1442:
1207:
2775:
4036:
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
2593:
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
3105:
3456:
1437:{\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}}}
3608:
1198:
2608:
3851:) 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.
2911:
2920:
1074:
3302:
3127:
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
968:
4214:
2451:
4441:
3858:
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
4627:
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
4320:
357:
4584:
3195:
432:
2336:
3505:
1099:
1212:
3755:
2141:
1912:
1643:
1747:
248:
2784:
2531:
2058:
3658:
1809:
847:
779:
604:
3893:
2589:
1572:
4485:
2223:
1967:
724:
3941:. 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
1526:
808:
890:
998:
3297:
3259:
3223:
681:
655:
3695:
1839:
3496:
2161:
1998:
1667:
4103:
3781:
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
2770:{\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 )}
4610:
4342:
4123:
909:
3100:{\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 )}
297:
3131:
366:
4072:
Debruijn illustrates the use of asymptotics in the following dialog between Dr. N.A., a
Numerical Analyst, and Dr. A.A., an Asymptotic Analyst:
4131:
3451:{\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}
3620:
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
2341:
200:
4353:
4225:
4693:
4496:
2236:
4957:
4788:
4004:
when identifying the causation of crash through count modeling with large number of crash counts in a given time and space.
4645:
19:
This article is about the behavior of functions as inputs approach infinity or some other limit value. For asymptotes in
538:
521:
goes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, in
3700:
2063:
1846:
1577:
976:
1672:
1481:
of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for
4917:
4897:
4873:
4849:
4822:
3603:{\displaystyle e^{-{\frac {1}{t}}}\operatorname {Ei} \left({\frac {1}{t}}\right)=\sum _{n=0}^{\infty }n!\;t^{n+1}}
4762:
2456:
1193:{\displaystyle \operatorname {Ai} (x)\sim {\frac {e^{-{\frac {2}{3}}x^{\frac {3}{2}}}}{2{\sqrt {\pi }}x^{1/4}}}}
895:
Such properties allow asymptotically equivalent functions to be freely exchanged in many algebraic expressions.
4663:
4009:
2011:
3623:
4889:
4732:
4013:
1756:
814:
3975:, asymptotics are used in analysis of long-run or large-sample behaviour of random variables and estimators.
4977:
4972:
1485:. The idea is that successive terms provide an increasingly accurate description of the order of growth of
4727:
4722:
971:
734:
620:
4648: – computational complexity as measured by the limiting behavior of resource usage for large inputs
4021:
4933:
4767:
4687:
4049:
3862:
2536:
1531:
4449:
2173:
1917:
476:
can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.
4831:
689:
3919:
2906:{\displaystyle xe^{x}E_{1}(x)\sim \sum _{n=0}^{\infty }{\frac {(-1)^{n}n!}{x^{n}}}\ (x\to \infty )}
503:. The way of passing to the limit is often not stated explicitly, if it is clear from the context.
1495:
784:
4025:
3968:
3777:
3772:
3766:
1750:
853:
174:
4621:
A.A.: Haven't I told you so? My estimate of 20% was not far off from the 14% of the real error.
4028:
governing fluid flow. In many cases, the asymptotic expansion is in power of a small parameter,
3983:
3840:
goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.
257:
3823:
A special case of an asymptotic distribution is when the late entries go to zero—that is, the
3264:
3228:
3994:
3938:
3927:
3911:
3202:
660:
634:
31:
3663:
1814:
4061:
4033:
3942:
3467:
3461:
2779:
2146:
1976:
1652:
1474:
1459:
1453:
465:
445:
162:
4592:
A.A.: It is almost the best thing I possibly can get. Why don't you take larger values of
4079:
8:
4045:
4041:
3953:
43:
4946:
4841:
4681:
4654:
4595:
4327:
4108:
4017:
3972:
4913:
4893:
4869:
4845:
4818:
4784:
4698:
4001:
2005:
522:
4927:
4053:
3979:
3957:
3923:
3115:
2001:
1202:
253:
49:
As an illustration, suppose that we are interested in the properties of a function
1069:{\displaystyle p(n)\sim {\frac {1}{4n{\sqrt {3}}}}e^{\pi {\sqrt {\frac {2n}{3}}}}}
4907:
4883:
4859:
4835:
4814:
4808:
4040:
Asymptotic expansions typically arise in the approximation of certain integrals (
3990:
506:
Although the above definition is common in the literature, it is problematic if
4675:
4057:
3934:
3499:
2915:
2603:
261:
4684: – Terms in a mathematical expression with the largest order of magnitude
3612:
Here, the right hand side is clearly not convergent for any non-zero value of
4966:
4669:
1078:
995:
as a sum of positive integers, where the order of addends is not considered.
4865:
4615:
N.A.: !!! I think it's better to ask my electronic computing machine.
190:
4064:
are another example of asymptotic expansions which often do not converge.
4020:
of real-world phenomena. An illustrative example is the derivation of the
479:
The same notation is also used for other ways of passing to a limit: e.g.
4783:. Dover books on advanced mathematics. New York: Dover publ. p. 19.
1478:
3915:
3844:
963:{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}
4952:
4639:
4209:{\displaystyle f(x)=x^{-1}+\mathrm {O} (x^{-2})\qquad (x\to \infty )}
3930:
3855:
2143:
One should however be careful that this is not a standard use of the
904:
24:
3460:
The integral on the left hand side can be expressed in terms of the
4672: – Dealing with applied mathematical systems in limiting cases
4642: – Limit of the tangent line at a point that tends to infinity
3961:
3757:
results in the asymptotic expansion given earlier in this article.
