22:
8550:, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He, however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption." §535.
1170:
4095:
2888:
1939:
2410:
2654:
3424:
4118:
envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the inverse of sine (arcsin). (From the general pedagogy web-site. For the notation, see
7140:
3753:
6927:
6751:
2141:
3101:
1702:
7885:
3220:
8321:(1952) . "§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions".
2666:
7271:
310:
3515:
2166:
6217:
5180:
392:
6111:
5793:
5087:
3162:
5551:
2421:
5634:
5712:
6341:
705:
6947:
5464:
2941:
7985:
5001:
4106:), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it; and further fractional iterates up to the 1/64th. The functions below the (
6762:
6583:
4725:
7630:
4863:
4803:
158:
5929:
5858:
4601:
8202:
6481:
4915:
7718:
6407:
7384:
2893:
This can be carried on indefinitely, although inefficiently, as the latter terms become increasingly complicated. A more systematic procedure is outlined in the following section on
7558:
5234:
1946:
1138:. It can be visualized as the behavior of a point-cloud or dust-cloud under repeated iteration. The invariant measure is an eigenstate of the Ruelle-Frobenius-Perron operator or
6260:
6010:
5969:
4652:
4531:
63:. In this process, starting from some initial object, the result of applying a given function is fed again into the function as input, and this process is repeated.
8771:
4422:
439:
7733:
4481:
4385:
248:
8688:
4451:
1934:{\displaystyle f^{n}(x)=f^{n}(a)+(x-a)\left.{\frac {d}{dx}}f^{n}(x)\right|_{x=a}+{\frac {(x-a)^{2}}{2}}\left.{\frac {d^{2}}{dx^{2}}}f^{n}(x)\right|_{x=a}+\cdots }
1075:
Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an
1521:
If a function is bijective (and so possesses an inverse function), then negative iterates correspond to function inverses and their compositions. For example,
8057:
1116:
315:
7188:
1134:
If one considers the evolution of a density distribution, rather than that of individual point dynamics, then the limiting behavior is given by the
8606:). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log
3419:{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdots }}}=f^{n}(1)=2-(\ln 2)^{n}+{\frac {(\ln 2)^{n+1}((\ln 2)^{n}-1)}{4(\ln 2-1)}}-\cdots }
8263:
6117:
3833:. Thus, if one can solve for one iterated function system, one also has solutions for all topologically conjugate systems. For example, the
8336:
2883:{\displaystyle f^{n}(x)=x+{\frac {(x-a)^{2}}{2}}(nf''(a))+{\frac {(x-a)^{3}}{6}}\left({\frac {3}{2}}n(n-1)f''(a)^{2}+nf'''(a)\right)+\cdots }
5097:
8434:
6018:
5718:
9324:
5009:
3114:
9271:
5474:
5557:
3748:{\displaystyle f^{n}(x)=1+b^{n}(x-1)+{\frac {1}{2}}b^{n}(b^{n}-1)(x-1)^{2}+{\frac {1}{3!}}b^{n}(b^{n}-1)(b^{n}-2)(x-1)^{3}+\cdots ~,}
8822:
8798:
8755:
8654:
7894:
suffices to determine the entire flow, given this exponential realization which automatically provides the general solution to the
9126:
7135:{\displaystyle {\frac {\delta f^{N}(x)}{\delta f(y)}}=f'(f^{N-1}(x)){\frac {\delta f^{N-1}(x)}{\delta f(y)}}+\delta (f^{N-1}(x)-y)}
5644:
4173:
2405:{\displaystyle f^{n}(x)=a+(x-a)f'(a)^{n}+{\frac {(x-a)^{2}}{2}}(f''(a)f'(a)^{n-1})\left(1+f'(a)+\cdots +f'(a)^{n-1}\right)+\cdots }
6266:
8633:(xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
5237:
619:
9329:
8419:
8251:
8193:
1357:
s domain can be extended sufficiently, cf. picture. The roots chosen are normally the ones belonging to the orbit under study.
