4669:
4186:
4664:{\displaystyle {\begin{aligned}{\mathcal {B}}A(t)&=\sum _{\ell =0}^{\infty }\left(\sum _{k=0}^{\infty }{\frac {{\big (}(2\ell +2)t{\big )}^{k}}{k!}}\right){\frac {(-1)^{\ell }}{(2\ell +1)!}}\\&=\sum _{\ell =0}^{\infty }e^{(2\ell +2)t}{\frac {(-1)^{\ell }}{(2\ell +1)!}}\\&=e^{t}\sum _{\ell =0}^{\infty }{\big (}e^{t}{\big )}^{2\ell +1}{\frac {(-1)^{\ell }}{(2\ell +1)!}}\\&=e^{t}\sin(e^{t}).\end{aligned}}}
1446:
3213:
3797:
6079:
1128:
2022:
There are always many different functions with any given asymptotic expansion. However, there is sometimes a best possible function, in the sense that the errors in the finite-dimensional approximations are as small as possible in some region. Watson's theorem and
Carleman's theorem show that Borel
1164:
2424:
Carleman's theorem gives a summation method for any asymptotic series whose terms do not grow too fast, as the sum can be defined to be the unique function with this asymptotic series in a suitable sector if it exists. Borel summation is slightly weaker than special case of this when
2976:
4175:
4842:
3974:
2930:
2315:
is given by the Borel sum of its asymptotic series. More precisely, the series for the Borel transform converges in a neighborhood of the origin, and can be analytically continued to the positive real axis, and the integral defining the Borel sum converges to
6433:
provides conditions under which convergence of one summation method implies convergence under another method. The principal
Tauberian theorem for Borel summation provides conditions under which the weak Borel method implies convergence of the series.
3606:
5902:
537:
3579:
2778:
979:
5467:
304:
3343:
5364:
6556:
98:
There are (at least) three slightly different methods called Borel summation. They differ in which series they can sum, but are consistent, meaning that if two of the methods sum the same series they give the same answer.
1441:{\displaystyle A(z)=\sum _{k=0}^{\infty }a_{k}z^{k}=\sum _{k=0}^{\infty }a_{k}\left(\int _{0}^{\infty }e^{-t}t^{k}dt\right){\frac {z^{k}}{k!}}=\int _{0}^{\infty }e^{-t}\sum _{k=0}^{\infty }a_{k}{\frac {(tz)^{k}}{k!}}dt,}
1858:
5137:
1932:
1636:
647:
1781:
1707:
895:
4973:
3208:{\displaystyle \lim _{t\rightarrow \infty }e^{-t}\sum _{n=0}^{\infty }{\frac {1-z^{n+1}}{1-z}}{\frac {t^{n}}{n!}}=\lim _{t\rightarrow \infty }{\frac {e^{-t}}{1-z}}{\big (}e^{t}-ze^{tz}{\big )}={\frac {1}{1-z}},}
2341:
Carleman's theorem shows that a function is uniquely determined by an asymptotic series in a sector provided the errors in the finite order approximations do not grow too fast. More precisely it states that if
6656:
745:
3448:
4010:
4687:
6747:
2224:
5169:. More can be said about the domain of convergence of the Borel sum, than that it is a star domain, which is referred to as the Borel polygon, and is determined by the singularities of the series
193:
6263:
3846:
5907:
5811:
5762:
4191:
414:
2642:
657:
Suppose that the Borel transform converges for all positive real numbers to a function growing sufficiently slowly that the following integral is well defined (as an improper integral), the
2789:
3837:
tends to 0 that is given by the original divergent series. This is a typical example of the fact that Borel summation will sometimes "correctly" sum divergent asymptotic expansions.
5859:
86:, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it called
6777:
in quantum field theory. In particular in 2-dimensional
Euclidean field theory the Schwinger functions can often be recovered from their perturbation series using Borel summation (
5502:
3792:{\displaystyle \int _{0}^{\infty }e^{-t}{\mathcal {B}}A(tz)\,dt=\int _{0}^{\infty }{\frac {e^{-t}}{1+tz}}\,dt={\frac {1}{z}}\cdot e^{1/z}\cdot \Gamma \left(0,{\frac {1}{z}}\right)}
37:, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see
788:
2289:
6419:
6074:{\displaystyle {\begin{aligned}A(z)&=\sum _{k=0}^{\infty }(\omega _{1}^{k}+\cdots +\omega _{m}^{k})z^{k}\\&=\sum _{i=1}^{m}{\frac {1}{1-\omega _{i}z}},\end{aligned}}}
5588:
2499:. In practice this generalization is of little use, as there are almost no natural examples of series summable by this method that cannot also be summed by Borel's method.
431:
3480:
6302:
6163:
5677:
5639:
5535:
2669:
1123:{\displaystyle \sum _{k=0}^{\infty }a_{k}z^{k}=A(z)<\infty \quad \Rightarrow \quad {\textstyle \sum }a_{k}z^{k}=A(z)\,\,({\boldsymbol {B}},\,{\boldsymbol {wB}}).}
41:, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by
5612:
5555:
1526:
of series which are divergent under weak Borel summation, but which are Borel summable. The following theorem characterises the equivalence of the two methods.
5390:
214:
3240:
5289:
6475:
6774:
1786:
7168:
7119:
7089:
7058:
5068:
1863:
1567:
578:
1715:
1641:
829:
4889:
6567:
4170:{\displaystyle A(z)=\sum _{k=0}^{\infty }\left(\sum _{\ell =0}^{\infty }{\frac {(-1)^{\ell }(2\ell +2)^{k}}{(2\ell +1)!}}\right)z^{k}.}
673:
4837:{\displaystyle \int _{0}^{\infty }e^{t}\sin(e^{2t})\,dt=\int _{1}^{\infty }\sin(u^{2})\,du={\sqrt {\frac {\pi }{8}}}-S(1)<\infty ,}
3359:
6667:
2120:
7222:
6952:
6947:
6430:
3969:{\displaystyle \lim _{t\rightarrow \infty }e^{-t}({\mathcal {B}}A)(zt)=\lim _{t\rightarrow \infty }{\frac {e^{-t}}{1+zt}}=0,}
5199:
has strictly positive radius of convergence, so that it is analytic in a non-trivial region containing the origin, and let
122:
7196:
6195:
973:
converges (in the standard sense), then the Borel sum and weak Borel sum also converge, and do so to the same value. i.e.
5775:
5722:
336:
7217:
7212:
7191:
2581:
83:
2925:{\displaystyle \int _{0}^{\infty }e^{-t}{\mathcal {B}}A(tz)\,dt=\int _{0}^{\infty }e^{-t}e^{tz}\,dt={\frac {1}{1-z}}}
3807:
905:
This is similar to Borel's integral summation method, except that the Borel transform need not converge for all
6884:
1954:
87:
2559:
in Watson's theorem cannot be replaced by any smaller number (unless the bound on the error is made smaller).
7186:
5828:
5683:
2031:
Watson's theorem gives conditions for a function to be the Borel sum of its asymptotic series. Suppose that
38:
6917:
5479:
6305:
2944:
2446:. More generally one can define summation methods slightly stronger than Borel's by taking the numbers
922:
757:
2235:
6376:
3828:
6894:, which can be used when the bounding function is of some general type (psi-type), instead of being
6942:
6927:
6891:
5560:
1141:
is easily seen by a change in order of integration, which is valid due to absolute convergence: if
532:{\displaystyle \lim _{t\rightarrow \infty }e^{-t}\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}A_{n}(z).}
114:
3574:{\displaystyle {\mathcal {B}}A(t)\equiv \sum _{k=0}^{\infty }\left(-t\right)^{k}={\frac {1}{1+t}}}
5641:
as a polygon is somewhat of a misnomer, since the set need not be polygonal at all; if, however,
926:
3985:, the equivalence theorem ensures that weak Borel summation has the same domain of convergence,
2773:{\displaystyle {\mathcal {B}}A(tz)\equiv \sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}t^{k}=e^{zt},}
7164:
7115:
7085:
7054:
6937:
6932:
6912:
6271:
6141:
5655:
5617:
5513:
3226:. Alternatively this can be seen by appealing to part 2 of the equivalence theorem, since for
912:
751:
7046:
7028:
6895:
4859:
2573:
959:
71:
67:
42:
7178:
7145:
7129:
7099:
7068:
7174:
7152:
7141:
7125:
7095:
7064:
6922:
6166:
1972:
2552:, but is not given by the Borel sum of its asymptotic series. This shows that the number
6907:
5597:
5540:
5462:{\displaystyle \Pi _{A}=\operatorname {cl} \left(\bigcap _{P\in S_{A}}\Pi _{P}\right).}
299:{\displaystyle {\mathcal {B}}A(t)\equiv \sum _{k=0}^{\infty }{\frac {a_{k}}{k!}}t^{k}.}
75:
34:
7206:
7157:
7016:
6801:
Borel summation requires that the coefficients do not grow too fast: more precisely,
3338:{\displaystyle \lim _{t\rightarrow \infty }e^{-t}({\mathcal {B}}A)(zt)=e^{t(z-1)}=0.}
6987:
7138:
Lectures on the theory of functions of a complex variable. I. Holomorphic functions
5359:{\displaystyle \Pi _{P}=\{z\in \mathbb {C} \,\colon \,Oz\cap L_{P}=\varnothing \},}
7079:
5159:
An immediate consequence is that the domain of convergence of the Borel sum is a
7107:
7075:
6781:, p. 461). Some of the singularities of the Borel transform are related to
5160:
6551:{\displaystyle {\textstyle \sum }a_{k}z_{0}^{k}=a(z_{0})\,({\boldsymbol {wB}})}
7050:
6786:
6829:
6782:
2950:
Considering instead the weak Borel transform, the partial sums are given by
5504:
denote the largest star domain on which there is an analytic extension of
50:
17:
1853:{\displaystyle \lim _{t\rightarrow \infty }e^{-t}{\mathcal {B}}A(zt)=0,}
7033:
6847:
5132:{\displaystyle {\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {B}}),}
4180:
After changing the order of summation, the Borel transform is given by
1927:{\displaystyle {\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {wB}})}
1631:{\displaystyle {\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {wB}})}
642:{\displaystyle {\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {wB}})}
1776:{\displaystyle {\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {B}})}
1702:{\displaystyle {\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {B}})}
890:{\displaystyle {\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {B}})}
4968:{\displaystyle \lim _{t\rightarrow \infty }e^{(z-1)t}\sin(e^{zt})=0}
900:
6343:
is dense in the unit circle, there can be no analytic extension of
6651:{\displaystyle a_{k}z_{0}^{k}=O(k^{-1/2}),\qquad \forall k\geq 0,}
6421:. In particular one sees that the Borel polygon is not polygonal.