2163:
symbol, and that it does not correspond to the definition given in
20:
4490:
A.A.: I can gain a little on some of my estimates. Now I find that
4666: – Study of convergence properties of statistical estimators
3199:
The expression on the left is valid on the entire complex plane
2446:{\displaystyle f-g_{1}-\cdots -g_{k-2}-g_{k-1}=g_{k}+o(g_{k}),}
4436:{\displaystyle |f(x)-x^{-1}|<57000x^{-2}\qquad (x>100).}
3464:. The integral on the right hand side, after the substitution
4958:
A paper on time series analysis using asymptotic distribution
4315:{\displaystyle |f(x)-x^{-1}|<8x^{-2}\qquad (x>10^{4}).}
441:
352:{\displaystyle f(x)\sim g(x)\quad ({\text{as }}x\to \infty )}
3918:, asymptotic theory provides limiting approximations of the
1473:
is in practice an expression of that function in terms of a
4579:{\displaystyle |f(x)-x^{-1}|<20x^{-2}\qquad (x>100).}
3190:{\displaystyle {\frac {1}{1-w}}=\sum _{n=0}^{\infty }w^{n}}
427:{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1.}
2331:{\displaystyle f-g_{1}-\cdots -g_{k-2}=g_{k-1}+o(g_{k-1})}
608:
This definition is equivalent to the prior definition if
4052:) or in the approximation of probability distributions (
3847:
function which cleanly approaches a constant value (the
3502:. Evaluating both, one obtains the asymptotic expansion
683:, then, under some mild conditions, the following hold:
4951: —home page of the journal, which is published by
4659:
Pages displaying short descriptions of redirect targets
4067:
178:
4701: – lemma on the asymptotic behavior of integrals
4598:
4499:
4452:
4356:
4330:
4228:
4134:
4111:
4082:
3865:
3703:
3666:
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2239:
2176:
2149:
2066:
2014:
1979:
1920:
1849:
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1210:
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1001:
912:
856:
817:
787:
737:
692:
663:
637:
541:
369:
300:
203:
4703:
Pages displaying wikidata descriptions as a fallback
4690: – Solution of a simplified form of an equation
4650:
Pages displaying wikidata descriptions as a fallback
4008:
Asymptotic analysis is a key tool for exploring the
2597:
161:
An example of an important asymptotic result is the
991:), gives the number of ways of writing the integer
4604:
4578:
4479:
4435:
4336:
4314:
4208:
4117:
4097:
3887:
3749:
3689:
3652:
3602:
3490:
3450:
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3253:
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3099:
2905:
2769:
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2445:
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2052:
1992:
1961:
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841:
802:
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718:
675:
649:
598:
426:
351:
242:
4810:From Divergent Power Series To Analytic Functions
4678: – Describes limiting behavior of a function
4446:N.A.: This is no news to me. I know already that
4347:A.A.: Why did you not say so? My evaluations give
3750:{\displaystyle \operatorname {Ei} (x)=-E_{1}(-x)}
2136:{\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k}).}
1907:{\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}}
1638:{\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}}
4964:
1742:{\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k})}
788:
371:
3225:, while the right hand side converges only for
243:{\displaystyle \pi (x)\sim {\frac {x}{\ln x}}.}
177:(which is not directly related to the constant
1085:), is a solution of the differential equation
16:Description of limiting behavior of a function
4925:
4857:
4745:
4618:Machine: f(100) = 0.01137 42259 34008 67153
3986:, considering the performance of algorithms.
3948:Examples of applications are the following.
4032:: in the boundary layer case, this is the
4016:differential equations which arise in the
3898:becomes arbitrarily small in magnitude as
3760:
3583:
2526:{\displaystyle g_{k}+o(g_{k})=o(g_{k-1}),}
4830:
4756:
4754:
3441:
3356:
2053:{\displaystyle f\sim g_{1}+\cdots +g_{k}}
360:
260:and is often expressed there in terms of
127:". This is often written symbolically as
4861:A Distributional Approach to Asymptotics
3653:{\displaystyle \operatorname {Ei} (1/t)}
2170:In the present situation, this relation
252:Asymptotic analysis is commonly used in
4929:Asymptotics and Mellin-Barnes Integrals
4694:Method of matched asymptotic expansions
4125:, with a relative error of at most 1%.
3956:, asymptotic analysis is used to build
3910:Asymptotic analysis is used in several
1804:{\displaystyle f-(g_{1}+\cdots +g_{k})}
1447:
1096:; it has many applications in physics.
842:{\displaystyle f\times a\sim g\times b}
4965:
4905:
4881:
4806:
4778:
4751:
4219:N.A.: I am sorry, I don't understand.
2225:actually follows from combining steps
4076:N.A.: I want to evaluate my function
4068:Asymptotic versus Numerical Analysis
3808:. An asymptotic distribution allows
4926:Paris, R. B.; Kaminsky, D. (2001),
4858:Estrada, R.; Kanwal, R. P. (2002),
4779:Bruijn, Nicolaas Govert de (1981).
4646:Asymptotic computational complexity
4589:N.A.: I asked for 1%, not for 20%.
1649:. In view of the definition of the
774:{\displaystyle \log(f)\sim \log(g)}
13:
4200:
4167:
3843:This is based on the notion of an
3572:
3413:
3382:
3316:
3299:and integrating both sides yields
3172:
3091:
2991:
2897:
2839:
2761:
2651:
599:{\displaystyle f(x)=g(x)(1+o(1)).}
381:
343:
93:becomes insignificant compared to
14:
4989:
4940:
3888:{\displaystyle y={\frac {1}{x}},}
3814:to range without bound, that is,
3122:
2598:Examples of asymptotic expansions
2584:{\displaystyle g_{k}=o(g_{k-1}).}
2004:. In that case, some authors may
1567:{\displaystyle f-g_{1}\sim g_{2}}
4657: – Concept in number theory
4480:{\displaystyle 0<f(100)<1}
2218:{\displaystyle g_{k}=o(g_{k-1})}
1962:{\displaystyle g_{k+1}=o(g_{k})}
1669:symbol, the last equation means
4557:
4414:
4286:
4190:
3905:
2164:
899:Examples of asymptotic formulas
719:{\displaystyle f^{r}\sim g^{r}}
328:
197:. Then the theorem states that
193:that are less than or equal to
4837:Asymptotic Methods in Analysis
4781:Asymptotic methods in analysis
4772:
4739:
4715:
4664:Asymptotic theory (statistics)
4570:
4558:
4534:
4514:
4508:
4501:
4468:
4462:
4427:
4415:
4391:
4371:
4365:
4358:
4306:
4287:
4263:
4243:
4237:
4230:
4203:
4197:
4191:
4187:
4171:
4144:
4138:
4092:
4086:
3744:
3735:
3716:
3710:
3647:
3633:
3241:
3233:
3094:
3088:
3082:
3067:
3050:
3033:
3018:
3006:
2996:
2963:
2957:
2900:
2894:
2888:
2857:
2847:
2817:
2811:
2764:
2758:
2752:
2666:
2654:
2575:
2556:
2517:
2498:
2489:
2476:
2437:
2424:
2325:
2306:
2212:
2193:
2127:
2114:
2105:
2073:
1956:
1943:
1798:
1766:
1736:
1723:
1714:
1682:
1350:
1344:
1339:
1333:
1242:
1236:
1231:
1225:
1115:
1109:
1011:
1005:
768:
762:
750:
744:
590:
587:
581:
569:
566:
560:
551:
545:
412:
406:
398:
392:
378:
346:
340:
329:
325:
319:
310:
304:
294:, we define a binary relation
213:
207:
1:
4890:American Mathematical Society
4800:
4763:Practical Applied Mathematics
4624:N.A.: !!! . . . !