468:
4345:-th iterate. The table below lists some that do. Note that all these expressions are valid even for non-integer and negative
6530:
5245:
1079:. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an
59:
another function with itself two or several times. The process of repeatedly applying the same function is called
21:
9334:
9319:
9165:
8328:
7904:
6922:{\displaystyle \left\{b+1,\prod _{i=a}^{b}g(i)\right\}\equiv \left(\{i,x\}\rightarrow \{i+1,xg(i)\}\right)^{b-a+1}\{a,1\}}
6746:{\displaystyle \left\{b+1,\sum _{i=a}^{b}g(i)\right\}\equiv \left(\{i,x\}\rightarrow \{i+1,x+g(i)\}\right)^{b-a+1}\{a,0\}}
2649:{\displaystyle f^{n}(x)=a+(x-a)f'(a)^{n}+{\frac {(x-a)^{2}}{2}}(f''(a)f'(a)^{n-1}){\frac {f'(a)^{n}-1}{f'(a)-1}}+\cdots }
1638:
One of several methods of finding a series formula for fractional iteration, making use of a fixed point, is as follows.
6550:
4925:
4658:
7569:
4811:
4733:
1677:
belonging to the reals. This, in some ways, is the most natural extra condition to place upon the fractional iterates.
1112:
1045:
73:
5869:
7727:, then, now written as a subscript, this amounts to Lie's celebrated exponential realization of a continuous group,
5801:
4537:
9309:
8968:
4327:
1041:
6422:
4167:
1021:
9067:
8052:
8047:
8032:
6518:, respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table.
4873:
3983:
743:
8159:
8138:
8002:
7641:
6570:
6349:
4134:) is equivalent to the algorithm of the preceding section, albeit, in practice, more powerful and systematic.
4084:
3096:{\displaystyle f^{n}(x)={\frac {D}{1-C}}+\left(x-{\frac {D}{1-C}}\right)C^{n}=C^{n}x+{\frac {1-C^{n}}{1-C}}D~,}
1158:
1052:
8547:
8282:
472:
7290:
8211:
8017:
7179:
880:
500:
1145:
In general, because repeated iteration corresponds to a shift, the transfer operator, and its adjoint, the
8151:
8037:
7491:
6558:
5190:
4338:
1361:
1281:
1076:
1064:
219:
6938:
6410:
6225:
5975:
4079:
no longer needs be integer or positive, and is a continuous "time" of evolution for the full orbit: the
3809:
2415:
1056:
755:
177:
8214:, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall: 8–26 .
5937:
4609:
9229:
9014:
8872:
8142:
8134:
4489:
4159:
1080:
464:
56:
39:
28:
1346:
does not denote a unique function, just as numbers have multiple algebraic roots. A trivial root of
1142:, corresponding to an eigenvalue of 1. Smaller eigenvalues correspond to unstable, decaying states.
475:
for it, but without giving a specific reference to the work of Bürmann, which remains undiscovered.
9288:
9210:
Curtright, T. L.; Zachos, C. K. (2010). "Chaotic maps, Hamiltonian flows and holographic methods".
8072:
8027:
8022:
7997:
1037:
9245:
9219:
9146:
9030:
9004:
8898:
8862:
8765:
8682:
8233:
8225:
8062:
1570:. Fractional negative iterates are defined analogously to fractional positive ones; for example,
1511:
1087:
9314:
9267:
8818:
8794:
8751:
8702:
8650:
8354:
8332:
8115:
6534:
4391:
4143:
1139:
1135:
579:
406:
189:
7880:{\displaystyle e^{t~{\frac {\partial ~~}{\partial h(x)}}}g(x)=g(h^{-1}(h(x)+t))=g(f_{t}(x)).}
4146:, that is, a matrix whose rows or columns sum to one, then the iterated system is known as a
2136:{\displaystyle f^{n}(x)=f^{n}(a)+(x-a)f'(a)f'(f(a))f'(f^{2}(a))\cdots f'(f^{n-1}(a))+\cdots }
1103:
The ideas of attraction and repulsion generalize similarly; one may categorize iterates into
9237:
9198:
9138:
9022:
8988:
8929:
8888:
8880:
8644:
8255:
8215:
8007:
7175:
6546:
4120:
4088:
1504:
1146:
911:
424:
173:
165:
8972:
8670:
8555:
8012:
6554:
4457:
4361:
4131:
1161:
provides general insight into many iterated functions, especially those leading to chaos.