7112:
Methods of modern mathematical physics. IV. Analysis of operators
3820:, so the original divergent series is Borel summable for all such
740:{\displaystyle \int _{0}^{\infty }e^{-t}{\mathcal {B}}A(tz)\,dt,}
419:
A weak form of Borel's summation method defines the Borel sum of
3443:{\displaystyle A(z)=\sum _{k=0}^{\infty }k!(-1)^{k}\cdot z^{k},}
4985:, and so the weak Borel sum converges on this smaller domain.
5237:
can be continued analytically along the open chord from 0 to
6890:
In the most general case, Borel summation is generalized by
6742:{\displaystyle \sum _{k=0}^{\infty }a_{k}z_{0}^{k}=a(z_{0})}
5022:, then it is also Borel summable at all points on the chord
4196:
3884:
3640:
3486:
3278:
2823:
2675:
1821:
901:
Borel's integral summation method with analytic continuation
763:
707:
220:
2023:
summation produces such a best possible sum of the series.
1451:
where the rightmost expression is exactly the Borel sum at
5472:
An alternative definition was used by Borel and Phragmén (
2219:{\displaystyle |f(z)-a_{0}-a_{1}z-\cdots -a_{n-1}z^{n-1}|}
1523:
45:, his teacher, he said in Latin, 'The Master forbids it'.
6308:
with the geometric series). It can however be shown that
6828:. There is a variation of Borel summation that replaces
1476:
imply that these methods provide analytic extensions to
6480:
5073:
1868:
1720:
1646:
1572:
1051:
834:
583:
6670:
6570:
6478:
6379:
6274:
6198:
6144:
5905:
5831:
5778:
5725:
5658:
5620:
5600:
5563:
5543:
5516:
5482:
5393:
5292:
5071:
4892:
4690:
4189:
4013:
3849:
3609:
3483:
3362:
3243:
2979:
2792:
2672:
2584:
2238:
2123:
2005:
method converges up to the domain of convergence for
1866:
1789:
1718:
1644:
1570:
1167:
982:
832:
760:
676:
581:
434:
339:
217:
188:{\displaystyle A(z)=\sum _{k=0}^{\infty }a_{k}z^{k},}
125:
6258:{\displaystyle A(z)=\sum _{k=0}^{\infty }z^{2^{k}},}
2530:
with an error bound of the form above in the region
7045:(2nd ed.), Berlin, New York: Springer-Verlag,
6373:from which (via the second definition) one obtains
5806:{\displaystyle z\in \mathbb {C} \setminus \Pi _{A}}
5757:{\displaystyle z\in \operatorname {int} (\Pi _{A})}
5686:provides convergence criteria for Borel summation.
4866:along chords, the Borel integral converges for all
2037:is a function satisfying the following conditions:
409:{\displaystyle A_{n}(z)=\sum _{k=0}^{n}a_{k}z^{k}.}
7156:
6741:
6650:
6550:
6413:
6296:
6257:
6157:
6073:
5853:
5805:
5756:
5671:
5633:
5606:
5582:
5549:
5529:
5496:
5461:
5358:
5131:
4967:
4836:
4663:
4169:
3968:
3791:
3573:
3442:
3337:
3207:
2924:
2772:
2636:
2283:
2218:
1926:
1852:
1775:
1701:
1630:
1440:
1122:
889:
782:
739:
641:
531:
408:
298:
187:
6855:, which allows the summation of some series with
5041:to the origin. Moreover, there exists a function
6356:. Subsequently the largest star domain to which
5473:
5369:the set of points which lie on the same side of
4894:
4000:The following example extends on that given in (
3911:
3851:
3245:
3095:
2981:
2637:{\displaystyle A(z)=\sum _{k=0}^{\infty }z^{k},}
1971:can be seen as the limiting case of generalized
1791:
1491:Nonequivalence of Borel and weak Borel summation
436:
4989:Existence results and the domain of convergence
2309:Then Watson's theorem says that in this region
32:
4544:
4526:
4301:
4272:
3176:
3140:
8:
7163:, vol. II, Cambridge University Press,
7136:Sansone, Giovanni; Gerretsen, Johan (1960),
5350:
5306:
3592:, which can be analytically continued to all
309:
6969:
6967:
4863:
2647:which converges (in the standard sense) to
208:to be its corresponding exponential series
6778:
5652:has only finitely many singularities then
2348:is analytic in the interior of the sector
2300:in the region (for some positive constant
7032:
6730:
6711:
6706:
6696:
6686:
6675:
6669:
6616:
6609:
6590:
6585:
6575:
6569:
6537:
6533:
6524:
6505:
6500:
6490:
6479:
6477:
6384:
6378:
6283:
6275:
6273:
6244:
6239:
6229:
6218:
6197:
6149:
6143:
6052:
6036:
6030:
6019:
5999:
5986:
5981:
5962:
5957:
5944:
5933:
5906:
5904:
5845:
5830:
5797:
5786:
5785:
5777:
5745:
5724:
5663:
5657:
5625:
5619:
5599:
5590:the interior of the circle with diameter
5574:
5562:
5542:
5521:
5515:
5490:
5489:
5481:
5445:
5433:
5422:
5398:
5392:
5338:
5324:
5320:
5316:
5315:
5297:
5291:
5118:
5114:
5093:
5083:
5072:
5070:
5052:analytic throughout the disk with radius
4947:
4913:
4897:
4891:
4798:
4788:
4779:
4760:
4755:
4741:
4729:
4710:
4700:
4695:
4689:
4645:
4626:
4580:
4564:
4549:
4543:
4542:
4535:
4525:
4524:
4518:
4507:
4497:
4451:
4435:
4411:
4401:
4390:
4344:
4328:
4306:
4300:
4299:
4271:
4270:
4267:
4261:
4250:
4235:
4224:
4195:
4194:
4190:
4188:
4158:
4117:
4092:
4076:
4070:
4059:
4044:
4033:
4012:
3932:
3926:
3914:
3883:
3882:
3870:
3854:
3848:
3774:
3747:
3743:
3726:
3716:
3691:
3685:
3679:
3674:
3660:
3639:
3638:
3629:
3619:
3614:
3608:
3553:
3544:
3521:
3510:
3485:
3484:
3482:
3431:
3418:
3393:
3382:
3361:
3308:
3277:
3276:
3264:
3248:
3242:
3184:
3175:
3174:
3165:
3149:
3139:
3138:
3116:
3110:
3098:
3075:
3069:
3043:
3030:
3024:
3013:
3000:
2984:
2978:
2904:
2894:
2885:
2872:
2862:
2857:
2843:
2822:
2821:
2812:
2802:
2797:
2791:
2758:
2745:
2725:
2719:
2713:
2702:
2674:
2673:
2671:
2625:
2615:
2604:
2583:
2275:
2270:
2261:
2243:
2237:
2211:
2199:
2183:
2161:
2148:
2124:
2122:
1913:
1909:
1888:
1878:
1867:
1865:
1820:
1819:
1810:
1794:
1788:
1765:
1761:
1740:
1730:
1719:
1717:
1691:
1687:
1666:
1656:
1645:
1643:
1617:
1613:
1592:
1582:
1571:
1569:
1412:
1396:
1390:
1380:
1369:
1356:
1346:
1341:
1318:
1312:
1295:
1282:
1272:
1267:
1252:
1242:
1231:
1218:
1208:
1198:
1187:
1166:
1106:
1105:
1097:
1093:
1092:
1071:
1061:
1050:
1018:
1008:
998:
987:
981:
962:summation methods, meaning that whenever
879:
875:
854:
844:
833:
831:
762:
761:
759:
727:
706:
705:
696:
686:
681:
675:
652:
628:
624:
603:
593:
582:
580:
511:
491:
485:
479:
468:
455:
439:
433:
397:
387:
377:
366:
344:
338:
287:
267:
261:
255:
244:
219:
218:
216:
176:
166:
156:
145:
124:
82:). It is particularly useful for summing
54:
6790:
5682:The following theorem, due to Borel and
6963:
6541:
6538:
5790:
5347:
5119:
1941:Relationship to other summation methods
1917:
1914:
1766:
1692:
1621:
1618:
1110:
1107:
1098:
880:
632:
629:
5854:{\displaystyle z\in \partial \Pi _{A}}
6773:Borel summation finds application in
6442:
6174:centred at the origin, and such that
6138:, and consequently the Borel polygon
5694:
5497:{\displaystyle S\subset \mathbb {C} }
4001:
3996:An example in which equivalence fails
2964:) = (1 − z)/(1 −
2935:which converges in the larger region
1534:
79:
29:Summation method for divergent series
7:
7041:Glimm, James; Jaffe, Arthur (1987),
7017:"Mémoire sur les séries divergentes"
5861:depends on the nature of the point.
