.
872:.
520:.
4521::
4487::
4479::
4402:n
4398:a
4394:s
4377:.
4370:s
4366:n
4360:n
4356:a
4343:1
4340:=
4337:n
4287:x
4254:x
4237:x
4220:!
4217:5
4211:5
4207:x
4201:+
4195:!
4192:3
4186:3
4182:x
4173:x
4170:=
4165:1
4162:+
4159:n
4156:2
4152:x
4145:!
4142:)
4139:1
4136:+
4133:n
4130:2
4127:(
4120:n
4116:)
4112:1
4106:(
4093:0
4090:=
4087:n
4079:=
4076:x
4042:z
4018:,
4013:n
4009:2
4002:i
3994:1
3988:n
3984:2
3975:1
3972:+
3966:)
3963:i
3960:(
3952:2
3941:=
3938:n
3927:n
3922:n
3918:2
3910:1
3904:n
3900:)
3896:1
3890:(
3884:=
3879:i
3875:a
3864:i
3860:z
3854:i
3850:a
3839:1
3836:=
3833:i
3808:)
3806:z
3804:(
3802:h
3797:g
3793:z
3789:z
3787:(
3785:g
3781:z
3779:(
3777:h
3772:h
3752:n
3748:z
3740:2
3736:n
3732:1
3720:1
3717:=
3714:n
3693:)
3691:z
3689:(
3687:g
3678:)
3676:z
3674:(
3672:f
3665:z
3647:,
3642:n
3638:z
3632:n
3629:1
3617:1
3614:=
3611:n
3591:z
3586:)
3584:z
3580:z
3578:(
3576:g
3555:,
3550:n
3546:z
3535:0
3532:=
3529:n
3509:z
3504:)
3502:z
3498:z
3496:(
3494:f
3476:r
3471:a
3467:z
3459:z
3453:r
3448:a
3436:π
3432:π
3428:i
3425:π
3419:.
3417:i
3414:π
3410:z
3403:π
3399:y
3395:y
3391:y
3387:y
3381:y
3377:i
3373:y
3367:z
3363:x
3359:x
3355:e
3351:e
3347:)
3345:y
3341:i
3337:y
3331:y
3327:y
3323:x
3306:,
3303:)
3300:)
3297:y
3294:(
3285:i
3282:+
3279:)
3276:y
3273:(
3264:(
3259:x
3255:e
3251:=
3246:y
3243:i
3239:e
3233:x
3229:e
3225:=
3220:z
3216:e
3202:)
3200:y
3196:i
3192:y
3188:e
3178:x
3174:z
3156:0
3153:=
3150:1
3142:z
3138:e
3116:z
3107:n
3103:B
3084:n
3080:z
3073:!
3070:n
3064:n
3060:B
3047:0
3044:=
3041:n
3033:=
3027:1
3019:z
3015:e
3010:z
2974:.
2968:+
2963:7
2958:7
2954:z
2943:5
2938:5
2934:z
2928:+
2923:3
2918:3
2914:z
2905:z
2902:=
2899:)
2896:z
2893:(
2855:i
2851:z
2849:(
2847:f
2830:.
2825:n
2822:2
2818:z
2812:n
2808:)
2804:1
2798:(
2788:0
2785:=
2782:n
2752:2
2748:z
2744:+
2741:1
2713:2
2709:z
2705:+
2702:1
2698:1
2693:=
2690:)
2687:z
2684:(
2681:f
2659:.
2642:2
2638:z
2634:+
2631:1
2627:1
2604:a
2596:f
2592:a
2588:a
2584:f
2565:.
2527:0
2524:=
2519:2
2515:n
2510:/
2506:1
2484:2
2480:n
2475:/
2471:1
2450:)
2441:n
2437:b
2431:n
2427:c
2420:1
2417:+
2414:n
2410:c
2404:+
2397:n
2393:c
2386:n
2382:b
2376:1
2370:n
2366:c
2358:(
2352:2
2349:1
2327:r
2323:/
2319:)
2316:1
2313:+
2310:p
2307:(
2261:p
2241:r
2237:/
2233:1
2213:0
2210:=
2207:n
2203:/
2199:1
2179:0
2176:=
2173:n
2169:/
2165:1
2145:n
2141:/
2137:1
2115:n
2111:b
2090:.
2084:,
2081:5
2078:,
2075:4
2072:,
2069:3
2066:=
2063:n
2054:2
2049:1
2043:n
2039:c
2030:2
2024:n
2020:c
2014:n
2010:c
2002:2
1997:n
1993:c
1984:1
1978:n
1974:c
1968:1
1965:+
1962:n
1958:c
1951:=
1946:2
1941:n
1937:b
1925:.
1909:r
1905:/
1901:1
1881:0
1878:=
1875:n
1871:/
1867:1
1840:=
1837:n
1817:0
1814:=
1811:n
1807:/
1803:1
1783:n
1779:/
1775:1
1753:1
1747:n
1743:c
1738:/
1732:n
1728:c
1707:r
1684:1
1678:n
1674:c
1669:/
1663:n
1659:c
1646:n
1638:=
1635:r
1631:/
1627:1
1604:1
1598:n
1594:c
1589:/
1583:n
1579:c
1566:n
1547:r
1533:r
1529:/
1525:1
1504:n
1482:n
1478:c
1450:/
1446:1
1443:=
1440:r
1420:2
1416:/
1412:1
1406:=
1375:.
1366:2
1363:+
1360:1
1355:)
1350:3
1342:+
1339:1
1336:(
1327:=
1324:)
1318:(
1315:f
1283:.
1279:|
1272:1
1269:+
1266:n
1262:c
1256:n
1252:c
1246:|
1234:n
1226:=
1216:|
1210:n
1206:c
1201:|
1194:|
1188:1
1185:+
1182:n
1178:c
1173:|
1158:n
1149:1
1140:|
1136:a
1130:z
1126:|
1089:|
1083:n
1079:)
1075:a
1069:z
1066:(
1061:n
1057:c
1052:|
1045:|
1039:1
1036:+
1033:n
1029:)
1025:a
1019:z
1016:(
1011:1
1008:+
1005:n
1001:c
996:|
981:n
950:.
946:|
939:1
936:+
933:n
929:c
923:n
919:c
913:|
901:n
893:=
890:r
866:f
862:r
835:n
829:|
823:n
819:c
814:|
799:n
790:1
785:=
782:r
769:a
765:z
761:C
757:C
735:|
731:a
725:z
721:|
716:)
710:n
704:|
698:n
694:c
689:|
682:(
670:n
662:=
656:n
650:|
644:n
640:)
636:a
630:z
627:(
622:n
618:c
613:|
598:n
590:=
587:C
545:n
541:c
518:z
511:z
507:r
503:a
499:z
481:}
463:n
459:)
455:a
449:z
446:(
441:n
437:c
426:0
423:=
420:n
408:|
400:|
396:a
390:z
386:|
381:{
374:=
371:r
345:.
342:r
335:|
331:a
325:z
321:|
294:r
287:|
283:a
277:z
273:|
239:r
231:z
225:n
220:n
216:c
202:a
182:,
177:n
173:)
169:a
163:z
160:(
155:n
151:c
140:0
137:=
134:n
126:=
123:)
120:z
117:(
114:f
101:f
20:)
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