1453:
1218:
2786:
4047:
ratio of the boundary layer thickness to a typical length scale of the problem. Indeed, applications of asymptotic analysis in mathematical modelling often center around a nondimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at
2604:
In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. This optimal partial sum will usually have more terms as the argument
3116:
3467:
1448:{\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{i\left(z-{\frac {2\pi \alpha -\pi }{4}}\right)}\\H_{\alpha }^{(2)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{-i\left(z-{\frac {2\pi \alpha -\pi }{4}}\right)}\end{aligned}}}
3619:
1209:
2619:
3862:) as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon.
2922:
2931:
1085:
3313:
3138:
Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. For example, we might start with the ordinary series
979:
4225:
2462:
4452:
3869:
is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation
4638:
Some days later, Miss N.A. wants to know the value of f(1000), but her machine would take a month of computation to give the answer. She returns to her
Asymptotic Colleague, and gets a fully satisfactory
4331:
368:
4595:
3206:
443:
2347:
3516:
1110:
1223:
3766:
2152:
1923:
1654:
1758:
259:
2795:
2542:
2069:
3669:
1820:
858:
790:
615:
3904:
2600:
1583:
4496:
2234:
1978:
735:
3952:. Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics, however. Non-asymptotic bounds are provided by methods of
1537:
819:
901:
1009:
3308:
3270:
3234:
692:
666:
3706:
1850:
3507:
2172:
2009:
1678:
4114:
3792:
is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables
2781:{\displaystyle {\frac {e^{x}}{x^{x}{\sqrt {2\pi x}}}}\Gamma (x+1)\sim 1+{\frac {1}{12x}}+{\frac {1}{288x^{2}}}-{\frac {139}{51840x^{3}}}-\cdots \ (x\to \infty )}
4621:
4353:
4134:
920:
3111:{\displaystyle {\sqrt {\pi }}xe^{x^{2}}\operatorname {erfc} (x)\sim 1+\sum _{n=1}^{\infty }(-1)^{n}{\frac {(2n-1)!!}{n!(2x^{2})^{n}}}\ (x\to \infty )}
308:
3142:
377:
4083:
Debruijn illustrates the use of asymptotics in the following dialog between Dr. N.A., a
Numerical Analyst, and Dr. A.A., an Asymptotic Analyst:
4142:
3462:{\displaystyle \int _{0}^{\infty }{\frac {e^{-{\frac {w}{t}}}}{1-w}}\,dw=\sum _{n=0}^{\infty }t^{n+1}\int _{0}^{\infty }e^{-u}u^{n}\,du}
3631:
small, and truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of
2352:
211:
4364:
4236:
4704:
4507:
2247:
4968:
4799:
4015:
when identifying the causation of crash through count modeling with large number of crash counts in a given time and space.
4656:
30:
This article is about the behavior of functions as inputs approach infinity or some other limit value. For asymptotes in
549:
532:
goes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, in
3711:
2074:
1857:
1588:
987:
1683:
1492:
of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for
4928:
4908:
4884:
4860:
4833:
17:
3614:{\displaystyle e^{-{\frac {1}{t}}}\operatorname {Ei} \left({\frac {1}{t}}\right)=\sum _{n=0}^{\infty }n!\;t^{n+1}}
4773:
2467:
1204:{\displaystyle \operatorname {Ai} (x)\sim {\frac {e^{-{\frac {2}{3}}x^{\frac {3}{2}}}}{2{\sqrt {\pi }}x^{1/4}}}}
906:
Such properties allow asymptotically equivalent functions to be freely exchanged in many algebraic expressions.
4674:
4020:
2022:
3634:
4900:
4743:
4024:
1767:
825:
3986:, asymptotics are used in analysis of long-run or large-sample behaviour of random variables and estimators.
4988:
4983:
1496:. The idea is that successive terms provide an increasingly accurate description of the order of growth of
4738:
4733:
982:
745:
631:
4659: – computational complexity as measured by the limiting behavior of resource usage for large inputs
4032:
4944:
4778:
4698:
4060:
3873:
2547:
1542:
4460:
2184:
1928:
487:
can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.
4842:
700:
3930:
2917:{\displaystyle xe^{x}E_{1}(x)\sim \sum _{n=0}^{\infty }{\frac {(-1)^{n}n!}{x^{n}}}\ (x\to \infty )}
514:. The way of passing to the limit is often not stated explicitly, if it is clear from the context.
1506:
795:
4036:
3979:
3788:
3783:
3777:
1761:
864:
185:
4632:
A.A.: Haven't I told you so? My estimate of 20% was not far off from the 14% of the real error.
4039:
governing fluid flow. In many cases, the asymptotic expansion is in power of a small parameter,
3994:
3851:
goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.
268:
3834:
A special case of an asymptotic distribution is when the late entries go to zero—that is, the
3275:
3239:
4005:
3949:
3938:
3922:
3213:
671:
645:
42:
3674:
1825:
4072:
4044:
3953:
3478:
3472:
2790:
2157:
1987:
1663:
1485:
1470:
1464:
476:
456:
173:
4603:
A.A.: It is almost the best thing I possibly can get. Why don't you take larger values of
4090:
8:
4056:
4052:
3964:
54:
4957:
4852:
4692:
4665:
4606:
4338:
4119:
4028:
3983:
4924:
4904:
4880:
4856:
4829:
4795:
4709:
4012:
2016:
533:
4938:
4064:
3990:
3968:
3934:
3126:
2012:
1213:
264:
60:
As an illustration, suppose that we are interested in the properties of a function
1080:{\displaystyle p(n)\sim {\frac {1}{4n{\sqrt {3}}}}e^{\pi {\sqrt {\frac {2n}{3}}}}}
4918:
4894:
4870:
4846:
4825:
4819:
4051:
Asymptotic expansions typically arise in the approximation of certain integrals (
4001:
517:
Although the above definition is common in the literature, it is problematic if
4686:
4068:
3945:
3510:
2926:
2614:
272:
4695: – Terms in a mathematical expression with the largest order of magnitude
3623:
Here, the right hand side is clearly not convergent for any non-zero value of
4977:
4680:
1089:
1006:
as a sum of positive integers, where the order of addends is not considered.
4876:
4626:
N.A.: !!! I think it's better to ask my electronic computing machine.
201:
4075:
are another example of asymptotic expansions which often do not converge.
4031:
of real-world phenomena. An illustrative example is the derivation of the
490:
The same notation is also used for other ways of passing to a limit: e.g.
4794:. Dover books on advanced mathematics. New York: Dover publ. p. 19.