1492:In symbols, it means we have
626:
267:
86:becomes very large, the term
3802:, for some positive integer
2594:approaches the limit value.
1521:{\displaystyle f\sim g_{1},}
803:{\displaystyle \lim g\neq 1}
517:is zero infinitely often as
42:, is a method of describing
7:
4885:Applied Asymptotic Analysis
4728:Encyclopedia of Mathematics
4632:
3498:, may be recognized as the
885:{\displaystyle f/a\sim g/b}
448:on the set of functions of
10:
4994:
4934:Cambridge University Press
4768:Cambridge University Press
4746:Estrada & Kanwal (2002
4688:Method of dominant balance
4050:method of steepest descent
3764:
1914:takes its full meaning if
1451:
983:, the partition function,
272:Formally, given functions
18:
462:asymptotically equivalent
163:prime number theorem
112:asymptotically equivalent
4709:
4022:boundary layer equations
3920:probability distribution
3292:{\displaystyle e^{-w/t}}
3254:{\displaystyle |w|<1}
2060:to denote the statement
972:Stirling's approximation
4026:Navier-Stokes equations
3969:mathematical statistics
3778:asymptotic distribution
3773:mathematical statistics
3767:Asymptotic distribution
3761:Asymptotic distribution
3218:{\displaystyle w\neq 1}
979:For a positive integer
676:{\displaystyle a\sim b}
650:{\displaystyle f\sim g}
623:of the limiting value.
175:prime-counting function
64:becomes very large. If
4906:Murray, J. D. (1984),
4882:Miller, P. D. (2006),
4630:
4606:
4587:
4580:
4481:
4444:
4437:
4338:
4324:N.A.: But my value of
4316:
4210:
4119:
4099:
4018:mathematical modelling
3984:analysis of algorithms
3889:
3751:
3691:
3690:{\displaystyle x=-1/t}
3654:
3616:. However, by keeping
3604:
3576:
3492:
3452:
3386:
3293:
3255:
3219:
3191:
3176:
3101:
2995:
2907:
2843:
2771:
2585:
2527:
2447:
2332:
2219:
2157:
2137:
2054:
1994:
1963:
1908:
1835:
1834:{\displaystyle g_{k}.}
1805:
1743:
1663:
1639:
1568:
1522:
1438:
1194:
1081:The Airy function, Ai(
1070:
964:
886:
843:
804:
775:
720:
677:
651:
600:
428:
353:
258:analysis of algorithms
244:
4723:"Asymptotic equality"
4607:
4581:
4492:
4482:
4438:
4349:
4339:
4317:
4211:
4120:
4100:
4074:
3995:statistical mechanics
3912:mathematical sciences
3890:
3752:
3692:
3655:
3605:
3556:
3493:
3491:{\displaystyle u=w/t}
3453:
3366:
3294:
3256:
3220:
3192:
3156:
3102:
2975:
2908:
2823:
2772:
2586:
2528:
2448:
2333:
2220:
2158:
2156:{\displaystyle \sim }
2138:
2055:
1995:
1993:{\displaystyle g_{k}}
1964:
1909:
1836:
1811:is much smaller than
1806:
1744:
1664:
1662:{\displaystyle \sim }
1640:
1569:
1523:
1439:
1195:
1071:
965:
887:
844:
805:
776:
721:
678:
652:
601:
444:. The relation is an
429:
354:
245:
32:mathematical analysis
4760:Howison, S. (2005),
4596:
4497:
4450:
4354:
4328:
4226:
4132:
4109:
4105:for large values of
4098:{\displaystyle f(x)}
4080:
4062:quantum field theory
3943:approximation theory
3863:
3701:
3664:
3624:
3506:
3468:
3462:exponential integral
3303:
3265:
3229:
3203:
3132:
2921:
2785:
2780:Exponential integral
2609:
2537:
2457:
2342:
2237:
2174:
2147:
2064:
2012:
1977:
1918:
1847:
1815:
1757:
1673:
1653:
1578:
1532:
1496:
1460:asymptotic expansion
1454:Asymptotic expansion
1448:Asymptotic expansion
1208:
1100:
999:
910:
854:
815:
785:
735:
690:
661:
635:
619:is not zero in some
539:
446:equivalence relation
367:
298:
201:
141:, which is read as "
4978:Mathematical series
4973:Asymptotic analysis
4948:Asymptotic Analysis
4909:Asymptotic Analysis
4807:Balser, W. (1994),
4046:saddle-point method
3993:, an example being
3954:applied mathematics
3417:
3320:
2233:−1; by subtracting
1343:
1235:
361:de Bruijn 1981
36:asymptotic analysis
4842:Dover Publications
4682:Leading-order term
4655:Asymptotic density
4602:
4576:
4477:
4433:
4334:
4312:
4206:
4115:
4095:
3973:probability theory
3885:
3747:
3687:
3650:
3600:
3488:
3448:
3403:
3306:
3289:
3251:
3215:
3187:
3097:
2903:
2767:
2581:
2523:
2443:
2328:
2215:
2153:
2133:
2050:
1990:
1973:, which means the
1959:
1904:
1831:
1801:
1739:
1659:
1635:
1564:
1518:
1434:
1432:
1323:
1215:
1190:
1066:
977:Partition function
960:
882:
839:
800:
771:
716:
673:
647:
596:
424:
385:
349:
240:
4790:978-0-486-64221-5
4605:{\displaystyle x}
4337:{\displaystyle x}
4118:{\displaystyle x}
4002:accident analysis
3958:numerical methods
3924:sample statistics
3880:
3547:
3525:
3354:
3339:
3261:. Multiplying by
3151:
3081:
3077:
2929:
2887:
2883:
2751:
2741:
2716:
2691:
2649:
2646:
2165:§ Definition
1751:little o notation
1421:
1375:
1374:
1310:
1267:
1266:
1188:
1167:
1153:
1139:
1062:
1061:
1037:
1034:
948:
933:
726:, for every real
523:little-o notation
416:
370:
335:
235:
189:is the number of
152:is asymptotic to
4985:
4936:
4922:
4902:
4878:
4854:
4832:de Bruijn, N. G.