5378:as the origin. The Borel polygon of
5277:which is perpendicular to the chord
4883:For the weak Borel sum we note that
563:, we say that the weak Borel sum of
310:Borel's exponential summation method
6749:, and the series converges for all
5210:denote the set of singularities of
2783:from which we obtain the Borel sum
2457:to be slightly larger, for example
6883:. This generalization is given by
6687:
6633:
6381:
6230:
6146:
5945:
5842:
5838:
5794:
5742:
5660:
5622:
5571:
5518:
5442:
5395:
5294:
4904:
4828:
4761:
4701:
4519:
4402:
4262:
4236:
4071:
4045:
3921:
3861:
3760:
3680:
3620:
3522:
3464:does not converge for any nonzero
3394:
3255:
3105:
3025:
2991:
2863:
2803:
2714:
2616:
1801:
1551:be a formal power series, and fix
1381:
1347:
1273:
1243:
1199:
1042:
999:
783:{\displaystyle {\mathcal {B}}A(z)}
687:
480:
446:
256:
157:
25:
2403:is zero provided that the series
1993:the domain of convergence of the
653:Borel's integral summation method
6362:can be analytically extended is
6314:does not converge for any point
5271:denote the line passing through
3813:This integral converges for all
3218:where, again, convergence is on
2284:{\displaystyle C^{n+1}n!|z|^{n}}
2113:with the property that the error
6632:
6414:{\displaystyle \Pi _{A}=B(0,1)}
6132: = 1, ...,
5891: = 1, ...,
3349:An alternating factorial series
2970:, and so the weak Borel sum is
1506:that is weak Borel summable at
1049:
1045:
814:, we say that the Borel sum of
6953:Van Wijngaarden transformation
6948:Abelian and tauberian theorems
6736:
6723:
6626:
6602:
6545:
6534:
6530:
6517:
6408:
6396:
6284:
6276:
6208:
6202:
5992:
5950:
5919:
5913:
5751:
5738:
5123:
5115:
5111:
5105:
4956:
4940:
4926:
4914:
4901:
4822:
4816:
4785:
4772:
4738:
4722:
4651:
4638:
4603:
4588:
4577:
4567:
4474:
4459:
4448:
4438:
4427:
4412:
4367:
4352:
4341:
4331:
4292:
4277:
4210:
4204:
4140:
4125:
4114:
4098:
4089:
4079:
4023:
4017:
3918:
3904:
3895:
3892:
3879:
3858:
3657:
3648:
3500:
3494:
3415:
3405:
3372:
3366:
3324:
3312:
3298:
3289:
3286:
3273:
3252:
3102:
2988:
2840:
2831:
2692:
2683:
2594:
2588:
2271:
2262:
2212:
2138:
2132:
2125:
2046:is holomorphic in some region
1921:
1910:
1906:
1900:
1838:
1829:
1798:
1770:
1762:
1758:
1752:
1696:
1688:
1684:
1678:
1625:
1614:
1610:
1604:
1409:
1399:
1177:
1171:
1114:
1094:
1089:
1083:
1046:
1036:
1030:
884:
876:
872:
866:
777:
771:
724:
715:
636:
625:
621:
615:
523:
517:
443:
356:
350:
234:
228:
135:
129:
1:
7159:The quantum theory of fields.
7114:, New York: Academic Press ,
5583:{\displaystyle P\in \Pi _{A}}
1522:. However, one can construct
793:If the integral converges at
6977:. AMS Chelsea, Rhode Island.
5474:Sansone & Gerretsen 1960
7192:Encyclopedia of Mathematics
7140:, P. Noordhoff, Groningen,
5679:will in fact be a polygon.
4873:(the integral diverges for
84:divergent asymptotic series
7239:
7021:Ann. Sci. Éc. Norm. Supér.
6181:is a midpoint of an edge.
4681:the Borel sum is given by
2522:has the asymptotic series
1991: → ∞
1516:is also Borel summable at
7185:Zakharov, A. A. (2001) ,
7051:10.1007/978-1-4612-4728-9
6789:in quantum field theory (
5537:is the largest subset of
3808:incomplete gamma function
3474:. The Borel transform is
2663:. The Borel transform is
2093:has an asymptotic series
7187:"Borel summation method"
6885:Mittag-Leffler summation
6337:. Since the set of such
6297:{\displaystyle |z|<1}
6165:is given by the regular
6158:{\displaystyle \Pi _{A}}
6094:. Seen as a function on
6089:(0,1) ⊂
5672:{\displaystyle \Pi _{A}}
5634:{\displaystyle \Pi _{A}}
5530:{\displaystyle \Pi _{A}}
3827:. This function has an
2947:of the original series.
2494: log log
1955:Mittag-Leffler summation
88:Mittag-Leffler summation
6775:perturbation expansions
5614:. Referring to the set
2649:1/(1 −
2391:in this region for all
2336:
2026:
1953:is the special case of
330:denote the partial sum
7223:Quantum chromodynamics
6779:Glimm & Jaffe 1987
6743:
6691:
6652:
6552:
6415:
6304:(for instance, by the
6298:
6259:
6234:
6159:
6075:
6035:
5949:
5855:
5807:
5758:
5673:
5635:
5608:
5584:
5551:
5531:
5498:
5463:
5360:
5154:, θ ∈
5133:
4969:
4838:
4665:
4523:
4406:
4266:
4240:
4171:
4075:
4049:
3970:
3793:
3600:. So the Borel sum is
3575:
3526:
3444:
3398:
3339:
3209:
3029:
2926:
2774:
2718:
2638:
2620:
2285:
2220:
1973:Euler summation method
1928:
1854:
1777:
1703:
1632:
1442:
1385:
1247:
1203:
1124:
1003:
923:analytically continued
911:, but converges to an
891:
784:
741:
643:
533:
484:
410:
382:
300:
260:
189:
161:
55:Reed & Simon (1978
47:
7084:, New York: Chelsea,
7076:Hardy, Godfrey Harold
6973:Hardy, G. H. (1992).
6812:has to be bounded by
6744:
6671:
6653:
6553:
6416:
6299:
6260:
6214:
6160:
6111:has singularities at
6076:
6015:
5929:
5856:
5808:
5759:
5674:
5636:
5609:
5585:
5552:
5532:
5499:
5464:
5361:
5134:
5009:is Borel summable at
4994:Summability on chords
4970:
4839:
4666:
4503:
4386:
4246:
4220:
4172:
4055:
4029:
3971:
3794:
3576:
3506:
3445:
3378:
3340:
3210:
3009:
2945:analytic continuation
2927:
2775:
2698:
2639:
2600:
2516:) = exp(–1/
2333:in the region above.