1489:
3926:
3855:
974:{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}
4963:
4650:
4220:{\displaystyle f(x)=x^{-1}+\mathrm {O} (x^{-2})\qquad (x\to \infty )}
3941:
3866:
2154:
One should however be careful that this is not a standard use of the
915:
35:
3471:
The integral on the left hand side can be expressed in terms of the
4683: – Dealing with applied mathematical systems in limiting cases
4653: – Limit of the tangent line at a point that tends to infinity
3972:
3768:
results in the asymptotic expansion given earlier in this article.
2174:
symbol, and that it does not correspond to the definition given in
31:
4501:
A.A.: I can gain a little on some of my estimates. Now I find that
4677: – Study of convergence properties of statistical estimators
3210:
The expression on the left is valid on the entire complex plane
2457:{\displaystyle f-g_{1}-\cdots -g_{k-2}-g_{k-1}=g_{k}+o(g_{k}),}
4447:{\displaystyle |f(x)-x^{-1}|<57000x^{-2}\qquad (x>100).}
3475:. The integral on the right hand side, after the substitution
4969:
A paper on time series analysis using asymptotic distribution
4326:{\displaystyle |f(x)-x^{-1}|<8x^{-2}\qquad (x>10^{4}).}
452:
363:{\displaystyle f(x)\sim g(x)\quad ({\text{as }}x\to \infty )}
3929:, asymptotic theory provides limiting approximations of the
1484:
is in practice an expression of that function in terms of a
4590:{\displaystyle |f(x)-x^{-1}|<20x^{-2}\qquad (x>100).}
3201:{\displaystyle {\frac {1}{1-w}}=\sum _{n=0}^{\infty }w^{n}}
438:{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1.}
2342:{\displaystyle f-g_{1}-\cdots -g_{k-2}=g_{k-1}+o(g_{k-1})}
619:
This definition is equivalent to the prior definition if
4063:) or in the approximation of probability distributions (
3858:
function which cleanly approaches a constant value (the
3513:. Evaluating both, one obtains the asymptotic expansion
694:, then, under some mild conditions, the following hold:
4962: —home page of the journal, which is published by
4670:
Pages displaying short descriptions of redirect targets
4078:
189:
4712: – lemma on the asymptotic behavior of integrals
4609:
4510:
4463:
4367:
4341:
4239:
4145:
4122:
4093:
3876:
3714:
3677:
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3519:
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3145:
2934:
2798:
2622:
2550:
2470:
2355:
2250:
2187:
2160:
2077:
2025:
1990:
1931:
1860:
1828:
1770:
1686:
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1591:
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1509:
1221:
1113:
1012:
923:
867:
828:
798:
748:
703:
674:
648:
552:
380:
311:
214:
4714:
Pages displaying wikidata descriptions as a fallback
4701: – Solution of a simplified form of an equation
4661:
Pages displaying wikidata descriptions as a fallback
4019:
Asymptotic analysis is a key tool for exploring the
2608:
172:
An example of an important asymptotic result is the
1002:), gives the number of ways of writing the integer
4615:
4589:
4490:
4446:
4347:
4325:
4219:
4128:
4108:
3898:
3760:
3700:
3663:
3613:
3501:
3461:
3302:
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3228:
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3110:
2916:
2780:
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2341:
2228:
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2146:
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2003:
1972:
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1079:
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909:
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784:
729:
686:
660:
609:
437:
362:
253:
4821:From Divergent Power Series To Analytic Functions
4689: – Describes limiting behavior of a function
4457:N.A.: This is no news to me. I know already that
4358:A.A.: Why did you not say so? My evaluations give
3761:{\displaystyle \operatorname {Ei} (x)=-E_{1}(-x)}
2147:{\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k}).}
1918:{\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}}
1649:{\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}}
4975:
1753:{\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k})}
799:
382:
3236:, while the right hand side converges only for
254:{\displaystyle \pi (x)\sim {\frac {x}{\ln x}}.}
188:(which is not directly related to the constant
1096:), is a solution of the differential equation
27:Description of limiting behavior of a function
4936:
4868:
4756:
4629:Machine: f(100) = 0.01137 42259 34008 67153
3997:, considering the performance of algorithms.
3959:Examples of applications are the following.
4043:: in the boundary layer case, this is the
4027:differential equations which arise in the
3909:becomes arbitrarily small in magnitude as
3771:
3594:
2537:{\displaystyle g_{k}+o(g_{k})=o(g_{k-1}),}
4841:
4767:
4765:
3452:
3367:
2064:{\displaystyle f\sim g_{1}+\cdots +g_{k}}
371:
271:and is often expressed there in terms of
138:". This is often written symbolically as
4872:A Distributional Approach to Asymptotics
3664:{\displaystyle \operatorname {Ei} (1/t)}
2181:In the present situation, this relation
263:Asymptotic analysis is commonly used in
4940:Asymptotics and Mellin-Barnes Integrals
4705:Method of matched asymptotic expansions
4136:, with a relative error of at most 1%.
3967:, asymptotic analysis is used to build
3921:Asymptotic analysis is used in several
1815:{\displaystyle f-(g_{1}+\cdots +g_{k})}
1458:
1107:; it has many applications in physics.
853:{\displaystyle f\times a\sim g\times b}
14:
4976:
4916:
4892:
4817:
4789:
4762:
4230:N.A.: I am sorry, I don't understand.
2236:actually follows from combining steps
4087:N.A.: I want to evaluate my function
4079:Asymptotic versus Numerical Analysis
3819:. An asymptotic distribution allows
4937:Paris, R. B.; Kaminsky, D. (2001),
4869:Estrada, R.; Kanwal, R. P. (2002),
4790:Bruijn, Nicolaas Govert de (1981).
4657:Asymptotic computational complexity
4600:N.A.: I asked for 1%, not for 20%.