1123:
1104:
970:
575:
9241:
9111:
9026:
4430:
31:
4, because that is the smallest positive exponent that produces the identity. Below is a
9233:
9018:
8876:
8570:—Three principal notations have been used to denote, say, the square of sin
8275:
8949:
8744:
8318:
8295:
7453:
7151:
4163:
3974:
Even in the absence of a strict homeomorphism, near a fixed point, here taken to be at
1150:
966:
711:
491:
8787:
8262:. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 .
977:
problem of finding the first periodic point in an orbit, and the period of the orbit.
9303:
9292:
9249:
9150:
9034:
8992:
8902:
8237:
8042:
7482:
5861:
4127:
3967:
3776:
3426:
which, taking just the first three terms, is correct to the first decimal place when
1696:
1060:
747:
32:
8724:
7156:
Iterated functions crop up in the series expansion of combined functions, such as
4177:
4147:
3838:
1108:
8590:, though the first is least likely to be misinterpreted. In the case of sin
8327:. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, USA:
8260:
A Collection of
Examples of the Applications of the Calculus of Finite Differences
1169:
1040:
that guarantee the existence of fixed points in various situations, including the
9293:"A Primer on the Elementary Theory of Infinite Compositions of Complex Functions"
9173:
8853:
Aldrovandi, R.; Freitas, L. P. (1998). "Continuous
Iteration of Dynamical Maps".
8322:
1126:
are points that move away, and never come back even close to where they started.
1154:
48:
8546:, etc., "as he then supposed for the first time. The work of a German Analyst,
7266:{\displaystyle v(x)=\left.{\frac {\partial f^{n}(x)}{\partial n}}\right|_{n=0}}
8706:
8358:
8119:
8099:
4331:
4110:) sine are six integral iterates below it, starting with the second iterate (
4094:
583:
9203:
9186:
305:{\displaystyle f^{0}~{\stackrel {\mathrm {def} }{=}}~\operatorname {id} _{X}}
8155:
8067:
6574:
3431:
1092:
974:
60:
8934:
8917:
8220:
8197:
6521:
Some of these examples are related among themselves by simple conjugacies.
8558:'s books, to remove the chief objection to them; Peirce wrote: "cos
8169:
6212:{\displaystyle g^{-1}{\Bigl (}a^{n}g(x)+{\frac {a^{n}-1}{a-1}}b{\Bigr )}}
4330:
of a group element on a set, then the iterated function corresponds to a
3834:
895:
169:
8963:
8867:
8731:. Monografie Matematyczne. Warszawa: PWN – Polish Scientific Publishers.