2286:
2221:
1986:in the sense that as
1929:
1855:
1778:
1704:
1633:
1443:
1365:
1227:
1183:
1125:
983:
892:
785:
742:
644:
542:If this converges at
534:
464:
411:
362:
301:
240:
190:
141:
6668:
6568:
6476:
6377:
6272:
6196:
6176:1 ∈
6142:
5903:
5886:-th roots of unity,
5829:
5776:
5723:
5656:
5618:
5598:
5561:
5541:
5514:
5480:
5391:
5290:
5069:
4890:
4871: ≤ 2
4688:
4187:
4011:
3847:
3829:asymptotic expansion
3607:
3481:
3360:
3353:Consider the series
3241:
2977:
2790:
2670:
2582:
2568:The geometric series
2236:
2121:
1864:
1787:
1716:
1642:
1568:
1165:
980:
830:
758:
674:
579:
432:
337:
215:
123:
7218:Summability methods
7213:Mathematical series
6943:Nachbin resummation
6892:Nachbin resummation
6716:
6595:
6510:
6467: ∈
6425:A Tauberian theorem
6319: ∈
6084:which converges on
5991:
5967:
5875: ∈
5254: ∈
5221: ∈
5017: ∈
4998:If a formal series
4864:convergence theorem
4765:
4705:
3684:
3624:
3469: ∈
2867:
2807:
2377:)| < |
2018:Uniqueness theorems
1556: ∈
1511: ∈
1351:
1277:
921:near 0 that can be
798: ∈
691:
547: ∈
115:formal power series
7034:10.24033/asens.463
7015:Borel, E. (1899),
6988:"Natural Boundary"
6918:Abel–Plana formula
6846:for some positive
6755:| < |
6739:
6702:
6648:
6581:
6548:
6496:
6484:
6411:
6294:
6268:converges for all
6255:
6189:The formal series
6155:
6071:
6069:
5977:
5953:
5851:
5803:
5754:
5669:
5631:
5604:
5580:
5557:such that for all
5547:
5527:
5494:
5459:
5440:
5356:
5283:. Define the sets
5216:. This means that
5129:
5077:
4965:
4908:
4834:
4751:
4691:
4661:
4659:
4167:
3966:
3925:
3865:
3789:
3670:
3610:
3590:| < 1
3571:
3440:
3335:
3259:
3232:) < 1
3224:) < 1
3205:
3109:
2995:
2941:) < 1
2922:
2853:
2793:
2770:
2661:| < 1
2634:
2536:)| <
2440:for some constant
2365:) > 0
2337:Carleman's theorem
2281:
2216:
2072:for some positive
1924:
1872:
1850:
1805:
1773:
1724:
1699:
1650:
1628:
1576:
1438:
1337:
1263:
1120:
1055:
927:positive real axis
887:
838:
780:
737:
677:
639:
587:
529:
450:
406:
296:
185:
7170:978-0-521-55002-4
7121:978-0-12-585004-9
7091:978-0-8218-2649-2
7060:978-0-387-96476-8
6938:Laplace transform
6933:Lambert summation
6431:Tauberian theorem
6062:
5772:divergent at all
5607:{\displaystyle S}
5550:{\displaystyle S}
5418:
5184:The Borel polygon
4983: < 1
4893:
4878: > 2
4808:
4807:
4610:
4481:
4374:
4321:
4147:
4004:, 8.5). Consider
3955:
3910:
3850:
3782:
3734:
3714:
3569:
3244:
3200:
3136:
3094:
3089:
3067:
2980:
2920:
2739:
2354:| <
2052:| <
1790:
1427:
1332:
1152:is convergent at
913:analytic function
752:Laplace transform
552:to some function
505:
435:
281:
62:In mathematics,
16:(Redirected from
7230:
7199:
7181:
7162:
7153:Weinberg, Steven
7148:
7132:
7102:
7081:Divergent Series
7071:
7037:
7036:
7003:
7002:
7000:
6998:
6984:
6978:
6975:Divergent Series
6971:
6928:Cesàro summation
6896:exponential type
6882:
6876:
6865:
6854:
6845:
6837:
6827:
6821:
6811:
6763:
6748:
6746:
6745:
6740:
6735:
6734:
6715:
6710:
6701:
6700:
6690:
6685:
6657:
6655:
6654:
6649:
6625:
6624:
6620:
6594:
6589:
6580:
6579:
6557:
6555:
6554:
6549:
6544:
6529:
6528:
6509:
6504:
6495:
6494:
6485:
6471:
6458:
6450:
6420:
6418:
6417:
6412:
6389:
6388:
6372:
6361:
6355:
6348:
6342:
6336:
6330:
6323:
6313:
6303:
6301:
6300:
6295:
6287:
6279:
6264:
6262:
6261:
6256:
6251:
6250:
6249:
6248:
6233:
6228:
6180:
6171:
6164:
6162:
6161:
6156:
6154:
6153:
6137:
6110:
6099:
6093:
6080:
6078:
6077:
6072:
6070:
6063:
6061:
6057:
6056:
6037:
6034:
6029:
6008:
6004:
6003:
5990:
5985:
5966:
5961:
5948:
5943:
5895:
5885:
5879:
5860:
5858:
5857:
5852:
5850:
5849:
5825:summability for
5824:
5812:
5810:
5809:
5804:
5802:
5801:
5789:
5771:
5763:
5761:
5760:
5755:
5750:
5749:
5719:summable at all
5718:
5710:
5678:
5676:
5675:
5670:
5668:
5667:
5651:
5640:
5638:
5637:
5632:
5630:
5629:
5613:
5611:
5610:
5605:
5594:is contained in
5589:
5587:
5586:
5581:
5579:
5578:
5556:
5554:
5553:
5548:
5536:
5534:
5533:
5528:
5526:
5525:
5509:
5503:
5501:
5500:
5495:
5493:
5468:
5466:
5465:
5460:
5455:
5451:
5450:
5449:
5439:
5438:
5437:
5403:
5402:
5383:
5377:
5365:
5363:
5362:
5357:
5343:
5342:
5319:
5302:
5301:
5282:
5276:
5270:
5261:
5248:
5242:
5236:
5230:
5215:
5209:
5198:
5179:
5168:
5155:
5138:
5136:
5135:
5130:
5122:
5098:
5097:
5088:
5087:
5078:
5061:
5051:
5040:
5031:
5021:
5008:
4984:
4974:
4972:
4971:
4966:
4955:
4954:
4933:
4932:
4907:
4879:
4872:
4860:Fresnel integral
4857:
4843:
4841:
4840:
4835:
4809:
4800:
4799:
4784:
4783:
4764:
4759:
4737:
4736:
4715:
4714:
4704:
4699:
4680:
4670:
4668:
4667:
4662:
4660:
4650:
4649:
4631:
4630:
4615:
4611:
4609:
4586:
4585:
4584:
4565:
4563:
4562:
4548:
4547:
4540:
4539:
4530:
4529:
4522:
4517:
4502:
4501:
4486:
4482:
4480:
4457:
4456:
4455:
4436:
4434:
4433:
4405:
4400:
4379:
4375:
4373:
4350:
4349:
4348:
4329:
4327:
4323:
4322:
4320:
4312:
4311:
4310:
4305:
4304:
4276:
4275:
4268:
4265:
4260:
4239:
4234:
4200:
4199:
4176:
4174:
4173:
4168:
4163:
4162:
4153:
4149:
4148:
4146:
4123:
4122:
4121:
4097:
4096:
4077:
4074:
4069:
4048:
4043:
3991:
3984:
3975:
3973:
3972:
3967:
3956:
3954:
3940:
3939:
3927:
3924:
3888:
3887:
3878:
3877:
3864:
3836:
3826:
3819:
3805:
3798:
3796:
3795:
3790:
3788:
3784:
3783:
3775:
3756:
3755:
3751:
3735:
3727:
3715:
3713:
3699:
3698:
3686:
3683:
3678:
3644:
3643:
3637:
3636:
3623:
3618:
3599:
3591:
3580:
3578:
3577:
3572:
3570:
3568:
3554:
3549:
3548:
3543:
3539:
3525:
3520:
3490:
3489:
3473:
3463:
3449:
3447:
3446:
3441:
3436:
3435:
3423:
3422:
3397:
3392:
3344:
3342:
3341:
3336:
3328:
3327:
3282:
3281:
3272:
3271:
3258:
3233:
3225:
3214:
3212:
3211:
3206:
3201:
3199:
3185:
3180:
3179:
3173:
3172:
3154:
3153:
3144:
3143:
3137:
3135:
3124:
3123:
3111:
3108:
3090:
3088:
3080:
3079:
3070:
3068:
3066:
3055:
3054:
3053:
3031:
3028:
3023:
3008:
3007:
2994:
2969:
2942:
2931:
2929:
2928:
2923:
2921:
2919:
2905:
2893:
2892:
2880:
2879:
2866:
2861:
2827:
2826:
2820:
2819:
2806:
2801:
2779:
2777:
2776:
2771:
2766:
2765:
2750:
2749:
2740:
2738:
2730:
2729:
2720:
2717:
2712:
2679:
2678:
2662:
2654:
2643:
2641:
2640:
2635:
2630:
2629:
2619:
2614:
2574:geometric series
2558:
2556:
2551:
2549:
2546: <
2540:
2529:
2528: + ...
2521:
2498:
2475:
2456:
2445:
2439:
2420:
2419: + ...
2402:
2396:
2390:
2366:
2358:
2347:
2332:
2326:
2314:
2305:
2299:
2290:
2288:
2287:
2282:
2280:
2279:
2274:
2265:
2254:
2253:
2225:
2223:
2222:
2217:
2215:
2210:
2209:
2194:
2193:
2166:
2165:
2153:
2152:
2128:
2112:
2111: + ...