1660:. In view of the definition of the
785:{\displaystyle \log(f)\sim \log(g)}
24:
4211:
4178:
3854:This is based on the notion of an
3583:
3424:
3393:
3327:
3310:and integrating both sides yields
3183:
3102:
3002:
2908:
2850:
2772:
2662:
610:{\displaystyle f(x)=g(x)(1+o(1)).}
392:
354:
104:becomes insignificant compared to
25:
5000:
4951:
3899:{\displaystyle y={\frac {1}{x}},}
3825:to range without bound, that is,
3133:
2609:Examples of asymptotic expansions
2595:{\displaystyle g_{k}=o(g_{k-1}).}
2015:. In that case, some authors may
1578:{\displaystyle f-g_{1}\sim g_{2}}
4668: – Concept in number theory
4491:{\displaystyle 0<f(100)<1}
2229:{\displaystyle g_{k}=o(g_{k-1})}
1973:{\displaystyle g_{k+1}=o(g_{k})}
1680:symbol, the last equation means
4568:
4425:
4297:
4201:
3916:
2175:
910:Examples of asymptotic formulas
730:{\displaystyle f^{r}\sim g^{r}}
339:
208:. Then the theorem states that
204:that are less than or equal to
4848:Asymptotic Methods in Analysis
4792:Asymptotic methods in analysis
4783:
4750:
4726:
4675:Asymptotic theory (statistics)
4581:
4569:
4545:
4525:
4519:
4512:
4479:
4473:
4438:
4426:
4402:
4382:
4376:
4369:
4317:
4298:
4274:
4254:
4248:
4241:
4214:
4208:
4202:
4198:
4182:
4155:
4149:
4103:
4097:
3755:
3746:
3727:
3721:
3658:
3644:
3252:
3244:
3105:
3099:
3093:
3078:
3061:
3044:
3029:
3017:
3007:
2974:
2968:
2911:
2905:
2899:
2868:
2858:
2828:
2822:
2775:
2769:
2763:
2677:
2665:
2586:
2567:
2528:
2509:
2500:
2487:
2448:
2435:
2336:
2317:
2223:
2204:
2138:
2125:
2116:
2084:
1967:
1954:
1809:
1777:
1747:
1734:
1725:
1693:
1361:
1355:
1350:
1344:
1253:
1247:
1242:
1236:
1126:
1120:
1022:
1016:
779:
773:
761:
755:
601:
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592:
580:
577:
571:
562:
556:
423:
417:
409:
403:
389:
357:
351:
340:
336:
330:
321:
315:
305:, we define a binary relation
224:
218:
13:
1:
4901:American Mathematical Society
4811:
4774:Practical Applied Mathematics
4635:N.A.: !!! . . . !
1503:In symbols, it means we have
637:
278:
97:becomes very large, the term
3813:, for some positive integer
2605:approaches the limit value.
1532:{\displaystyle f\sim g_{1},}
814:{\displaystyle \lim g\neq 1}
528:is zero infinitely often as
53:, is a method of describing
7:
4896:Applied Asymptotic Analysis
4739:Encyclopedia of Mathematics
4643:
3509:, may be recognized as the
896:{\displaystyle f/a\sim g/b}
459:on the set of functions of
10:
5005:
4945:Cambridge University Press
4779:Cambridge University Press
4757:Estrada & Kanwal (2002
4699:Method of dominant balance
4061:method of steepest descent
3775:
1925:takes its full meaning if
1462:
994:, the partition function,
283:Formally, given functions
29:
473:asymptotically equivalent
174:prime number theorem
123:asymptotically equivalent
4720:
4033:boundary layer equations
3931:probability distribution
3303:{\displaystyle e^{-w/t}}
3265:{\displaystyle |w|<1}
2071:to denote the statement
983:Stirling's approximation
4037:Navier-Stokes equations
3980:mathematical statistics
3789:asymptotic distribution
3784:mathematical statistics
3778:Asymptotic distribution
3772:Asymptotic distribution
3229:{\displaystyle w\neq 1}
990:For a positive integer
687:{\displaystyle a\sim b}
661:{\displaystyle f\sim g}
634:of the limiting value.
186:prime-counting function
75:becomes very large. If
4917:Murray, J. D. (1984),
4893:Miller, P. D. (2006),
4641:
4617:
4598:
4591:
4492:
4455:
4448:
4349:
4335:N.A.: But my value of
4327:
4221:
4130:
4110:
4029:mathematical modelling
3995:analysis of algorithms
3900:
3762:
3702:
3701:{\displaystyle x=-1/t}
3665:
3627:. However, by keeping
3615:
3587:
3503:
3463:
3397:
3304:
3266:
3230:
3202:
3187:
3112:
3006:
2918:
2854:
2782:
2596:
2538:
2458:
2343:
2230:
2168:
2148:
2065:
2005:
1974:
1919:
1846:
1845:{\displaystyle g_{k}.}
1816:
1754:
1674:
1650:
1579:
1533:
1449:
1205:
1092:The Airy function, Ai(
1081:
975:
897:
854:
815:
786:
731:
688:
662:
611:
439:
364:
269:analysis of algorithms
255:
4734:"Asymptotic equality"
4618:
4592:
4503:
4493:
4449:
4360:
4350:
4328:
4222:
4131:
4111:
4085:
4006:statistical mechanics
3923:mathematical sciences
3901:
3763:
3703:
3666:
3616:
3567:
3504:
3502:{\displaystyle u=w/t}
3464:
3377:
3305:
3267:
3231:
3203:
3167:
3113:
2986:
2919:
2834:
2783:
2597:
2539:
2459:
2344:
2231:
2169:
2167:{\displaystyle \sim }
2149:
2066:
2006:
2004:{\displaystyle g_{k}}
1975:
1920:
1847:
1822:is much smaller than
1817:
1755:
1675:
1673:{\displaystyle \sim }
1651:
1580:
1534:
1450:
1206:
1082:
976:
898:
855:
816:
787:
732:
689:
663:
612:
455:. The relation is an
440:
365:
256:
43:mathematical analysis
4771:Howison, S. (2005),
4607:
4508:
4461:
4365:
4339:
4237:
4143:
4120:
4116:for large values of
4109:{\displaystyle f(x)}
4091:
4073:quantum field theory
3954:approximation theory
3874:
3712:
3675:
3635:
3517:
3479:
3473:exponential integral
3314:
3276:
3240:
3214:
3143:
2932:
2796:
2791:Exponential integral
2620:
2548:
2468:
2353:
2248:
2185:
2158:
2075:
2023:
1988:
1929:
1858:
1826:
1768:
1684:
1664:
1589:
1543:
1507:
1471:asymptotic expansion
1465:Asymptotic expansion
1459:Asymptotic expansion
1219:
1111:
1010:
921:
865:
826:
796:
746:
701:
672:
646:
630:is not zero in some
550:
457:equivalence relation
378:
309:
212:
152:, which is read as "
4989:Mathematical series
4984:Asymptotic analysis
4959:Asymptotic Analysis
4920:Asymptotic Analysis
4818:Balser, W. (1994),
4057:saddle-point method
4004:, an example being
3965:applied mathematics
3428:
3331:
2244:−1; by subtracting
1354:
1246:
372:de Bruijn 1981
47:asymptotic analysis
4853:Dover Publications
4693:Leading-order term
4666:Asymptotic density
4613:
4587:
4488:
4444:
4345:
4323:
4217:
4126:
4106:
3984:probability theory
3896:
3758:
3698:
3661:
3611:
3499:
3459:
3414:
3317:
3300:
3262:
3226:
3198:
3108:
2914:
2778:
2592:
2534:
2454:
2339:
2226:
2164:
2144:
2061:
2001:
1984:, which means the
1970:
1915:
1842:
1812:
1750:
1670:
1646:
1575:
1529:
1445:
1443:
1334:
1226:
1201:
1077:
988:Partition function
971:
893:
850:
811:
782:
727:
684:
658:
607:
435:
396:
360:
251:
4801:978-0-486-64221-5
4616:{\displaystyle x}
4348:{\displaystyle x}
4129:{\displaystyle x}
4013:accident analysis
3969:numerical methods
3935:sample statistics
3891:
3558:
3536:
3365:
3350:
3272:. Multiplying by
3162:
3092:
3088:
2940:
2898:
2894:
2762:
2752:
2727:
2702:
2660:
2657:
2176:§ Definition
1762:little o notation
1432:
1386:
1385:
1321:
1278:
1277:
1199:
1178:
1164:
1150:
1073:
1072:
1048:
1045:
959:
944:
737:, for every real
534:little-o notation
427:
381:
346:
246:
200:is the number of
163:is asymptotic to
16:(Redirected from
4996:
4947:
4933:
4913:
4889:
4865:
4843:de Bruijn, N. G.