4827:
4795:
4794:
4776:
4770:
4758:
4749:
4743:
4737:
4736:
4719:
4704:
4660:
4651:
4611:
4609:
4608:
4603:
4585:
4583:
4582:
4577:
4556:
4555:
4537:
4532:
4531:
4504:
4486:
4484:
4483:
4478:
4442:
4440:
4439:
4434:
4413:
4412:
4394:
4389:
4388:
4361:
4343:
4341:
4340:
4335:
4321:
4319:
4318:
4313:
4305:
4304:
4285:
4284:
4266:
4261:
4260:
4233:
4215:
4213:
4212:
4207:
4186:
4185:
4170:
4162:
4161:
4124:
4122:
4121:
4116:
4104:
4102:
4101:
4096:
4054:Edgeworth series
4042:Laplace's method
4031:
3991:physical systems
3989:The behavior of
3980:computer science
3928:likelihood ratio
3894:
3892:
3891:
3886:
3881:
3873:
3839:
3833:
3819:
3813:
3807:
3801:
3791:
3756:
3754:
3753:
3748:
3734:
3733:
3697:and noting that
3696:
3694:
3693:
3688:
3683:
3659:
3657:
3656:
3651:
3643:
3609:
3607:
3606:
3601:
3599:
3598:
3575:
3570:
3552:
3548:
3540:
3528:
3527:
3526:
3518:
3497:
3495:
3494:
3489:
3484:
3457:
3455:
3454:
3449:
3440:
3439:
3430:
3429:
3416:
3411:
3402:
3401:
3385:
3380:
3355:
3353:
3342:
3341:
3340:
3332:
3322:
3319:
3314:
3298:
3296:
3295:
3290:
3288:
3287:
3283:
3260:
3258:
3257:
3252:
3244:
3236:
3224:
3222:
3221:
3216:
3196:
3194:
3193:
3188:
3186:
3185:
3175:
3170:
3152:
3150:
3136:
3116:double factorial
3113:
3106:
3104:
3103:
3098:
3079:
3078:
3076:
3075:
3074:
3065:
3064:
3042:
3016:
3014:
3013:
2994:
2989:
2950:
2949:
2948:
2947:
2930:
2925:
2912:
2910:
2909:
2904:
2885:
2884:
2882:
2881:
2872:
2865:
2864:
2845:
2842:
2837:
2810:
2809:
2800:
2799:
2776:
2774:
2773:
2768:
2749:
2742:
2740:
2739:
2738:
2722:
2717:
2715:
2714:
2713:
2697:
2692:
2690:
2679:
2650:
2648:
2647:
2636:
2634:
2633:
2623:
2622:
2613:
2590:
2588:
2587:
2582:
2574:
2573:
2549:
2548:
2532:
2530:
2529:
2524:
2516:
2515:
2488:
2487:
2469:
2468:
2452:
2450:
2449:
2444:
2436:
2435:
2417:
2416:
2404:
2403:
2385:
2384:
2360:
2359:
2337:
2335:
2334:
2329:
2324:
2323:
2299:
2298:
2280:
2279:
2255:
2254:
2224:
2222:
2221:
2216:
2211:
2210:
2186:
2185:
2162:
2160:
2159:
2154:
2142:
2140:
2139:
2134:
2126:
2125:
2104:
2103:
2085:
2084:
2059:
2057:
2056:
2051:
2049:
2048:
2030:
2029:
2002:asymptotic scale
1999:
1997:
1996:
1991:
1989:
1988:
1968:
1966:
1965:
1960:
1955:
1954:
1936:
1935:
1913:
1911:
1910:
1905:
1903:
1902:
1890:
1889:
1865:
1864:
1840:
1838:
1837:
1832:
1827:
1826:
1810:
1808:
1807:
1802:
1797:
1796:
1778:
1777:
1748:
1746:
1745:
1740:
1735:
1734:
1713:
1712:
1694:
1693:
1668:
1666:
1665:
1660:
1644:
1642:
1641:
1636:
1634:
1633:
1621:
1620:
1596:
1595:
1573:
1571:
1570:
1565:
1563:
1562:
1550:
1549:
1527:
1525:
1524:
1519:
1514:
1513:
1488:
1484:
1472:
1443:
1441:
1440:
1435:
1433:
1429:
1428:
1427:
1423:
1422:
1417:
1400:
1376:
1373:
1362:
1361:
1342:
1331:
1318:
1317:
1316:
1312:
1311:
1306:
1289:
1268:
1265:
1254:
1253:
1234:
1223:
1203:Hankel functions
1199:
1197:
1196:
1191:
1189:
1187:
1186:
1185:
1181:
1168:
1163:
1157:
1156:
1155:
1154:
1146:
1140:
1132:
1122:
1095:
1075:
1073:
1072:
1067:
1065:
1064:
1063:
1057:
1049:
1048:
1038:
1036:
1035:
1030:
1018:
969:
967:
966:
961:
959:
958:
953:
949:
941:
934:
923:
891:
889:
888:
883:
878:
864:
848:
846:
845:
840:
809:
807:
806:
801:
780:
778:
777:
772:
729:
725:
723:
722:
717:
715:
714:
702:
701:
682:
680:
679:
674:
656:
654:
653:
648:
618:
605:
603:
602:
597:
534:
520:
516:
502:
500:
492:
485:
475:
471:
459:
455:
452:; the functions
451:
439:
433:
431:
430:
425:
417:
415:
401:
387:
384:
359:if and only if (
358:
356:
355:
350:
336:
333:
293:
282:
254:computer science
249:
247:
246:
241:
236:
234:
220:
196:
188:
172:
157:
151:
140:
126:
119:
109:
98:
92:
85:
81:
63:
59:
38:, also known as
4993:
4992:
4988:
4987:
4986:
4984:
4983:
4982:
4963:
4962:
4943:
4920:
4900:
4876:
4852:
4825:
4815:Springer-Verlag
4803:
4798:
4791:
4777:
4773:
4759:
4752:
4744:
4740:
4721:
4720:
4716:
4712:
4707:
4702:
4658:
4649:
4635:
4597:
4594:
4593:
4548:
4544:
4533:
4524:
4520:
4500:
4498:
4495:
4494:
4451:
4448:
4447:
4405:
4401:
4390:
4381:
4377:
4357:
4355:
4352:
4351:
4329:
4326:
4325:
4300:
4296:
4277:
4273:
4262:
4253:
4249:
4229:
4227:
4224:
4223:
4178:
4174:
4166:
4154:
4150:
4133:
4130:
4129:
4110:
4107:
4106:
4081:
4078:
4077:
4070:
4029:
3960:to approximate
3908:
3872:
3864:
3861:
3860:
3835:
3832:
3824:
3815:
3809:
3803:
3793:
3790:
3782:
3769:
3763:
3729:
3725:
3702:
3699:
3698:
3679:
3665:
3662:
3661:
3660:. Substituting
3639:
3625:
3622:
3621:
3588:
3584:
3571:
3560:
3539:
3535:
3517:
3513:
3509:
3507:
3504:
3503:
3480:
3469:
3466:
3465:
3435:
3431:
3422:
3418:
3412:
3407:
3391:
3387:
3381:
3370:
3343:
3331:
3327:
3323:
3321:
3315:
3310:
3304:
3301:
3300:
3279:
3272:
3268:
3266:
3263:
3262:
3240:
3232:
3230:
3227:
3226:
3204:
3201:
3200:
3181:
3177:
3171:
3160:
3140:
3135:
3133:
3130:
3129:
3125:
3108:
3070:
3066:
3060:
3056:
3043:
3017:
3015:
3009:
3005:
2990:
2979:
2943:
2939:
2938:
2934:
2924:
2922:
2919:
2918:
2877:
2873:
2860:
2856:
2846:
2844:
2838:
2827:
2805:
2801:
2795:
2791:
2786:
2783:
2782:
2734:
2730:
2726:
2721:
2709:
2705:
2701:
2696:
2683:
2678:
2635:
2629:
2625:
2624:
2618:
2614:
2612:
2610:
2607:
2606:
2600:
2563:
2559:
2544:
2540:
2538:
2535:
2534:
2505:
2501:
2483:
2479:
2464:
2460:
2458:
2455:
2454:
2431:
2427:
2412:
2408:
2393:
2389:
2374:
2370:
2355:
2351:
2343:
2340:
2339:
2313:
2309:
2288:
2284:
2269:
2265:
2250:
2246:
2238:
2235:
2234:
2200:
2196:
2181:
2177:
2175:
2172:
2171:
2148:
2145:
2144:
2121:
2117:
2099:
2095:
2080:
2076:
2065:
2062:
2061:
2044:
2040:
2025:
2021:
2013:
2010:
2009:
1984:
1980:
1978:
1975:
1974:
1950:
1946:
1925:
1921:
1919:
1916:
1915:
1898:
1894:
1879:
1875:
1860:
1856:
1848:
1845:
1844:
1822:
1818:
1816:
1813:
1812:
1792:
1788:
1773:
1769:
1758:
1755:
1754:
1730:
1726:
1708:
1704:
1689:
1685:
1674:
1671:
1670:
1654:
1651:
1650:
1645:for each fixed
1629:
1625:
1610:
1606:
1591:
1587:
1579:
1576:
1575:
1558:
1554:
1545:
1541:
1533:
1530:
1529:
1509:
1505:
1497:
1494:
1493:
1486:
1482:
1463:
1456:
1450:
1431:
1430:
1401:
1399:
1392:
1388:
1381:
1377:
1366:
1360:
1353:
1332:
1327:
1320:
1319:
1290:
1288:
1281:
1277:
1273:
1269:
1258:
1252:
1245:
1224:
1219:
1211:
1209:
1206:
1205:
1177:
1173:
1169:
1162:
1158:
1145:
1141:
1131:
1127:
1123:
1121:
1101:
1098:
1097:
1086:
1050:
1047:
1043:
1039:
1029:
1022:
1017:
1000:
997:
996:
954:
940:
936:
935:
922:
911:
908:
907:
901:
874:
860:
855:
852:
851:
816:
813:
812:
786:
783:
782:
736:
733:
732:
727:
710:
706:
697:
693:
691:
688:
687:
662:
659:
658:
636:
633:
632:
629:
609:
540:
537:
536:
535:if and only if
526:
518:
507:
496:
494:
487:
480:
473:
469:
460:are said to be
457:
453:
449:
437:
402:
388:
386:
374:
368:
365:
364:
332:
299:
296:
295:
284:
273:
270:
256:as part of the
224:
219:
202:
199:
198:
194:
182:
166:
153:
142:
128:
121:
115:
110:is said to be "
100:
99:. The function
94:
87:
83:
65:
61:
50:
28:
17:
12:
11:
5:
4991:
4981:
4980:
4975:
4961:
4960:
4955:
4942:
4941:External links
4939:
4938:
4937:
4923:
4918:
4903:
4898:
4879:
4874:
4855:
4850:
4828:
4823:
4802:
4799:
4797:
4796:
4789:
4771:
4750:
4738:
4713:
4711:
4708:
4706:
4705:
4699:Watson's lemma
4696:
4691:
4685:
4679:
4676:Big O notation
4673:
4667:
4661:
4652:
4643:
4636:
4634:
4631:
4601:
4575:
4572:
4569:
4566:
4563:
4560:
4554:
4551:
4547:
4543:
4540:
4536:
4530:
4527:
4523:
4519:
4516:
4513:
4510:
4507:
4503:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4432:
4429:
4426:
4423:
4420:
4417:
4411:
4408:
4404:
4400:
4397:
4393:
4387:
4384:
4380:
4376:
4373:
4370:
4367:
4364:
4360:
4333:
4311:
4308:
4303:
4299:
4295:
4292:
4289:
4283:
4280:
4276:
4272:
4269:
4265:
4259:
4256:
4252:
4248:
4245:
4242:
4239:
4236:
4232:
4205:
4202:
4199:
4196:
4193:
4189:
4184:
4181:
4177:
4173:
4169:
4165:
4160:
4157:
4153:
4149:
4146:
4143:
4140:
4137:
4114:
4094:
4091:
4088:
4085:
4069:
4066:
4058:Feynman graphs
4034:nondimensional
4024:from the full
4006:
4005:
3998:
3987:
3976:
3965:
3935:expected value
3926:, such as the
3907:
3904:
3884:
3879:
3876:
3871:
3868:
3828:
3786:
3765:Main article:
3762:
3759:
3746:
3743:
3740:
3737:
3732:
3728:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3686:
3682:
3678:
3675:
3672:
3669:
3649:
3646:
3642:
3638:
3635:
3632:
3629:
3597:
3594:
3591:
3587:
3582:
3579:
3574:
3569:
3566:
3563:
3559:
3555:
3551:
3546:
3543:
3538:
3534:
3531:
3524:
3521:
3516:
3512:
3500:gamma function
3487:
3483:
3479:
3476:
3473:
3447:
3444:
3438:
3434:
3428:
3425:
3421:
3415:
3410:
3406:
3400:
3397:
3394:
3390:
3384:
3379:
3376:
3373:
3369:
3365:
3362:
3359:
3352:
3349:
3346:
3338:
3335:
3330:
3326:
3318:
3313:
3309:
3286:
3282:
3278:
3275:
3271:
3250:
3247:
3243:
3239:
3235:
3214:
3211:
3208:
3184:
3180:
3174:
3169:
3166:
3163:
3159:
3155:
3149:
3146:
3143:
3139:
3124:
3123:Worked example
3121:
3120:
3119:
3096:
3093:
3090:
3087:
3084:
3073:
3069:
3063:
3059:
3055:
3052:
3049:
3046:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3012:
3008:
3004:
3001:
2998:
2993:
2988:
2985:
2982:
2978:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2946:
2942:
2937:
2933:
2928:
2916:Error function
2913:
2902:
2899:
2896:
2893:
2890:
2880:
2876:
2871:
2868:
2863:
2859:
2855:
2852:
2849:
2841:
2836:
2833:
2830:
2826:
2822:
2819:
2816:
2813:
2808:
2804:
2798:
2794:
2790:
2777:
2766:
2763:
2760:
2757:
2754:
2748:
2745:
2737:
2733:
2729:
2725:
2720:
2712:
2708:
2704:
2700:
2695:
2689:
2686:
2682:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2645:
2642:
2639:
2632:
2628:
2621:
2617:
2604:Gamma function
2599:
2596:
2580:
2577:
2572:
2569:
2566:
2562:
2558:
2555:
2552:
2547:
2543:
2522:
2519:
2514:
2511:
2508:
2504:
2500:
2497:
2494:
2491:
2486:
2482:
2478:
2475:
2472:
2467:
2463:
2442:
2439:
2434:
2430:
2426:
2423:
2420:
2415:
2411:
2407:
2402:
2399:
2396:
2392:
2388:
2383:
2380:
2377:
2373:
2369:
2366:
2363:
2358:
2354:
2350:
2347:
2327:
2322:
2319:
2316:
2312:
2308:
2305:
2302:
2297:
2294:
2291:
2287:
2283:
2278:
2275:
2272:
2268:
2264:
2261:
2258:
2253:
2249:
2245:
2242:
2214:
2209:
2206:
2203:
2199:
2195:
2192:
2189:
2184:
2180:
2152:
2132:
2129:
2124:
2120:
2116:
2113:
2110:
2107:
2102:
2098:
2094:
2091:
2088:
2083:
2079:
2075:
2072:
2069:
2047:
2043:
2039:
2036:
2033:
2028:
2024:
2020:
2017:
1987:
1983:
1958:
1953:
1949:
1945:
1942:
1939:
1934:
1931:
1928:
1924:
1901:
1897:
1893:
1888:
1885:
1882:
1878:
1874:
1871:
1868:
1863:
1859:
1855:
1852:
1830:
1825:
1821:
1800:
1795:
1791:
1787:
1784:
1781:
1776:
1772:
1768:
1765:
1762:
1738:
1733:
1729:
1725:
1722:
1719:
1716:
1711:
1707:
1703:
1700:
1697:
1692:
1688:
1684:
1681:
1678:
1658:
1632:
1628:
1624:
1619:
1616:
1613:
1609:
1605:
1602:
1599:
1594:
1590:
1586:
1583:
1561:
1557:
1553:
1548:
1544:
1540:
1537:
1517:
1512:
1508:
1504:
1501:
1462:of a function
1452:Main article:
1449:
1446:
1445:
1444:
1426:
1420:
1416:
1413:
1410:
1407:
1404:
1398:
1395:
1391:
1387:
1384:
1380:
1372:
1369:
1365:
1359:
1356:
1354:
1352:
1349:
1346:
1341:
1338:
1335:
1330:
1326:
1322:
1321:
1315:
1309:
1305:
1302:
1299:
1296:
1293:
1287:
1284:
1280:
1276:
1272:
1264:
1261:
1257:
1251:
1248:
1246:
1244:
1241:
1238:
1233:
1230:
1227:
1222:
1218:
1214:
1213:
1200:
1184:
1180:
1176:
1172:
1166:
1161:
1152:
1149:
1144:
1138:
1135:
1130:
1126:
1120:
1117:
1114:
1111:
1108:
1105:
1076:
1060:
1056:
1053:
1046:
1042:
1033:
1028:
1025:
1021:
1016:
1013:
1010:
1007:
1004:
974:
957:
952:
947:
944:
939:
932:
929:
926:
921:
918:
915:
900:
897:
893:
892:
881:
877:
873:
870:
867:
863:
859:
849:
838:
835:
832:
829:
826:
823:
820:
810:
799:
796:
793:
790:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
730:
713:
709:
705:
700:
696:
672:
669:
666:
646:
643:
640:
628:
625:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
550:
547:
544:
423:
420:
414:
411:
408:
405:
400:
397:
394:
391:
383:
380:
377:
373:
348:
345:
342:
339:
331:
327:
324:
321:
318:
315:
312:
309:
306:
303:
269:
266:
262:big O notation
239:
233:
230:
227:
223:
218:
215:
212:
209:
206:
15:
9:
6:
4:
3:
2:
4990:
4979:
4976:
4974:
4971:
4970:
4968:
4959:
4956:
4954:
4950:
4949:
4945:
4944:
4935:
4931:
4930:
4924:
4921:
4919:9781461211228
4915:
4911:
4910:
4904:
4901:
4899:9780821840788
4895:
4891:
4887:
4886:
4880:
4877:
4875:9780817681302
4871:
4867:
4863:
4862:
4856:
4853:
4851:9780486642215
4847:
4843:
4839:
4838:
4833:
4829:
4826:
4824:9783540485940
4820:
4816:
4812:
4811:
4805:
4804:
4792:
4786:
4782:
4775:
4769:
4765:
4764:
4757:
4755:
4747:
4742:
4734:
4730:
4729:
4724:
4718:
4714:
4700:
4697:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4670:Asymptotology
4668:
4665:
4662:
4656:
4653:
4647:
4644:
4641:
4638:
4637:
4629:
4625:
4622:
4619:
4616:
4613:
4599:
4590:
4586:
4573:
4567:
4564:
4561:
4552:
4549:
4545:
4541:
4538:
4528:
4525:
4521:
4517:
4511:
4505:
4491:
4488:
4474:
4471:
4465:
4459:
4456:
4453:
4443:
4430:
4424:
4421:
4418:
4409:
4406:
4402:
4398:
4395:
4385:
4382:
4378:
4374:
4368:
4362:
4348:
4345:
4344:is only 100.