2092:
2083:
2077:
2071:
2066:
2056:
2045:
2036:
2027:Watson's theorem
2012:
2004:
1992:
1985:
1970:
1960:
1952:
1933:
1931:
1930:
1925:
1920:
1893:
1892:
1883:
1882:
1873:
1859:
1857:
1856:
1851:
1825:
1824:
1818:
1817:
1804:
1782:
1780:
1779:
1774:
1769:
1745:
1744:
1735:
1734:
1725:
1708:
1706:
1705:
1700:
1695:
1671:
1670:
1661:
1660:
1651:
1637:
1635:
1634:
1629:
1624:
1597:
1596:
1587:
1586:
1577:
1560:
1550:
1521:
1515:
1505:
1486:
1475:
1467:
1456:
1447:
1445:
1444:
1439:
1428:
1426:
1418:
1417:
1416:
1397:
1395:
1394:
1384:
1379:
1364:
1363:
1350:
1345:
1333:
1331:
1323:
1322:
1313:
1311:
1307:
1300:
1299:
1290:
1289:
1276:
1271:
1257:
1256:
1246:
1241:
1223:
1222:
1213:
1212:
1202:
1197:
1157:
1151:
1140:
1129:
1127:
1126:
1121:
1113:
1101:
1076:
1075:
1066:
1065:
1056:
1023:
1022:
1013:
1012:
1002:
997:
972:
957:
949:
933:Basic properties
920:
910:
896:
894:
893:
888:
883:
859:
858:
849:
848:
839:
825:
819:
813:
802:
789:
787:
786:
781:
767:
766:
746:
744:
743:
738:
711:
710:
704:
703:
690:
685:
666:
648:
646:
645:
640:
635:
608:
607:
598:
597:
588:
574:
568:
562:
551:
538:
536:
535:
530:
516:
515:
506:
504:
496:
495:
486:
483:
478:
463:
462:
449:
424:
415:
413:
412:
407:
402:
401:
392:
391:
381:
376:
349:
348:
329:
305:
303:
302:
297:
292:
291:
282:
280:
272:
271:
262:
259:
254:
224:
223:
207:
194:
192:
191:
186:
181:
180:
171:
170:
160:
155:
112:
74:, introduced by
72:divergent series
68:summation method
58:
21:
7238:
7237:
7233:
7232:
7231:
7229:
7228:
7227:
7203:
7202:
7184:
7171:
7151:
7135:
7122:
7106:Reed, Michael;
7105:
7092:
7074:
7061:
7043:Quantum physics
7040:
7014:
7011:
7006:
6996:
6994:
6986:
6985:
6981:
6972:
6965:
6961:
6923:Euler summation
6904:
6878:
6867:
6864:
6856:
6850:
6839:
6832:
6823:
6813:
6810:
6802:
6799:
6797:Generalizations
6771:
6761:
6750:
6726:
6692:
6666:
6665:
6605:
6571:
6566:
6565:
6520:
6486:
6474:
6473:
6466:
6460:
6452:
6446:
6427:
6380:
6375:
6374:
6363:
6357:
6350:
6344:
6338:
6332:
6325:
6315:
6309:
6306:comparison test
6270:
6269:
6240:
6235:
6194:
6193:
6187:
6175:
6167:
6145:
6140:
6139:
6127:
6117:
6112:
6101:
6095:
6085:
6068:
6067:
6048:
6041:
6006:
6005:
5995:
5922:
5901:
5900:
5896:, and consider
5887:
5881:
5874:
5870:
5867:
5841:
5827:
5826:
5818:
5793:
5774:
5773:
5765:
5741:
5721:
5720:
5712:
5701:
5659:
5654:
5653:
5642:
5621:
5616:
5615:
5596:
5595:
5570:
5559:
5558:
5539:
5538:
5517:
5512:
5511:
5505:
5478:
5477:
5441:
5429:
5417:
5413:
5394:
5389:
5388:
5379:
5375:
5370:
5334:
5293:
5288:
5287:
5278:
5272:
5268:
5263:
5259:
5250:
5244:
5238:
5232:
5231:if and only if
5229:
5217:
5211:
5208:
5200:
5189:
5186:
5170:
5164:
5153:
5143:
5089:
5079:
5067:
5066:
5060:
5053:
5042:
5039:
5033:
5030:
5023:
5016:
5010:
4999:
4996:
4991:
4979:
4978:holds only for
4943:
4909:
4888:
4887:
4874:
4867:
4848:
4775:
4725:
4706:
4686:
4685:
4675:
4658:
4657:
4641:
4622:
4613:
4612:
4587:
4576:
4566:
4541:
4531:
4493:
4484:
4483:
4458:
4447:
4437:
4407:
4377:
4376:
4351:
4340:
4330:
4313:
4298:
4269:
4245:
4241:
4213:
4185:
4184:
4154:
4124:
4113:
4088:
4078:
4054:
4050:
4009:
4008:
3998:
3986:
3980:
3941:
3928:
3866:
3845:
3844:
3832:
3821:
3814:
3803:
3767:
3763:
3739:
3700:
3687:
3625:
3605:
3604:
3593:
3585:
3558:
3532:
3528:
3527:
3479:
3478:
3465:
3454:
3427:
3414:
3358:
3357:
3351:
3304:
3260:
3239:
3238:
3227:
3219:
3189:
3161:
3145:
3125:
3112:
3081:
3071:
3056:
3039:
3032:
2996:
2975:
2974:
2959:
2951:
2936:
2909:
2881:
2868:
2808:
2788:
2787:
2754:
2741:
2731:
2721:
2668:
2667:
2656:
2648:
2621:
2580:
2579:
2570:
2565:
2554:
2553:
2547:
2542:
2531:
2524:0 + 0
2523:
2508:
2505:
2485:
2477:
2466:
2458:
2455:
2447:
2441:
2434:
2426:
2418:
2412: + 1/
2411:
2404:
2398:
2392:
2385:
2368:
2360:
2349:
2343:
2339:
2328:
2317:
2310:
2301:
2295:
2269:
2239:
2234:
2233:
2195:
2179:
2157:
2144:
2119:
2118:
2107:
2100:
2094:
2088:
2087:In this region
2079:
2073:
2064:
2058:
2047:
2041:
2032:
2029:
2020:
2006:
1994:
1987:
1975:
1964:
1959:α = 1
1958:
1946:
1943:
1884:
1874:
1862:
1861:
1806:
1785:
1784:
1736:
1726:
1714:
1713:
1662:
1652:
1640:
1639:
1588:
1578:
1566:
1565:
1552:
1541:
1517:
1507:
1496:
1493:
1477:
1469:
1461:
1452:
1419:
1408:
1398:
1386:
1352:
1324:
1314:
1291:
1278:
1262:
1258:
1248:
1214:
1204:
1163:
1162:
1153:
1142:
1134:
1067:
1057:
1014:
1004:
978:
977:
963:
951:
943:
940:
935:
916:
906:
903:
850:
840:
828:
827:
821:
815:
804:
794:
756:
755:
692:
672:
671:
662:
655:
599:
589:
577:
576:
570:
564:
553:
543:
507:
497:
487:
451:
430:
429:
420:
393:
383:
340:
335:
334:
323:
315:
312:
283:
273:
263:
213:
212:
203:
200:Borel transform
198:and define the
172:
162:
121:
120:
103:
102:Throughout let
96:
76:Émile Borel
64:Borel summation
60:
49:
30:
23:
22:
15:
12:
11:
5:
7236:
7234:
7226:
7225:
7220:
7215:
7205:
7204:
7201:
7200:
7182:
7169:
7149:
7133:
7120:
7103:
7090:
7072:
7059:
7038:
7010:
7007:
7005:
7004:
6979:
6962:
6960:
6957:
6956:
6955:
6950:
6945:
6940:
6935:
6930:
6925:
6920:
6915:
6913:Abel's theorem
6910:
6908:Abel summation
6903:
6900:
6860:
6806:
6798:
6795:
6770:
6767:
6766:
6765:
6759:
6738:
6733:
6729:
6725:
6722:
6719:
6714:
6709:
6705:
6699:
6695:
6689:
6684:
6681:
6678:
6674:
6661:
6660:
6659:
6658:
6647:
6644:
6641:
6638:
6635:
6631:
6628:
6623:
6619:
6615:
6612:
6608:
6604:
6601:
6598:
6593:
6588:
6584:
6578:
6574:
6560:
6559:
6547:
6543:
6540:
6536:
6532:
6527:
6523:
6519:
6516:
6513:
6508:
6503:
6499:
6493:
6489:
6483:
6464:
6426:
6423:
6410:
6407:
6404:
6401:
6398:
6395:
6392:
6387:
6383:
6329: = 1
6293:
6290:
6286:
6282:
6278:
6266:
6265:
6254:
6247:
6243:
6238:
6232:
6227:
6224:
6221:
6217:
6213:
6210:
6207:
6204:
6201:
6186:
6183:
6152:
6148:
6123:
6119: = {
6115:
6082:
6081:
6066:
6060:
6055:
6051:
6047:
6044:
6040:
6033:
6028:
6025:
6022:
6018:
6014:
6011:
6009:
6007:
6002:
5998:
5994:
5989:
5984:
5980:
5976:
5973:
5970:
5965:
5960:
5956:
5952:
5947:
5942:
5939:
5936:
5932:
5928:
5925:
5923:
5921:
5918:
5915:
5912:
5909:
5908:
5872:
5866:
5863:
5848:
5844:
5840:
5837:
5834:
5815:
5814:
5800:
5796:
5792:
5788:
5784:
5781:
5753:
5748:
5744:
5740:
5737:
5734:
5731:
5728:
5698:
5666:
5662:
5628:
5624:
5603:
5577:
5573:
5569:
5566:
5546:
5524:
5520:
5492:
5488:
5485:
5470:
5469:
5458:
5454:
5448:
5444:
5436:
5432:
5428:
5425:
5421:
5416:
5412:
5409:
5406:
5401:
5397:
5373:
5367:
5366:
5355:
5352:
5349:
5346:
5341:
5337:
5333:
5330:
5327:
5323:
5318:
5314:
5311:
5308:
5305:
5300:
5296:
5266:
5257:
5225:
5204:
5185:
5182:
5151:
5147: = θ
5140:
5139:
5128:
5125:
5121:
5117:
5113:
5110:
5107:
5104:
5101:
5096:
5092:
5086:
5082:
5076:
5058:
5037:
5028:
5014:
4995:
4992:
4990:
4987:
4976:
4975:
4964:
4961:
4958:
4953:
4950:
4946:
4942:
4939:
4936:
4931:
4928:
4925:
4922:
4919:
4916:
4912:
4906:
4903:
4900:
4896:
4845:
4844:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4806:
4803:
4797:
4794:
4791:
4787:
4782:
4778:
4774:
4771:
4768:
4763:
4758:
4754:
4750:
4747:
4744:
4740:
4735:
4732:
4728:
4724:
4721:
4718:
4713:
4709:
4703:
4698:
4694:
4679: = 2
4672:
4671:
4656:
4653:
4648:
4644:
4640:
4637:
4634:
4629:
4625:
4621:
4618:
4616:
4614:
4608:
4605:
4602:
4599:
4596:
4593:
4590:
4583:
4579:
4575:
4572:
4569:
4561:
4558:
4555:
4552:
4546:
4538:
4534:
4528:
4521:
4516:
4513:
4510:
4506:
4500:
4496:
4492:
4489:
4487:
4485:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4454:
4450:
4446:
4443:
4440:
4432:
4429:
4426:
4423:
4420:
4417:
4414:
4410:
4404:
4399:
4396:
4393:
4389:
4385:
4382:
4380:
4378:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4347:
4343:
4339:
4336:
4333:
4326:
4319:
4316:
4309:
4303:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4274:
4264:
4259:
4256:
4253:
4249:
4244:
4238:
4233:
4230:
4227:
4223:
4219:
4216:
4214:
4212:
4209:
4206:
4203:
4198:
4193:
4192:
4178:
4177:
4166:
4161:
4157:
4152:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4120:
4116:
4112:
4109:
4106:
4103:
4100:
4095:
4091:
4087:
4084:
4081:
4073:
4068:
4065:
4062:
4058:
4053:
4047:
4042:
4039:
4036:
4032:
4028:
4025:
4022:
4019:
4016:
3997:
3994:
3990: ≥ 0
3977:
3976:
3965:
3962:
3959:
3953:
3950:
3947:
3944:
3938:
3935:
3931:
3923:
3920:
3917:
3913:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3886:
3881:
3876:
3873:
3869:
3863:
3860:
3857:
3853:
3840:Again, since
3818: ≥ 0
3800:
3799:
3787:
3781:
3778:
3773:
3770:
3766:
3762:
3759:
3754:
3750:
3746:
3742:
3738:
3733:
3730:
3725:
3722:
3719:
3712:
3709:
3706:
3703:
3697:
3694:
3690:
3682:
3677:
3673:
3669:
3666:
3663:
3659:
3656:
3653:
3650:
3647:
3642:
3635:
3632:
3628:
3622:
3617:
3613:
3598: ≥ 0
3582:
3581:
3567:
3564:
3561:
3557:
3552:
3547:
3542:
3538:
3535:
3531:
3524:
3519:
3516:
3513:
3509:
3505:
3502:
3499:
3496:
3493:
3488:
3451:
3450:
3439:
3434:
3430:
3426:
3421:
3417:
3413:
3410:
3407:
3404:
3401:
3396:
3391:
3388:
3385:
3381:
3377:
3374:
3371:
3368:
3365:
3350:
3347:
3346:
3345:
3334:
3331:
3326:
3323:
3320:
3317:
3314:
3311:
3307:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3280:
3275:
3270:
3267:
3263:
3257:
3254:
3251:
3247:
3216:
3215:
3204:
3198:
3195:
3192:
3188:
3183:
3178:
3171:
3168:
3164:
3160:
3157:
3152:
3148:
3142:
3134:
3131:
3128:
3122:
3119:
3115:
3107:
3104:
3101:
3097:
3093:
3087:
3084:
3078:
3074:
3065:
3062:
3059:
3052:
3049:
3046:
3042:
3038:
3035:
3027:
3022:
3019:
3016:
3012:
3006:
3003:
2999:
2993:
2990:
2987:
2983:
2955:
2933:
2932:
2918:
2915:
2912:
2908:
2903:
2900:
2897:
2891:
2888:
2884:
2878:
2875:
2871:
2865:
2860:
2856:
2852:
2849:
2846:
2842:
2839:
2836:
2833:
2830:
2825:
2818:
2815:
2811:
2805:
2800:
2796:
2781:
2780:
2769:
2764:
2761:
2757:
2753:
2748:
2744:
2737:
2734:
2728:
2724:
2716:
2711:
2708:
2705:
2701:
2697:
2694:
2691:
2688:
2685:
2682:
2677:
2645:
2644:
2633:
2628:
2624:
2618:
2613:
2610:
2607:
2603:
2599:
2596:
2593:
2590:
2587:
2569:
2566:
2564:
2561:
2504:
2501:
2481:
2462:
2451:
2430:
2416:
2409:
2381:
2338:
2335:
2292:
2291:
2278:
2273:
2268:
2264:
2260:
2257:
2252:
2249:
2246:
2242:
2229:is bounded by
2227:
2226:
2214:
2208:
2205:
2202:
2198:
2192:
2189:
2186:
2182:
2178:
2175:
2172:
2169:
2164:
2160:
2156:
2151:
2147:
2143:
2140:
2137:
2134:
2131:
2127:
2115:
2114:
2105:
2098:
2085:
2028:
2025:
2019:
2016:
2015:
2014:
1962:
1942:
1939:
1938:
1937:
1936:
1935:
1923:
1919:
1916:
1912:
1908:
1905:
1902:
1899:
1896:
1891:
1887:
1881:
1877:
1871:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1823:
1816:
1813:
1809:
1803:
1800:
1797:
1793:
1772:
1768:
1764:
1760:
1757:
1754:
1751:
1748:
1743:
1739:
1733:
1729:
1723:
1710:
1698:
1694:
1690:
1686:
1683:
1680:
1677:
1674:
1669:
1665:
1659:
1655:
1649:
1627:
1623:
1620:
1616:
1612:
1609:
1606:
1603:
1600:
1595:
1591:
1585:
1581:
1575:
1538:
1492:
1489:
1460:Regularity of
1449:
1448:
1437:
1434:
1431:
1425:
1422:
1415:
1411:
1407:
1404:
1401:
1393:
1389:
1383:
1378:
1375:
1372:
1368:
1362:
1359:
1355:
1349:
1344:
1340:
1336:
1330:
1327:
1321:
1317:
1310:
1306:
1303:
1298:
1294:
1288:
1285:
1281:
1275:
1270:
1266:
1261:
1255:
1251:
1245:
1240:
1237:
1234:
1230:
1226:
1221:
1217:
1211:
1207:
1201:
1196:
1193:
1190:
1186:
1182:
1179:
1176:
1173:
1170:
1133:Regularity of
1131:
1130:
1119:
1116:
1112:
1109:
1104:
1100:
1096:
1091:
1088:
1085:
1082:
1079:
1074:
1070:
1064:
1060:
1054:
1048:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1021:
1017:
1011:
1007:
1001:
996:
993:
990:
986:
939:
936:
934:
931:
902:
899:
886:
882:
878:
874:
871:
868:
865:
862:
857:
853:
847:
843:
837:
779:
776:
773:
770:
765:
748:
747:
736:
733:
730:
726:
723:
720:
717:
714:
709:
702:
699:
695:
689:
684:
680:
654:
651:
638:
634:
631:
627:
623:
620:
617:
614:
611:
606:
602:
596:
592:
586:
540:
539:
528:
525:
522:
519:
514:
510:
503:
500:
494:
490:
482:
477:
474:
471:
467:
461:
458:
454:
448:
445:
442:
438:
417:
416:
405:
400:
396:
390:
386:
380:
375:
372:
369:
365:
361:
358:
355:
352:
347:
343:
319:
311:
308:
307:
306:
295:
290:
286:
279:
276:
270:
266:
258:
253:
250:
247:
243:
239:
236:
233:
230:
227:
222:
196:
195:
184:
179:
175:
169:
165:
159:
154:
151:
148:
144:
140:
137:
134:
131:
128:
95:
92:
39:Mittag-Leffler
31:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7235:
7224:
7221:
7219:
7216:
7214:
7211:
7210:
7208:
7198:
7194:
7193:
7188:
7183:
7180:
7176:
7172:
7166:
7161:
7160:
7154:
7150:
7147:
7143:
7139:
7134:
7131:
7127:
7123:
7117:
7113:
7109:
7104:
7101:
7097:
7093:
7087:
7083:
7082:
7077:
7073:
7070:
7066:
7062:
7056:
7052:
7048:
7044:
7039:
7035:
7030:
7026:
7022:
7018:
7013:
7012:
7008:
6993:
6989:
6983:
6980:
6976:
6970:
6968:
6964:
6958:
6954:
6951:
6949:
6946:
6944:
6941:
6939:
6936:
6934:
6931:
6929:
6926:
6924:
6921:
6919:
6916:
6914:
6911:
6909:
6906:
6905:
6901:
6899:
6897:
6893:
6888:
6886:
6881:
6875:
6871:
6863:
6859:
6853:
6849:
6843:
6835:
6831:
6826:
6820:
6816:
6809:
6805:
6796:
6794:
6792:
6791:Weinberg 2005
6788:
6784:
6780:
6776:
6768:
6758:
6754:
6731:
6727:
6720:
6717:
6712:
6707:
6703:
6697:
6693:
6682:
6679:
6676:
6672:
6663:
6662:
6645:
6642:
6639:
6636:
6629:
6621:
6617:
6613:
6610:
6606:
6599:
6596:
6591:
6586:
6582:
6576:
6572:
6564:
6563:
6562:
6561:
6525:
6521:
6514:
6511:
6506:
6501:
6497:
6491:
6487:
6481:
6470:
6463:
6456:
6449:
6444:
6440:
6437:
6436:
6435:
6432:
6424:
6422:
6405:
6402:
6399:
6393:
6390:
6385:
6370:
6367: =
6366:
6360:
6353:
6347:
6341:
6335:
6328:
6322:
6318:
6312:
6307:
6291:
6288:
6280:
6252:
6245:
6241:
6236:
6225:
6222:
6219:
6215:
6211:
6205:
6199:
6192:
6191:
6190:
6184:
6182:
6179:
6173:
6170:
6150:
6135:
6131:
6128: :
6126:
6122:
6118:
6108:
6104:
6098:
6092:
6088:
6064:
6058:
6053:
6049:
6045:
6042:
6038:
6031:
6026:
6023:
6020:
6016:
6012:
6010:
6000:
5996:
5987:
5982:
5978:
5974:
5971:
5968:
5963:
5958:
5954:
5940:
5937:
5934:
5930:
5926:
5924:
5916:
5910:
5899:
5898:
5897:
5894:
5890:
5884:
5878:
5864:
5862:
5846:
5835:
5832:
5822:
5798:
5782:
5779:
5769:
5746:
5735:
5732:
5729:
5726:
5716:
5708:
5704:
5699:
5696:
5692:
5689:
5688:
5687:
5685:
5680:
5664:
5649:
5645:
5626:
5601:
5593:
5575:
5567:
5564:
5544:
5522:
5508:
5486:
5483:
5475:
5456:
5452:
5446:
5434:
5430:
5426:
5423:
5419:
5414:
5410:
5407:
5404:
5399:
5387:
5386:
5385:
5382:
5376:
5353:
5344:
5339:
5335:
5331:
5328:
5325:
5321:
5312:
5309:
5303:
5298:
5286:
5285:
5284:
5281:
5275:
5269:
5260:
5253:
5247:
5243:, but not to
5241:
5235:
5228:
5224:
5220:
5214:
5207:
5203:
5196:
5192:
5188:Suppose that
5183:
5181:
5177:
5173:
5167:
5162:
5157:
5150:
5146:
5126:
5108:
5102:
5099:
5094:
5090:
5084:
5080:
5074:
5065:
5064:
5063:
5057:
5049:
5045:
5036:
5027:
5020:
5013:
5006:
5002:
4993:
4988:
4986:
4982:
4962:
4959:
4951:
4948:
4944:
4937:
4934:
4929:
4923:
4920:
4917:
4910:
4898:
4886:
4885:
4884:
4881:
4877:
4870:
4865:
4861:
4855:
4851:
4831:
4825:
4819:
4813:
4810:
4804:
4801:
4795:
4792:
4789:
4780:
4776:
4769:
4766:
4756:
4752:
4748:
4745:
4742:
4733:
4730:
4726:
4719:
4716:
4711:
4707:
4696:
4692:
4684:
4683:
4682:
4678:
4654:
4646:
4642:
4635:
4632:
4627:
4623:
4619:
4617:
4606:
4600:
4597:
4594:
4591:
4581:
4573:
4570:
4559:
4556:
4553:
4550:
4536:
4532:
4514:
4511:
4508:
4504:
4498:
4494:
4490:
4488:
4477:
4471:
4468:
4465:
4462:
4452:
4444:
4441:
4430:
4424:
4421:
4418:
4415:
4408:
4397:
4394:
4391:
4387:
4383:
4381:
4370:
4364:
4361:
4358:
4355:
4345:
4337:
4334:
4324:
4317:
4314:
4307:
4295:
4289:
4286:
4283:
4280:
4257:
4254:
4251:
4247:
4242:
4231:
4228:
4225:
4221:
4217:
4215:
4207:
4201:
4183:
4182:
4181:
4164:
4159:
4155:
4150:
4143:
4137:
4134:
4131:
4128:
4118:
4110:
4107:
4104:
4101:
4093:
4085:
4082:
4066:
4063:
4060:
4056:
4051:
4040:
4037:
4034:
4030:
4026:
4020:
4014:
4007:
4006:
4005:
4003:
3995:
3993:
3989:
3983:
3963:
3960:
3957:
3951:
3948:
3945:
3942:
3936:
3933:
3929:
3915:
3907:
3901:
3898:
3889:
3874:
3871:
3867:
3855:
3843:
3842:
3841:
3838:
3835:
3830:
3825:
3817:
3811:
3809:
3785:
3779:
3776:
3771:
3768:
3764:
3757:
3752:
3748:
3744:
3740:
3736:
3731:
3728:
3723:
3720:
3717:
3710:
3707:
3704:
3701:
3695:
3692:
3688:
3675:
3671:
3667:
3664:
3661:
3654:
3651:
3645:
3633:
3630:
3626:
3615:
3611:
3603:
3602:
3601:
3597:
3589:
3565:
3562:
3559:
3555:
3550:
3545:
3540:
3536:
3533:
3529:
3517:
3514:
3511:
3507:
3503:
3497:
3491:
3477:
3476:
3475:
3472:
3468:
3461:
3457:
3437:
3432:
3428:
3424:
3419:
3411:
3408:
3402:
3399:
3389:
3386:
3383:
3379:
3375:
3369:
3363:
3356:
3355:
3354:
3348:
3332:
3329:
3321:
3318:
3315:
3309:
3305:
3301:
3295:
3292:
3283:
3268:
3265:
3261:
3249:
3237:
3236:
3235:
3231:
3223:
3202:
3196:
3193:
3190:
3186:
3181:
3169:
3166:
3162:
3158:
3155:
3150:
3146:
3132:
3129:
3126:
3120:
3117:
3113:
3099:
3091:
3085:
3082:
3076:
3072:
3063:
3060:
3057:
3050:
3047:
3044:
3040:
3036:
3033:
3020:
3017:
3014:
3010:
3004:
3001:
2997:
2985:
2973:
2972:
2971:
2967:
2963:
2958:
2954:
2948:
2946:
2940:
2916:
2913:
2910:
2906:
2901:
2898:
2895:
2889:
2886:
2882:
2876:
2873:
2869:
2858:
2854:
2850:
2847:
2844:
2837:
2834:
2828:
2816:
2813:
2809:
2798:
2794:
2786:
2785:
2784:
2767:
2762:
2759:
2755:
2751:
2746:
2742:
2735:
2732:
2726:
2722:
2709:
2706:
2703:
2699:
2695:
2689:
2686:
2680:
2666:
2665:
2664:
2660:
2652:
2631:
2626:
2622:
2611:
2608:
2605:
2601:
2597:
2591:
2585:
2578:
2577:
2576:
2575:
2572:Consider the
2567:
2562:
2560:
2545:
2539:
2535:
2527:
2519:
2515:
2511:
2507:The function
2502:
2500:
2497:
2493:
2489:
2484:
2480:
2474:
2470:
2465:
2461:
2454:
2450:
2444:
2438:
2433:
2429:
2422:
2415:
2408:
2401:
2395:
2388:
2384:
2380:
2376:
2372:
2364:
2357:
2353:
2346:
2334:
2331:
2324:
2320:
2313:
2307:
2304:
2298:
2276:
2266:
2258:
2255:
2250:
2247:
2244:
2240:
2232:
2231:
2230:
2206:
2203:
2200:
2196:
2190:
2187:
2184:
2180:
2176:
2173:
2170:
2167:
2162:
2158:
2154:
2149:
2145:
2141:
2135:
2129:
2117:
2116:
2110:
2104:
2101: +
2097:
2091:
2086:
2082:
2076:
2070:
2062:
2055:
2051:
2044:
2040:
2039:
2038:
2035:
2024:
2017:
2010:
2002:
1998:
1990:
1983:
1979:
1974:
1968:
1963:
1956:
1950:
1945:
1944:
1940:
1903:
1897:
1894:
1889:
1885:
1879:
1875:
1869:
1847:
1844:
1841:
1835:
1832:
1826:
1814:
1811:
1807:
1795:
1755:
1749:
1746:
1741:
1737:
1731:
1727:
1721:
1711:
1681:
1675:
1672:
1667:
1663:
1657:
1653:
1647:
1607:
1601:
1598:
1593:
1589:
1583:
1579:
1573:
1563:
1562:
1559:
1555:
1548:
1544:
1539:
1536:
1532:
1529:
1528:
1527:
1525:
1520:
1514:
1510:
1503:
1499:
1490:
1488:
1484:
1480:
1473:
1465:
1458:
1455:
1435:
1432:
1429:
1423:
1420:
1413:
1405:
1402:
1391:
1387:
1376:
1373:
1370:
1366:
1360:
1357:
1353:
1342:
1338:
1334:
1328:
1325:
1319:
1315:
1308:
1304:
1301:
1296:
1292:
1286:
1283:
1279:
1268:
1264:
1259:
1253:
1249:
1238:
1235:
1232:
1228:
1224:
1219:
1215:
1209:
1205:
1194:
1191:
1188:
1184:
1180:
1174:
1168:
1161:
1160:
1159:
1156:
1149:
1145:
1138:
1117:
1102:
1086:
1080:
1077:
1072:
1068:
1062:
1058:
1052:
1039:
1033:
1027:
1024:
1019:
1015:
1009:
1005:
994:
991:
988:
984:
976:
975:
974:
970:
966:
961:
955:
947:
937:
932:
930:
928:
924:
919:
914:
909:
898:
869:
863:
860:
855:
851:
845:
841:
835:
824:
820:converges at
818:
811:
807:
801:
797:
791:
774:
768:
753:
750:representing
734:
731:
728:
721:
718:
712:
700:
697:
693:
682:
678:
670:
669:
668:
665:
660:
650:
618:
612:
609:
604:
600:
594:
590:
584:
573:
569:converges at
567:
560:
556:
550:
546:
526:
520:
512:
508:
501:
498:
492:
488:
475:
472:
469:
465:
459:
456:
452:
440:
428:
427:
426:
423:
403:
398:
394:
388:
384:
378:
373:
370:
367:
363:
359:
353:
345:
341:
333:
332:
331:
327:
322:
318:
293:
288:
284:
277:
274:
268:
264:
251:
248:
245:
241:
237:
231:
225:
211:
210:
209:
206:
201:
182:
177:
173:
167:
163:
152:
149:
146:
142:
138:
132:
126:
119:
118:
117:
116:
110:
106:
100:
93:
91:
89:
85:
81:
77:
73:
69:
65:
59:
57:, p. 38)
56:
52:
46:
44:
40:
36:
27:
19:
7190:
7158:
7137:
7111:
7108:Simon, Barry
7080:
7042:
7024:
7023:, Series 3,
7020:
6995:. Retrieved
6991:
6982:
6974:
6889:
6879:
6873:
6869:
6861:
6857:
6851:
6841:
6833:
6824:
6818:
6814:
6807:
6803:
6800:
6772:
6769:Applications
6756:
6752:
6468:
6461:
6459:summable at
6454:
6447:
6445:, 9.13). If
6438:
6428:
6368:
6364:
6358:
6351:
6345:
6339:
6333:
6326:
6320:
6316:
6310:
6267:
6188:
6177:
6168:
6133:
6129:
6124:
6120:
6113:
6106:
6102:
6096:
6090:
6086:
6083:
5892:
5888:
5882:
5876:
5868:
5820:
5816:
5767:
5714:
5706:
5702:
5690:
5681:
5647:
5643:
5591:
5506:
5476:, 8.3). Let
5471:
5380:
5371:
5368:
5279:
5273:
5264:
5255:
5251:
5249:itself. For
5245:
5239:
5233:
5226:
5222:
5218:
5212:
5205:
5201:
5194:
5190:
5187:
5175:
5171:
5165:
5158:
5148:
5144:
5141:
5055:
5047:
5043:
5034:
5025:
5018:
5011:
5004:
5000:
4997:
4980:
4977:
4882:
4875:
4868:
4853:
4849:
4846:
4676:
4673:
4179:
3999:
3987:
3981:
3978:
3839:
3833:
3823:
3815:
3812:
3801:
3595:
3587:
3583:
3470:
3466:
3459:
3455:
3452:
3352:
3229:
3221:
3217:
2965:
2961:
2956:
2952:
2949:
2943:, giving an
2938:
2934:
2782:
2658:
2650:
2646:
2571:
2543:
2537:
2533:
2525:
2517:
2513:
2509:
2506:
2495:
2491:
2487:
2482:
2478:
2472:
2468:
2463:
2459:
2452:
2448:
2442:
2436:
2431:
2427:
2423:
2413:
2406:
2399:
2393:
2386:
2382:
2378:
2374:
2370:
2362:
2355:
2351:
2344:
2340:
2329:
2322:
2318:
2311:
2308:
2302:
2296:
2293:
2228:
2108:
2102:
2095:
2089:
2080:
2074:
2068:
2060:
2053:
2049:
2042:
2033:
2030:
2021:
2008:
2000:
1996:
1988:
1981:
1977:
1966:
1948:
1557:
1553:
1546:
1542:
1530:
1518:
1512:
1508:
1501:
1497:
1494:
1482:
1478:
1471:
1463:
1459:
1453:
1450:
1154:
1147:
1143:
1136:
1132:
968:
964:
953:
945:
942:The methods
941:
917:
907:
904:
826:, and write
822:
816:
809:
805:
799:
795:
792:
749:
667:is given by
663:
658:
656:
575:, and write
571:
565:
558:
554:
548:
544:
541:
421:
418:
325:
320:
316:
313:
204:
199:
197:
108:
104:
101:
97:
63:
61:
53:, quoted by
48:
33:
26:
6866:bounded by
6787:renormalons
6349:outside of
5880:denote the
5700:The series
5384:is the set
5161:star domain
5032:connecting
1495:Any series
43:Weierstrass
7207:Categories
7009:References
6997:19 October
6830:factorials
6783:instantons
6443:Hardy 1992
6324:such that
5817:Note that
5695:Hardy 1992
5062:such that
4862:. Via the
4002:Hardy 1992
2421:diverges.
1535:Hardy 1992
938:Regularity
925:along the
94:Definition
7197:EMS Press
7078:(1992) ,
7027:: 9–131,
6992:MathWorld
6877:for some
6822:for some
6793:, 20.7).
6688:∞
6673:∑
6640:≥
6634:∀
6611:−
6482:∑
6382:Π
6331:for some
6231:∞
6216:∑
6185:Example 2
6147:Π
6050:ω
6046:−
6017:∑
5979:ω
5972:⋯
5955:ω
5946:∞
5931:∑
5865:Example 1
5843:Π
5839:∂
5836:∈
5795:Π
5791:∖
5783:∈
5764:, and is
5743:Π
5736:
5730:∈
5661:Π
5623:Π
5572:Π
5568:∈
5519:Π
5487:⊂
5443:Π
5427:∈
5420:⋂
5411:
5396:Π
5348:∅
5332:∩
5322::
5313:∈
5295:Π
5075:∑
4938:
4921:−
4905:∞
4902:→
4829:∞
4811:−
4802:π
4770:
4762:∞
4753:∫
4720:
4702:∞
4693:∫
4636:
4595:ℓ
4582:ℓ
4571:−
4554:ℓ
4520:∞
4509:ℓ
4505:∑
4466:ℓ
4453:ℓ
4442:−
4419:ℓ
4403:∞
4392:ℓ
4388:∑
4359:ℓ
4346:ℓ
4335:−
4284:ℓ
4263:∞
4248:∑
4237:∞
4226:ℓ
4222:∑
4132:ℓ
4105:ℓ
4094:ℓ
4083:−
4072:∞
4061:ℓ
4057:∑
4046:∞
4031:∑
3934:−
3922:∞
3919:→
3872:−
3862:∞
3859:→
3761:Γ
3758:⋅
3737:⋅
3693:−
3681:∞
3672:∫
3631:−
3621:∞
3612:∫
3534:−
3523:∞
3508:∑
3504:≡
3425:⋅
3409:−
3395:∞
3380:∑
3319:−
3266:−
3256:∞
3253:→
3194:−
3156:−
3130:−
3118:−
3106:∞
3103:→
3061:−
3037:−
3026:∞
3011:∑
3002:−
2992:∞
2989:→
2914:−
2874:−
2864:∞
2855:∫
2814:−
2804:∞
2795:∫
2715:∞
2700:∑
2696:≡
2617:∞
2602:∑
2471:log
2204:−
2188:−
2177:−
2174:⋯
2171:−
2155:−
2142:−
2078:and
1870:∑
1812:−
1802:∞
1799:→
1722:∑
1648:∑
1574:∑
1382:∞
1367:∑
1358:−
1348:∞
1339:∫
1284:−
1274:∞
1265:∫
1244:∞
1229:∑
1200:∞
1185:∑
1053:∑
1047:⇒
1043:∞
1000:∞
985:∑
958:are both
836:∑
698:−
688:∞
679:∫
659:Borel sum
585:∑
481:∞
466:∑
457:−
447:∞
444:→
364:∑
257:∞
242:∑
238:≡
158:∞
143:∑
113:denote a
18:Borel sum
7155:(2005),
7110:(1978),
6902:See also
5684:Phragmén
5142:for all
3979:for all
2563:Examples
2541:for any
2294:for all
2063:)| <
1561:, then:
1537:, 8.5)).
1524:examples
803:to some
51:Mark Kac
7179:2148467
7146:0113988
7130:0493421
7100:0030620
7069:0887102
6848:integer
6439:Theorem
5697:, 8.8).
5691:Theorem
5510:, then
4858:is the
3806:is the
3802:(where
2503:Example
2467:=
2397:, then
1638:, then
1531:Theorem
1158:, then
960:regular
78: (
7177:
7167:
7144:
7128:
7118:
7098:
7088:
7067:
7057:
5262:, let
4847:where
3822:
3594:
1783:, and
425:to be
6959:Notes
6838:with
6664:then
6558:, and
6371:(0,1)
6354:(0,1)
3453:then
2532:|arg(
2067:/2 +
2059:|arg(
1957:with
1860:then
66:is a
35:Borel
7165:ISBN
7116:ISBN
7086:ISBN
7055:ISBN
6999:2016
6785:and
6289:<
6172:-gon
5869:Let
4826:<
3584:for
2655:for
2490:log
2367:and
2327:for
1540:Let
1468:and
1040:<
950:and
314:Let
80:1899
70:for
7047:doi
7029:doi
6451:is
5733:int
5711:is
5163:in
4935:sin
4895:lim
4880:).
4767:sin
4717:sin
4674:At
4633:sin
3912:lim
3852:lim
3831:as
3810:).
3246:lim
3228:Re(
3220:Re(
3096:lim
2982:lim
2937:Re(
2476:or
2361:Re(
2306:).
1792:lim
1712:If
1564:If
915:of
754:of
661:of
437:lim
202:of
7209::
7195:,
7189:,
7175:MR
7173:,
7142:MR
7126:MR
7124:,
7096:MR
7094:,
7065:MR
7063:,
7053:,
7025:16
7019:,
6990:.
6966:^
6898:.
6887:.
6872:)!
6870:kn
6844:)!
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6455:wB
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5592:OP
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5280:OP
5180:.
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3992:.
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954:wB
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5650:)
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5453:)
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5415:(
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3804:Γ
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3780:z
3777:1
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3769:0
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3676:0
3668:=
3665:t
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3649:(
3646:A
3641:B
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3406:(
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3390:0
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3376:=
3373:)
3370:z
3367:(
3364:A
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3322:1
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3250:t
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3141:(
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3092:=
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2966:z
2962:z
2960:(
2957:N
2953:A
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2007:(
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