4838:
4806:
4805:
4787:
4781:
4769:
4760:
4754:
4748:
4747:
4730:
4715:
4671:
4662:
4622:
4620:
4619:
4614:
4596:
4594:
4593:
4588:
4567:
4566:
4548:
4543:
4542:
4515:
4497:
4495:
4494:
4489:
4453:
4451:
4450:
4445:
4424:
4423:
4405:
4400:
4399:
4372:
4354:
4352:
4351:
4346:
4332:
4330:
4329:
4324:
4316:
4315:
4296:
4295:
4277:
4272:
4271:
4244:
4226:
4224:
4223:
4218:
4197:
4196:
4181:
4173:
4172:
4135:
4133:
4132:
4127:
4115:
4113:
4112:
4107:
4065:Edgeworth series
4053:Laplace's method
4042:
4002:physical systems
4000:The behavior of
3991:computer science
3939:likelihood ratio
3905:
3903:
3902:
3897:
3892:
3884:
3850:
3844:
3830:
3824:
3818:
3812:
3802:
3767:
3765:
3764:
3759:
3745:
3744:
3708:and noting that
3707:
3705:
3704:
3699:
3694:
3670:
3668:
3667:
3662:
3654:
3620:
3618:
3617:
3612:
3610:
3609:
3586:
3581:
3563:
3559:
3551:
3539:
3538:
3537:
3529:
3508:
3506:
3505:
3500:
3495:
3468:
3466:
3465:
3460:
3451:
3450:
3441:
3440:
3427:
3422:
3413:
3412:
3396:
3391:
3366:
3364:
3353:
3352:
3351:
3343:
3333:
3330:
3325:
3309:
3307:
3306:
3301:
3299:
3298:
3294:
3271:
3269:
3268:
3263:
3255:
3247:
3235:
3233:
3232:
3227:
3207:
3205:
3204:
3199:
3197:
3196:
3186:
3181:
3163:
3161:
3147:
3127:double factorial
3124:
3117:
3115:
3114:
3109:
3090:
3089:
3087:
3086:
3085:
3076:
3075:
3053:
3027:
3025:
3024:
3005:
3000:
2961:
2960:
2959:
2958:
2941:
2936:
2923:
2921:
2920:
2915:
2896:
2895:
2893:
2892:
2883:
2876:
2875:
2856:
2853:
2848:
2821:
2820:
2811:
2810:
2787:
2785:
2784:
2779:
2760:
2753:
2751:
2750:
2749:
2733:
2728:
2726:
2725:
2724:
2708:
2703:
2701:
2690:
2661:
2659:
2658:
2647:
2645:
2644:
2634:
2633:
2624:
2601:
2599:
2598:
2593:
2585:
2584:
2560:
2559:
2543:
2541:
2540:
2535:
2527:
2526:
2499:
2498:
2480:
2479:
2463:
2461:
2460:
2455:
2447:
2446:
2428:
2427:
2415:
2414:
2396:
2395:
2371:
2370:
2348:
2346:
2345:
2340:
2335:
2334:
2310:
2309:
2291:
2290:
2266:
2265:
2235:
2233:
2232:
2227:
2222:
2221:
2197:
2196:
2173:
2171:
2170:
2165:
2153:
2151:
2150:
2145:
2137:
2136:
2115:
2114:
2096:
2095:
2070:
2068:
2067:
2062:
2060:
2059:
2041:
2040:
2013:asymptotic scale
2010:
2008:
2007:
2002:
2000:
1999:
1979:
1977:
1976:
1971:
1966:
1965:
1947:
1946:
1924:
1922:
1921:
1916:
1914:
1913:
1901:
1900:
1876:
1875:
1851:
1849:
1848:
1843:
1838:
1837:
1821:
1819:
1818:
1813:
1808:
1807:
1789:
1788:
1759:
1757:
1756:
1751:
1746:
1745:
1724:
1723:
1705:
1704:
1679:
1677:
1676:
1671:
1655:
1653:
1652:
1647:
1645:
1644:
1632:
1631:
1607:
1606:
1584:
1582:
1581:
1576:
1574:
1573:
1561:
1560:
1538:
1536:
1535:
1530:
1525:
1524:
1499:
1495:
1483:
1454:
1452:
1451:
1446:
1444:
1440:
1439:
1438:
1434:
1433:
1428:
1411:
1387:
1384:
1373:
1372:
1353:
1342:
1329:
1328:
1327:
1323:
1322:
1317:
1300:
1279:
1276:
1265:
1264:
1245:
1234:
1214:Hankel functions
1210:
1208:
1207:
1202:
1200:
1198:
1197:
1196:
1192:
1179:
1174:
1168:
1167:
1166:
1165:
1157:
1151:
1143:
1133:
1106:
1086:
1084:
1083:
1078:
1076:
1075:
1074:
1068:
1060:
1059:
1049:
1047:
1046:
1041:
1029:
980:
978:
977:
972:
970:
969:
964:
960:
952:
945:
934:
902:
900:
899:
894:
889:
875:
859:
857:
856:
851:
820:
818:
817:
812:
791:
789:
788:
783:
740:
736:
734:
733:
728:
726:
725:
713:
712:
693:
691:
690:
685:
667:
665:
664:
659:
629:
616:
614:
613:
608:
545:
531:
527:
513:
511:
503:
496:
486:
482:
470:
466:
463:; the functions
462:
450:
444:
442:
441:
436:
428:
426:
412:
398:
395:
370:if and only if (
369:
367:
366:
361:
347:
344:
304:
293:
265:computer science
260:
258:
257:
252:
247:
245:
231:
207:
199:
183:
168:
162:
151:
137:
130:
120:
109:
103:
96:
92:
74:
70:
49:, also known as
21:
18:Asymptotic limit
5004:
5003:
4999:
4998:
4997:
4995:
4994:
4993:
4974:
4973:
4954:
4931:
4911:
4887:
4863:
4836:
4826:Springer-Verlag
4814:
4809:
4802:
4788:
4784:
4770:
4763:
4755:
4751:
4732:
4731:
4727:
4723:
4718:
4713:
4669:
4660:
4646:
4608:
4605:
4604:
4559:
4555:
4544:
4535:
4531:
4511:
4509:
4506:
4505:
4462:
4459:
4458:
4416:
4412:
4401:
4392:
4388:
4368:
4366:
4363:
4362:
4340:
4337:
4336:
4311:
4307:
4288:
4284:
4273:
4264:
4260:
4240:
4238:
4235:
4234:
4189:
4185:
4177:
4165:
4161:
4144:
4141:
4140:
4121:
4118:
4117:
4092:
4089:
4088:
4081:
4040:
3971:to approximate
3919:
3883:
3875:
3872:
3871:
3846:
3843:
3835:
3826:
3820:
3814:
3804:
3801:
3793:
3780:
3774:
3740:
3736:
3713:
3710:
3709:
3690:
3676:
3673:
3672:
3671:. Substituting
3650:
3636:
3633:
3632:
3599:
3595:
3582:
3571:
3550:
3546:
3528:
3524:
3520:
3518:
3515:
3514:
3491:
3480:
3477:
3476:
3446:
3442:
3433:
3429:
3423:
3418:
3402:
3398:
3392:
3381:
3354:
3342:
3338:
3334:
3332:
3326:
3321:
3315:
3312:
3311:
3290:
3283:
3279:
3277:
3274:
3273:
3251:
3243:
3241:
3238:
3237:
3215:
3212:
3211:
3192:
3188:
3182:
3171:
3151:
3146:
3144:
3141:
3140:
3136:
3119:
3081:
3077:
3071:
3067:
3054:
3028:
3026:
3020:
3016:
3001:
2990:
2954:
2950:
2949:
2945:
2935:
2933:
2930:
2929:
2888:
2884:
2871:
2867:
2857:
2855:
2849:
2838:
2816:
2812:
2806:
2802:
2797:
2794:
2793:
2745:
2741:
2737:
2732:
2720:
2716:
2712:
2707:
2694:
2689:
2646:
2640:
2636:
2635:
2629:
2625:
2623:
2621:
2618:
2617:
2611:
2574:
2570:
2555:
2551:
2549:
2546:
2545:
2516:
2512:
2494:
2490:
2475:
2471:
2469:
2466:
2465:
2442:
2438:
2423:
2419:
2404:
2400:
2385:
2381:
2366:
2362:
2354:
2351:
2350:
2324:
2320:
2299:
2295:
2280:
2276:
2261:
2257:
2249:
2246:
2245:
2211:
2207:
2192:
2188:
2186:
2183:
2182:
2159:
2156:
2155:
2132:
2128:
2110:
2106:
2091:
2087:
2076:
2073:
2072:
2055:
2051:
2036:
2032:
2024:
2021:
2020:
1995:
1991:
1989:
1986:
1985:
1961:
1957:
1936:
1932:
1930:
1927:
1926:
1909:
1905:
1890:
1886:
1871:
1867:
1859:
1856:
1855:
1833:
1829:
1827:
1824:
1823:
1803:
1799:
1784:
1780:
1769:
1766:
1765:
1741:
1737:
1719:
1715:
1700:
1696:
1685:
1682:
1681:
1665:
1662:
1661:
1656:for each fixed
1640:
1636:
1621:
1617:
1602:
1598:
1590:
1587:
1586:
1569:
1565:
1556:
1552:
1544:
1541:
1540:
1520:
1516:
1508:
1505:
1504:
1497:
1493:
1474:
1467:
1461:
1442:
1441:
1412:
1410:
1403:
1399:
1392:
1388:
1377:
1371:
1364:
1343:
1338:
1331:
1330:
1301:
1299:
1292:
1288:
1284:
1280:
1269:
1263:
1256:
1235:
1230:
1222:
1220:
1217:
1216:
1188:
1184:
1180:
1173:
1169:
1156:
1152:
1142:
1138:
1134:
1132:
1112:
1109:
1108:
1097:
1061:
1058:
1054:
1050:
1040:
1033:
1028:
1011:
1008:
1007:
965:
951:
947:
946:
933:
922:
919:
918:
912:
885:
871:
866:
863:
862:
827:
824:
823:
797:
794:
793:
747:
744:
743:
738:
721:
717:
708:
704:
702:
699:
698:
673:
670:
669:
647:
644:
643:
640:
620:
551:
548:
547:
546:if and only if
537:
529:
518:
507:
505:
498:
491:
484:
480:
471:are said to be
468:
464:
460:
448:
413:
399:
397:
385:
379:
376:
375:
343:
310:
307:
306:
295:
284:
281:
267:as part of the
235:
230:
213:
210:
209:
205:
193:
177:
164:
153:
139:
132:
126:
121:is said to be "
111:
110:. The function
105:
98:
94:
76:
72:
61:
39:
28:
23:
22:
15:
12:
11:
5:
5002:
4992:
4991:
4986:
4972:
4971:
4966:
4953:
4952:External links
4950:
4949:
4948:
4934:
4929:
4914:
4909:
4890:
4885:
4866:
4861:
4839:
4834:
4813:
4810:
4808:
4807:
4800:
4782:
4761:
4749:
4724:
4722:
4719:
4717:
4716:
4710:Watson's lemma
4707:
4702:
4696:
4690:
4687:Big O notation
4684:
4678:
4672:
4663:
4654:
4647:
4645:
4642:
4612:
4586:
4583:
4580:
4577:
4574:
4571:
4565:
4562:
4558:
4554:
4551:
4547:
4541:
4538:
4534:
4530:
4527:
4524:
4521:
4518:
4514:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4466:
4443:
4440:
4437:
4434:
4431:
4428:
4422:
4419:
4415:
4411:
4408:
4404:
4398:
4395:
4391:
4387:
4384:
4381:
4378:
4375:
4371:
4344:
4322:
4319:
4314:
4310:
4306:
4303:
4300:
4294:
4291:
4287:
4283:
4280:
4276:
4270:
4267:
4263:
4259:
4256:
4253:
4250:
4247:
4243:
4216:
4213:
4210:
4207:
4204:
4200:
4195:
4192:
4188:
4184:
4180:
4176:
4171:
4168:
4164:
4160:
4157:
4154:
4151:
4148:
4125:
4105:
4102:
4099:
4096:
4080:
4077:
4069:Feynman graphs
4045:nondimensional
4035:from the full
4017:
4016:
4009:
3998:
3987:
3976:
3946:expected value
3937:, such as the
3918:
3915:
3895:
3890:
3887:
3882:
3879:
3839:
3797:
3776:Main article:
3773:
3770:
3757:
3754:
3751:
3748:
3743:
3739:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3697:
3693:
3689:
3686:
3683:
3680:
3660:
3657:
3653:
3649:
3646:
3643:
3640:
3608:
3605:
3602:
3598:
3593:
3590:
3585:
3580:
3577:
3574:
3570:
3566:
3562:
3557:
3554:
3549:
3545:
3542:
3535:
3532:
3527:
3523:
3511:gamma function
3498:
3494:
3490:
3487:
3484:
3458:
3455:
3449:
3445:
3439:
3436:
3432:
3426:
3421:
3417:
3411:
3408:
3405:
3401:
3395:
3390:
3387:
3384:
3380:
3376:
3373:
3370:
3363:
3360:
3357:
3349:
3346:
3341:
3337:
3329:
3324:
3320:
3297:
3293:
3289:
3286:
3282:
3261:
3258:
3254:
3250:
3246:
3225:
3222:
3219:
3195:
3191:
3185:
3180:
3177:
3174:
3170:
3166:
3160:
3157:
3154:
3150:
3135:
3134:Worked example
3132:
3131:
3130:
3107:
3104:
3101:
3098:
3095:
3084:
3080:
3074:
3070:
3066:
3063:
3060:
3057:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3023:
3019:
3015:
3012:
3009:
3004:
2999:
2996:
2993:
2989:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2957:
2953:
2948:
2944:
2939:
2927:Error function
2924:
2913:
2910:
2907:
2904:
2901:
2891:
2887:
2882:
2879:
2874:
2870:
2866:
2863:
2860:
2852:
2847:
2844:
2841:
2837:
2833:
2830:
2827:
2824:
2819:
2815:
2809:
2805:
2801:
2788:
2777:
2774:
2771:
2768:
2765:
2759:
2756:
2748:
2744:
2740:
2736:
2731:
2723:
2719:
2715:
2711:
2706:
2700:
2697:
2693:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2656:
2653:
2650:
2643:
2639:
2632:
2628:
2615:Gamma function
2610:
2607:
2591:
2588:
2583:
2580:
2577:
2573:
2569:
2566:
2563:
2558:
2554:
2533:
2530:
2525:
2522:
2519:
2515:
2511:
2508:
2505:
2502:
2497:
2493:
2489:
2486:
2483:
2478:
2474:
2453:
2450:
2445:
2441:
2437:
2434:
2431:
2426:
2422:
2418:
2413:
2410:
2407:
2403:
2399:
2394:
2391:
2388:
2384:
2380:
2377:
2374:
2369:
2365:
2361:
2358:
2338:
2333:
2330:
2327:
2323:
2319:
2316:
2313:
2308:
2305:
2302:
2298:
2294:
2289:
2286:
2283:
2279:
2275:
2272:
2269:
2264:
2260:
2256:
2253:
2225:
2220:
2217:
2214:
2210:
2206:
2203:
2200:
2195:
2191:
2163:
2143:
2140:
2135:
2131:
2127:
2124:
2121:
2118:
2113:
2109:
2105:
2102:
2099:
2094:
2090:
2086:
2083:
2080:
2058:
2054:
2050:
2047:
2044:
2039:
2035:
2031:
2028:
1998:
1994:
1969:
1964:
1960:
1956:
1953:
1950:
1945:
1942:
1939:
1935:
1912:
1908:
1904:
1899:
1896:
1893:
1889:
1885:
1882:
1879:
1874:
1870:
1866:
1863:
1841:
1836:
1832:
1811:
1806:
1802:
1798:
1795:
1792:
1787:
1783:
1779:
1776:
1773:
1749:
1744:
1740:
1736:
1733:
1730:
1727:
1722:
1718:
1714:
1711:
1708:
1703:
1699:
1695:
1692:
1689:
1669:
1643:
1639:
1635:
1630:
1627:
1624:
1620:
1616:
1613:
1610:
1605:
1601:
1597:
1594:
1572:
1568:
1564:
1559:
1555:
1551:
1548:
1528:
1523:
1519:
1515:
1512:
1473:of a function
1463:Main article:
1460:
1457:
1456:
1455:
1437:
1431:
1427:
1424:
1421:
1418:
1415:
1409:
1406:
1402:
1398:
1395:
1391:
1383:
1380:
1376:
1370:
1367:
1365:
1363:
1360:
1357:
1352:
1349:
1346:
1341:
1337:
1333:
1332:
1326:
1320:
1316:
1313:
1310:
1307:
1304:
1298:
1295:
1291:
1287:
1283:
1275:
1272:
1268:
1262:
1259:
1257:
1255:
1252:
1249:
1244:
1241:
1238:
1233:
1229:
1225:
1224:
1211:
1195:
1191:
1187:
1183:
1177:
1172:
1163:
1160:
1155:
1149:
1146:
1141:
1137:
1131:
1128:
1125:
1122:
1119:
1116:
1087:
1071:
1067:
1064:
1057:
1053:
1044:
1039:
1036:
1032:
1027:
1024:
1021:
1018:
1015:
985:
968:
963:
958:
955:
950:
943:
940:
937:
932:
929:
926:
911:
908:
904:
903:
892:
888:
884:
881:
878:
874:
870:
860:
849:
846:
843:
840:
837:
834:
831:
821:
810:
807:
804:
801:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
741:
724:
720:
716:
711:
707:
683:
680:
677:
657:
654:
651:
639:
636:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
434:
431:
425:
422:
419:
416:
411:
408:
405:
402:
394:
391:
388:
384:
359:
356:
353:
350:
342:
338:
335:
332:
329:
326:
323:
320:
317:
314:
280:
277:
273:big O notation
250:
244:
241:
238:
234:
229:
226:
223:
220:
217:
26:
9:
6:
4:
3:
2:
5001:
4990:
4987:
4985:
4982:
4981:
4979:
4970:
4967:
4965:
4961:
4960:
4956:
4955:
4946:
4942:
4941:
4935:
4932:
4930:9781461211228
4926:
4922:
4921:
4915:
4912:
4910:9780821840788
4906:
4902:
4898:
4897:
4891:
4888:
4886:9780817681302
4882:
4878:
4874:
4873:
4867:
4864:
4862:9780486642215
4858:
4854:
4850:
4849:
4844:
4840:
4837:
4835:9783540485940
4831:
4827:
4823:
4822:
4816:
4815:
4803:
4797:
4793:
4786:
4780:
4776:
4775:
4768:
4766:
4758:
4753:
4745:
4741:
4740:
4735:
4729:
4725:
4711:
4708:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4681:Asymptotology
4679:
4676:
4673:
4667:
4664:
4658:
4655:
4652:
4649:
4648:
4640:
4636:
4633:
4630:
4627:
4624:
4610:
4601:
4597:
4584:
4578:
4575:
4572:
4563:
4560:
4556:
4552:
4549:
4539:
4536:
4532:
4528:
4522:
4516:
4502:
4499:
4485:
4482:
4476:
4470:
4467:
4464:
4454:
4441:
4435:
4432:
4429:
4420:
4417:
4413:
4409:
4406:
4396:
4393:
4389:
4385:
4379:
4373:
4359:
4356:
4355:is only 100.