4331:
4322:
4309:
4301:
4297:
4293:
4290:
4281:
4278:
4274:
4270:
4267:
4257:
4254:
4250:
4246:
4240:
4234:
4220:
4217:
4194:
4182:
4179:
4175:
4163:
4158:
4155:
4151:
4147:
4141:
4135:
4126:
4112:
4089:
4083:
4073:
4065:
4063:
4059:
4055:
4051:
4047:
4043:
4038:
4035:
4027:
4023:
4019:
4015:
4011:
4003:
3999:
3996:
3992:
3988:
3985:
3981:
3977:
3974:
3970:
3966:
3963:
3959:
3955:
3951:
3950:
3949:
3946:
3944:
3940:
3936:
3932:
3929:
3925:
3921:
3917:
3913:
3903:
3901:
3897:
3882:
3877:
3874:
3869:
3866:
3857:
3852:
3850:
3846:
3841:
3838:
3831:
3827:
3821:
3820:is infinite.
3818:
3812:
3806:
3800:
3796:
3789:
3785:
3780:
3779:
3774:
3768:
3758:
3741:
3738:
3730:
3726:
3722:
3719:
3713:
3707:
3704:
3684:
3680:
3676:
3673:
3670:
3667:
3644:
3640:
3636:
3630:
3627:
3619:
3615:
3610:
3595:
3592:
3589:
3585:
3580:
3577:
3567:
3564:
3561:
3557:
3553:
3549:
3544:
3541:
3536:
3532:
3529:
3522:
3519:
3514:
3510:
3501:
3485:
3481:
3477:
3474:
3471:
3463:
3458:
3445:
3442:
3436:
3432:
3426:
3423:
3419:
3408:
3404:
3398:
3395:
3392:
3388:
3377:
3374:
3371:
3367:
3363:
3360:
3357:
3350:
3347:
3344:
3336:
3333:
3328:
3324:
3311:
3307:
3284:
3280:
3276:
3273:
3269:
3248:
3245:
3237:
3212:
3209:
3206:
3197:
3182:
3178:
3167:
3164:
3161:
3157:
3153:
3147:
3144:
3141:
3137:
3117:
3111:
3085:
3071:
3061:
3057:
3053:
3047:
3044:
3039:
3036:
3030:
3027:
3024:
3021:
3010:
3002:
2999:
2986:
2983:
2980:
2976:
2972:
2969:
2966:
2960:
2954:
2951:
2944:
2940:
2935:
2931:
2926:
2917:
2914:
2891:
2878:
2874:
2869:
2866:
2861:
2853:
2850:
2834:
2831:
2828:
2824:
2820:
2814:
2806:
2802:
2796:
2792:
2788:
2781:
2778:
2755:
2746:
2743:
2735:
2731:
2727:
2723:
2718:
2710:
2706:
2702:
2698:
2693:
2687:
2684:
2680:
2675:
2672:
2669:
2663:
2660:
2657:
2643:
2640:
2637:
2630:
2626:
2619:
2615:
2605:
2602:
2601:
2595:
2591:
2578:
2570:
2567:
2564:
2560:
2553:
2550:
2545:
2541:
2520:
2512:
2509:
2506:
2502:
2495:
2492:
2484:
2480:
2473:
2470:
2465:
2461:
2440:
2432:
2428:
2421:
2418:
2413:
2409:
2405:
2400:
2397:
2394:
2390:
2386:
2381:
2378:
2375:
2371:
2367:
2364:
2361:
2356:
2352:
2348:
2345:
2320:
2317:
2314:
2310:
2303:
2300:
2295:
2292:
2289:
2285:
2281:
2276:
2273:
2270:
2266:
2262:
2259:
2256:
2251:
2247:
2243:
2240:
2232:
2228:
2207:
2204:
2201:
2197:
2190:
2187:
2182:
2178:
2168:
2166:
2150:
2130:
2122:
2118:
2111:
2108:
2100:
2096:
2092:
2089:
2086:
2081:
2077:
2070:
2067:
2045:
2041:
2037:
2034:
2031:
2026:
2022:
2018:
2015:
2007:
2003:
1985:
1981:
1972:
1951:
1947:
1940:
1937:
1932:
1929:
1926:
1922:
1899:
1895:
1891:
1886:
1883:
1880:
1876:
1872:
1869:
1866:
1861:
1857:
1853:
1850:
1843:The relation
1841:
1828:
1823:
1819:
1793:
1789:
1785:
1782:
1779:
1774:
1770:
1763:
1760:
1752:
1731:
1727:
1720:
1717:
1709:
1705:
1701:
1698:
1695:
1690:
1686:
1679:
1676:
1656:
1648:
1630:
1626:
1622:
1617:
1614:
1611:
1607:
1603:
1600:
1597:
1592:
1588:
1584:
1581:
1559:
1555:
1551:
1546:
1542:
1538:
1535:
1515:
1510:
1506:
1502:
1499:
1490:
1480:
1476:
1470:
1466:
1461:
1455:
1424:
1418:
1414:
1411:
1408:
1405:
1402:
1396:
1393:
1389:
1385:
1382:
1378:
1370:
1367:
1363:
1357:
1355:
1347:
1336:
1328:
1324:
1313:
1307:
1303:
1300:
1297:
1294:
1291:
1285:
1282:
1278:
1274:
1270:
1262:
1259:
1255:
1249:
1247:
1239:
1228:
1220:
1216:
1204:
1201:
1182:
1178:
1174:
1170:
1164:
1159:
1150:
1147:
1142:
1136:
1133:
1128:
1124:
1118:
1112:
1106:
1103:
1093:
1089:
1084:
1080:
1079:Airy function
1077:
1058:
1054:
1051:
1044:
1040:
1031:
1026:
1023:
1019:
1014:
1008:
1002:
994:
990:
986:
982:
978:
975:
973:
955:
950:
945:
942:
937:
930:
927:
924:
919:
916:
913:
906:
903:
902:
896:
879:
875:
871:
868:
865:
861:
857:
850:
836:
833:
830:
827:
824:
821:
818:
811:
797:
794:
791:
765:
759:
756:
753:
747:
741:
738:
731:
711:
707:
703:
698:
694:
686:
685:
684:
670:
667:
664:
644:
641:
638:
624:
622:
621:neighbourhood
616:
612:
606:
593:
584:
578:
575:
572:
563:
557:
554:
548:
542:
533:
529:
524:
514:
510:
504:
499:
490:
483:
477:
467:
463:
447:
443:
434:
421:
418:
409:
403:
395:
389:
375:
362:
337:
322:
316:
313:
307:
301:
291:
287:
280:
276:
265:
263:
259:
255:
250:
237:
231:
228:
225:
221:
216:
210:
204:
192:
191:prime numbers
186:
180:
176:
170:
164:
159:
156:
149:
145:
139:
135:
131:
124:
118:
113:
107:
103:
97:
91:
80:
76:
72:
68:
57:
53:
47:
45:
41:
37:
33:
26:
22:
4947:
4928:
4912:, Springer,
4908:
4884:
4860:
4836:
4809:
4780:
4774:
4761:
4741:
4726:
4717:
4626:
4623:
4620:
4617:
4614:
4591:
4588:
4493:
4489:
4445:
4350:
4346:
4323:
4221:
4218:
4127:
4075:
4071:
4039:
4007:
3947:
3909:
3906:Applications
3899:
3895:
3853:
3848:
3842:
3836:
3829:
3825:
3822:
3816:
3810:
3804:
3798:
3794:
3787:
3783:
3776:
3770:
3617:
3613:
3611:
3459:
3198:
3126:
3109:
2592:
2230:
2226:
2169:
1970:
1842:
1646:
1491:
1479:partial sums
1468:
1464:
1457:
1091:
1087:
1082:
992:
988:
984:
980:
894:
630:
614:
610:
607:
531:
527:
512:
508:
505:
497:
488:
481:
478:
461:
435:
289:
285:
278:
274:
271:
251:
184:
168:
160:
154:
147:
143:
137:
133:
129:
122:
116:
111:
105:
101:
95:
89:
78:
74:
70:
66:
55:
51:
48:
39:
35:
29:
3902:increases.
3834:go to 0 as
436:The symbol
173:denote the
40:asymptotics
4967:Categories
4866:Birkhäuser
4801:References
3964:solutions.
3916:statistics
3845:asymptotic
627:Properties
525:, is that
501:| → 0
268:Definition
82:, then as
46:behavior.
4953:IOS Press
4733:EMS Press
4640:Asymptote
4550:−
4526:−
4518:−
4407:−
4383:−
4375:−
4279:−
4255:−
4247:−
4201:∞
4198:→
4180:−
4156:−
3931:statistic
3856:asymptote
3849:asymptote
3739:−
3723:−
3708:
3674:−
3631:
3573:∞
3558:∑
3533:
3515:−
3424:−
3414:∞
3405:∫
3383:∞
3368:∑
3348:−
3329:−
3317:∞
3308:∫
3274:−
3210:≠
3173:∞
3158:∑
3145:−
3092:∞
3089:→
3028:−
3000:−
2992:∞
2977:∑
2967:∼
2955:
2927:π
2898:∞
2895:→
2851:−
2840:∞
2825:∑
2821:∼
2762:∞
2759:→
2747:⋯
2744:−
2719:−
2670:∼
2652:Γ
2641:π
2568:−
2510:−
2453:one gets
2398:−
2387:−
2379:−
2368:−
2365:⋯
2362:−
2349:−
2318:−
2293:−
2274:−
2263:−
2260:⋯
2257:−
2244:−
2205:−
2151:∼
2090:⋯
2071:−
2035:⋯
2019:∼
2006:abusively
1892:∼
1884:−
1873:−
1870:⋯
1867:−
1854:−
1783:⋯
1764:−
1699:⋯
1680:−
1657:∼
1623:∼
1615:−
1604:−
1601:⋯
1598:−
1585:−
1552:∼
1539:−
1528:but also
1503:∼
1415:π
1412:−
1409:α
1406:π
1397:−
1383:−
1368:π
1358:∼
1329:α
1304:π
1301:−
1298:α
1295:π
1286:−
1260:π
1250:∼
1221:α
1165:π
1129:−
1119:∼
1107:
1045:π
1015:∼
970:—this is
928:π
920:∼
905:Factorial
869:∼
834:×
828:∼
822:×
795:≠
760:
754:∼
742:
704:∼
668:∼
642:∼
382:∞
379:→
344:∞
341:→
314:∼
229:
217:∼
205:π
25:Asymptote
4834:(1981),
4633:See also
4010:ordinary
3962:equation
3939:deviance
3933:and the
3797:= 1, …,
2000:form an
1969:for all
1753:, i.e.,
1088:y″
363:, §1.4)
334:as
277: (
181:), i.e.
132: (
54: (
44:limiting
21:geometry
4748:, §1.2)
4735:, 2001
4056:). The
4014:partial
3982:in the
3937:of the
3114:is the
1749:in the
440:is the
4916:
4896:
4872:
4848:
4821:
4787:
4628:reply.
4222:A.A.:
4128:A.A.:
4037:hand.
3107:where
3080:
2886:
2750:
2008:write
1477:, the
1475:series
495:|
466:domain
464:. The
165:. Let
23:, see
4710:Notes
4399:57000
3914:. In
3775:, an
2728:51840
2533:i.e.
2338:from
442:tilde
120:, as
4914:ISBN
4894:ISBN
4870:ISBN
4846:ISBN
4819:ISBN
4785:ISBN
4565:>
4539:<
4472:<
4457:<
4422:>
4396:<
4294:>
4268:<
4012:and
3971:and
3792:for
3246:<
2952:erfc
2229:and
1574:and
657:and
472:and
456:and
283:and
136:) ~
73:) =
4568:100
4466:100
4425:100
4060:in
4000:In
3978:In
3967:In
3952:In
3922:of
3854:An
3771:In
2724:139
2703:288
1458:An
1094:= 0
789:lim
781:if
757:log
739:log
631:If
491:↓ 0
484:→ 0
468:of
372:lim
158:".
125:→ ∞
114:to
77:+ 3
60:as
30:In
4969::
4932:,
4892:,
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