1086:
When the points of the orbit converge to one or more limits, the set of
730:
In general, for arbitrary general (negative, non-integer, etc.) indices
9142:
8893:
3217:
expanded around the fixed point value of 2 is then an infinite series,
1025:
of the iterated sequence. The set of fixed points is often denoted as
605:
In general, the following identity holds for all non-negative integers
8229:
5175:{\displaystyle \alpha ={\frac {2ax+b\pm {\sqrt {(2ax+b)^{2}-16}}}{4}}}
8962:
Kimura, Tosihusa (1971). "On the
Iteration of Analytic Functions",
8884:
8554:— The use of Herschel's notation underwent a slight change in
4080:
4041:
Thus, its iteration orbit, or flow, under suitable provisions (e.g.,
387:{\displaystyle f^{n+1}~{\stackrel {\mathrm {def} }{=}}~f\circ f^{n},}
6106:{\displaystyle g^{-1}{\Bigl (}a\ g(x)+b{\Bigr )}\ (a\neq 1\vee b=0)}
1481:, and so forth, all based on the principle, mentioned earlier, that
9066:, not necessarily integer, where Ψ is the solution of the relevant
8835:
8256:"Part III. Section I. Examples of the Direct Method of Differences"
5788:{\displaystyle g^{-1}{\Bigl (}h^{n}{\bigl (}g(x){\bigr )}{\Bigr )}}
1360:
Fractional iteration of a function can be defined: for instance, a
9224:
9009:
8622:) are of frequent occurrence in analysis. The notation sin
5082:{\displaystyle {\frac {2\alpha ^{2^{n}}+2\alpha ^{-2^{n}}-b}{2a}}}
4093:
1168:
3157:{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdots }}}}
66:
For example, on the image on the right:
8462:), but he justifies his own notation by pointing out that since
5546:{\displaystyle \alpha ={\frac {a+d+{\sqrt {(a-d)^{2}+4bc}}}{2}}}
3815:
Clearly, topological conjugacy is preserved under iteration, as
5629:{\displaystyle \beta ={\frac {a+d-{\sqrt {(a-d)^{2}+4bc}}}{2}}}
8918:"Analysis of Carleman Representation of Analytical Recursions"
8526:.=c. Some years later Herschel explained that in 1813 he used
6502:
are the only cases that have a closed-form solution. Choosing
1284:
of the functional roots of the identity map. For example, for
851:
also holds, analogous to the property of exponentiation that
9163:
Brand, Louis, "A sequence defined by a difference equation,"
5707:{\displaystyle g^{-1}{\Big (}h{\bigl (}g(x){\bigr )}{\Big )}}
4239:
A nonchaotic case Schröder also illustrated with his method,
9187:"Semigroups of analytic functions and composition operators"
6336:{\displaystyle {\sqrt {a^{n}x^{2}+{\frac {a^{n}-1}{a-1}}b}}}
8711:
Encyclopédie des sciences mathématiques pures et appliquées
7209:
6941:
of an iterated function is given by the recursive formula:
1857:
1767:
1495:. This idea can be generalized so that the iteration count
700:{\displaystyle f^{m}\circ f^{n}=f^{n}\circ f^{m}=f^{m+n}~.}
4048:), amounts to the conjugate of the orbit of the monomial,
1280:
has multiple solutions, which is normally the case, as in
484:
may refer to both iteration (composition) of the function
8594:
two interpretations suggest themselves; first, sin
8203:
6573:
can be defined in terms of iterated functions. These are
6553:, which in turn anchor the study of such broad topics as
4126:
This method (perturbative determination of the principal
1642:
First determine a fixed point for the function such that
8793:. Universitext: Tracts in Mathematics. Springer-Verlag.
4073:
functional iteration has been reduced to multiplication!
9264:
Lectures on
Functional Equations and Their Applications
9047:
For explicit instance, example 2 above amounts to just
8995:(2009). "Evolution Profiles and Functional Equations".
8677:. Vol. I (new ed.). Boston, USA. p. 203.
8630:) has been widely used and is now the prevailing one.
8514: V=∫ V, we may write similarly sin.
5459:{\displaystyle {\frac {a}{c}}+{\frac {bc-ad}{c}}\left}
7907:
7736:
7644:
7572:
7494:
7293:
7191:
6950:
6765:
6586:
6425:
6352:
6269:
6228:
6120:
6021:
5978:
5940:
5872:
5804:
5721:
5647:
5560:
5477:
5248:
5193:
5100:
5012:
4928:
4876:
4814:
4736:
4661:
4612:
4540:
4492:
4460:
4433:
4394:
4364:
4355:
3518:
3223:
3117:
2944:
2669:
2424:
2169:
1949:
1705:
622:
427:
318:
251:
76:
16:
Result of repeatedly applying a mathematical function
8552:
Persistence of rival notations for inverse function.