4342:
4333:
4320:
4312:
4308:
4304:
4301:
4292:
4289:
4285:
4281:
4278:
4268:
4265:
4261:
4257:
4251:
4245:
4231:
4228:
4205:
4193:
4190:
4186:
4174:
4169:
4166:
4162:
4158:
4152:
4146:
4137:
4123:
4100:
4094:
4084:
4076:
4074:
4070:
4066:
4062:
4058:
4054:
4049:
4046:
4038:
4034:
4030:
4026:
4022:
4014:
4010:
4007:
4003:
3999:
3996:
3992:
3988:
3985:
3981:
3977:
3974:
3970:
3966:
3962:
3961:
3960:
3957:
3955:
3951:
3947:
3943:
3940:
3936:
3932:
3928:
3924:
3914:
3912:
3908:
3893:
3888:
3885:
3880:
3877:
3868:
3863:
3861:
3857:
3852:
3849:
3842:
3838:
3832:
3831:is infinite.
3829:
3823:
3817:
3811:
3807:
3800:
3796:
3791:
3790:
3785:
3779:
3769:
3752:
3749:
3741:
3737:
3733:
3730:
3724:
3718:
3715:
3695:
3691:
3687:
3684:
3681:
3678:
3655:
3651:
3647:
3641:
3638:
3630:
3626:
3621:
3606:
3603:
3600:
3596:
3591:
3588:
3578:
3575:
3572:
3568:
3564:
3560:
3555:
3552:
3547:
3543:
3540:
3533:
3530:
3525:
3521:
3512:
3496:
3492:
3488:
3485:
3482:
3474:
3469:
3456:
3453:
3447:
3443:
3437:
3434:
3430:
3419:
3415:
3409:
3406:
3403:
3399:
3388:
3385:
3382:
3378:
3374:
3371:
3368:
3361:
3358:
3355:
3347:
3344:
3339:
3335:
3322:
3318:
3295:
3291:
3287:
3284:
3280:
3259:
3256:
3248:
3223:
3220:
3217:
3208:
3193:
3189:
3178:
3175:
3172:
3168:
3164:
3158:
3155:
3152:
3148:
3128:
3122:
3096:
3082:
3072:
3068:
3064:
3058:
3055:
3050:
3047:
3041:
3038:
3035:
3032:
3021:
3013:
3010:
2997:
2994:
2991:
2987:
2983:
2980:
2977:
2971:
2965:
2962:
2955:
2951:
2946:
2942:
2937:
2928:
2925:
2902:
2889:
2885:
2880:
2877:
2872:
2864:
2861:
2845:
2842:
2839:
2835:
2831:
2825:
2817:
2813:
2807:
2803:
2799:
2792:
2789:
2766:
2757:
2754:
2746:
2742:
2738:
2734:
2729:
2721:
2717:
2713:
2709:
2704:
2698:
2695:
2691:
2686:
2683:
2680:
2674:
2671:
2668:
2654:
2651:
2648:
2641:
2637:
2630:
2626:
2616:
2613:
2612:
2606:
2602:
2589:
2581:
2578:
2575:
2571:
2564:
2561:
2556:
2552:
2531:
2523:
2520:
2517:
2513:
2506:
2503:
2495:
2491:
2484:
2481:
2476:
2472:
2451:
2443:
2439:
2432:
2429:
2424:
2420:
2416:
2411:
2408:
2405:
2401:
2397:
2392:
2389:
2386:
2382:
2378:
2375:
2372:
2367:
2363:
2359:
2356:
2331:
2328:
2325:
2321:
2314:
2311:
2306:
2303:
2300:
2296:
2292:
2287:
2284:
2281:
2277:
2273:
2270:
2267:
2262:
2258:
2254:
2251:
2243:
2239:
2218:
2215:
2212:
2208:
2201:
2198:
2193:
2189:
2179:
2177:
2161:
2141:
2133:
2129:
2122:
2119:
2111:
2107:
2103:
2100:
2097:
2092:
2088:
2081:
2078:
2056:
2052:
2048:
2045:
2042:
2037:
2033:
2029:
2026:
2018:
2014:
1996:
1992:
1983:
1962:
1958:
1951:
1948:
1943:
1940:
1937:
1933:
1910:
1906:
1902:
1897:
1894:
1891:
1887:
1883:
1880:
1877:
1872:
1868:
1864:
1861:
1854:The relation
1852:
1839:
1834:
1830:
1804:
1800:
1796:
1793:
1790:
1785:
1781:
1774:
1771:
1763:
1742:
1738:
1731:
1728:
1720:
1716:
1712:
1709:
1706:
1701:
1697:
1690:
1687:
1667:
1659:
1641:
1637:
1633:
1628:
1625:
1622:
1618:
1614:
1611:
1608:
1603:
1599:
1595:
1592:
1570:
1566:
1562:
1557:
1553:
1549:
1546:
1526:
1521:
1517:
1513:
1510:
1501:
1491:
1487:
1481:
1477:
1472:
1466:
1435:
1429:
1425:
1422:
1419:
1416:
1413:
1407:
1404:
1400:
1396:
1393:
1389:
1381:
1378:
1374:
1368:
1366:
1358:
1347:
1339:
1335:
1324:
1318:
1314:
1311:
1308:
1305:
1302:
1296:
1293:
1289:
1285:
1281:
1273:
1270:
1266:
1260:
1258:
1250:
1239:
1231:
1227:
1215:
1212:
1193:
1189:
1185:
1181:
1175:
1170:
1161:
1158:
1153:
1147:
1144:
1139:
1135:
1129:
1123:
1117:
1114:
1104:
1100:
1095:
1091:
1090:Airy function
1088:
1069:
1065:
1062:
1055:
1051:
1042:
1037:
1034:
1030:
1025:
1019:
1013:
1005:
1001:
997:
993:
989:
986:
984:
966:
961:
956:
953:
948:
941:
938:
935:
930:
927:
924:
917:
914:
913:
907:
890:
886:
882:
879:
876:
872:
868:
861:
847:
844:
841:
838:
835:
832:
829:
822:
808:
805:
802:
776:
770:
767:
764:
758:
752:
749:
742:
722:
718:
714:
709:
705:
697:
696:
695:
681:
678:
675:
655:
652:
649:
635:
633:
632:neighbourhood
627:
623:
617:
604:
595:
589:
586:
583:
574:
568:
565:
559:
553:
544:
540:
535:
525:
521:
515:
510:
501:
494:
488:
478:
474:
458:
454:
445:
432:
429:
420:
414:
406:
400:
386:
373:
348:
333:
327:
324:
318:
312:
302:
298:
291:
287:
276:
274:
270:
266:
261:
248:
242:
239:
236:
232:
227:
221:
215:
203:
202:prime numbers
197:
191:
187:
181:
175:
170:
167:
160:
156:
150:
146:
142:
135:
129:
124:
118:
114:
108:
102:
91:
87:
83:
79:
68:
64:
58:
56:
52:
48:
44:
37:
33:
19:
4958:
4939:
4923:, Springer,
4919:
4895:
4871:
4847:
4820:
4791:
4785:
4772:
4752:
4737:
4728:
4637:
4634:
4631:
4628:
4625:
4602:
4599:
4504:
4500:
4456:
4361:
4357:
4334:
4232:
4229:
4138:
4086:
4082:
4050:
4018:
3958:
3920:
3917:Applications
3910:
3906:
3864:
3859:
3853:
3847:
3840:
3836:
3833:
3827:
3821:
3815:
3809:
3805:
3798:
3794:
3787:
3781:
3628:
3624:
3622:
3470:
3209:
3137:
3120:
2603:
2241:
2237:
2180:
1981:
1853:
1657:
1502:
1490:partial sums
1479:
1475:
1468:
1102:
1098:
1093:
1003:
999:
995:
991:
905:
641:
625:
621:
618:
542:
538:
523:
519:
516:
508:
499:
492:
489:
472:
446:
300:
296:
289:
285:
282:
262:
195:
179:
171:
165:
158:
154:
148:
144:
140:
133:
127:
122:
116:
112:
106:
100:
89:
85:
81:
77:
66:
62:
59:
50:
46:
40:
3913:increases.
3845:go to 0 as
447:The symbol
184:denote the
51:asymptotics
4978:Categories
4877:Birkhäuser
4812:References
3975:solutions.
3927:statistics
3856:asymptotic
638:Properties
536:, is that
512:| → 0
279:Definition
93:, then as
57:behavior.
4964:IOS Press
4744:EMS Press
4651:Asymptote
4561:−
4537:−
4529:−
4418:−
4394:−
4386:−
4290:−
4266:−
4258:−
4212:∞
4209:→
4191:−
4167:−
3942:statistic
3867:asymptote
3860:asymptote
3750:−
3734:−
3719:
3685:−
3642:
3584:∞
3569:∑
3544:
3526:−
3435:−
3425:∞
3416:∫
3394:∞
3379:∑
3359:−
3340:−
3328:∞
3319:∫
3285:−
3221:≠
3184:∞
3169:∑
3156:−
3103:∞
3100:→
3039:−
3011:−
3003:∞
2988:∑
2978:∼
2966:
2938:π
2909:∞
2906:→
2862:−
2851:∞
2836:∑
2832:∼
2773:∞
2770:→
2758:⋯
2755:−
2730:−
2681:∼
2663:Γ
2652:π
2579:−
2521:−
2464:one gets
2409:−
2398:−
2390:−
2379:−
2376:⋯
2373:−
2360:−
2329:−
2304:−
2285:−
2274:−
2271:⋯
2268:−
2255:−
2216:−
2162:∼
2101:⋯
2082:−
2046:⋯
2030:∼
2017:abusively
1903:∼
1895:−
1884:−
1881:⋯
1878:−
1865:−
1794:⋯
1775:−
1710:⋯
1691:−
1668:∼
1634:∼
1626:−
1615:−
1612:⋯
1609:−
1596:−
1563:∼
1550:−
1539:but also
1514:∼
1426:π
1423:−
1420:α
1417:π
1408:−
1394:−
1379:π
1369:∼
1340:α
1315:π
1312:−
1309:α
1306:π
1297:−
1271:π
1261:∼
1232:α
1176:π
1140:−
1130:∼
1118:
1056:π
1026:∼
981:—this is
939:π
931:∼
916:Factorial
880:∼
845:×
839:∼
833:×
806:≠
771:
765:∼
753:
715:∼
679:∼
653:∼
393:∞
390:→
355:∞
352:→
325:∼
240:
228:∼
216:π
36:Asymptote
4845:(1981),
4644:See also
4021:ordinary
3973:equation
3950:deviance
3944:and the
3808:= 1, …,
2011:form an
1980:for all
1764:, i.e.,
1099:y″
374:, §1.4)
345:as
288: (
192:), i.e.
143: (
65: (
55:limiting
32:geometry
4759:, §1.2)
4746:, 2001
4067:). The
4025:partial
3993:in the
3948:of the
3125:is the
1760:in the
451:is the
4927:
4907:
4883:
4859:
4832:
4798:
4639:reply.
4233:A.A.:
4139:A.A.:
4048:hand.
3118:where
3091:
2897:
2761:
2019:write
1488:, the
1486:series
506:|
477:domain
475:. The
176:. Let
34:, see
4721:Notes
4410:57000
3925:. In
3786:, an
2739:51840
2544:i.e.
2349:from
453:tilde
131:, as
4925:ISBN
4905:ISBN
4881:ISBN
4857:ISBN
4830:ISBN
4796:ISBN
4576:>
4550:<
4483:<
4468:<
4433:>
4407:<
4305:>
4279:<
4023:and
3982:and
3803:for
3257:<
2963:erfc
2240:and
1585:and
668:and
483:and
467:and
294:and
147:) ~
84:) =
4579:100
4477:100
4436:100
4071:in
4011:In
3989:In
3978:In
3963:In
3933:of
3865:An
3782:In
2735:139
2714:288
1469:An
1105:= 0
800:lim
792:if
768:log
750:log
642:If
502:↓ 0
495:→ 0
479:of
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4008:.
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1941:+
1938:k
1934:g
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243:x
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155:f
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113:f
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99:3
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80:(
78:f
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69:)
67:n
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38:.
20:)
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