8450:, but what is usually written thus, arc (cos.=
7980:{\displaystyle f_{t}(f_{\tau }(x))=f_{t+\tau }(x)~.}
4142:
If the function is linear and can be described by a
3512:, expand around the fixed point 1 to get the series
1165:
Fractional iterates and flows, and negative iterates
1122:
Other limiting behaviors are possible; for example,
9112:
Evolution surfaces and Schröder functional methods.
8916:Berkolaiko, G.; Rabinovich, S.; Havlin, S. (1998).
188:The formal definition of an iterated function on a
9212:Journal of Physics A: Mathematical and Theoretical
8786:
8743:
8586:. The prevailing notation at present is sin
7979:
7879:
7712:
7624:
7552:
7378:
7265:
7152:Shift operator § Functions of a real variable
7134:
6921:
6745:
6475:
6401:
6335:
6254:
6211:
6105:
6004:
5963:
5923:
5852:
5787:
5706:
5628:
5545:
5458:
5228:
5174:
5081:
4996:{\displaystyle ax^{2}+bx+{\frac {b^{2}-2b-8}{4a}}}
4995:
4909:
4857:
4797:
4719:
4646:
4595:
4525:
4475:
4445:
4416:
4379:
3747:
3418:
3156:
3095:
2882:
2648:
2404:
2135:
1933:
710:This is structurally identical to the property of
699:
433:
386:
304:
152:
25:Iterated transformations of the object on the left
6204:
6136:
6068:
6037:
5780:
5737:
5699:
5663:
4720:{\displaystyle a^{\frac {b^{n}-1}{b-1}}x^{b^{n}}}
3168:times (and possibly the interpolated values when
8198:"On a Remarkable Application of Cotes's Theorem"
7625:{\displaystyle h(x)=\int {\frac {1}{v(x)}}\,dx.}
6549:, iterated functions occur as a special case of
4858:{\displaystyle {\frac {2\alpha ^{2^{n}}-b}{2a}}}
4798:{\displaystyle ax^{2}+bx+{\frac {b^{2}-2b}{4a}}}
153:{\displaystyle L=F(K),\ M=F\circ F(K)=F^{2}(K).}
8742:Kuczma, M., Choczewski B., and Ger, R. (1990).
8454:)." He admits that some authors use cos.
5924:{\displaystyle g^{-1}{\bigl (}g(x)+nb{\bigr )}}
4071:in this expression serves as a plain exponent:
3775:are two iterated functions, and there exists a
1063:applied to an iterated fixed point is known as
8713:(in French). Vol. I. p. 195. Part I.
8446:must not be understood to signify 1/cos.
5853:{\displaystyle g^{-1}{\bigl (}g(x)+b{\bigr )}}
4596:{\displaystyle a^{n}x+{\frac {a^{n}-1}{a-1}}b}
1503:, a sort of continuous "time" of a continuous
750:. On a logarithmic scale, this reduces to the
27:On top is a clockwise rotation by 90°. It has
8313:
8311:
8309:
5916:
5888:
5845:
5820:
5773:
5754:
5692:
5673:
1510:In such cases, one refers to the system as a
507:to denote the compositional meaning, writing
8:
8770:: CS1 maint: multiple names: authors list (
6916:
6904:
6878:
6848:
6842:
6830:
6740:
6728:
6702:
6669:
6663:
6651:
4337:Most functions do not have explicit general
4162:. Well-known iterated functions include the
9266:(Dover Books on Mathematics, 2006), Ch. 6,
8058:Infinite compositions of analytic functions
6529:Iterated functions can be studied with the
6476:{\displaystyle T_{mn}=\cos(m^{n}\arccos x)}
4176:, in 1870, worked out special cases of the
4102:), in the first half-period. Half-iterate (
3468:, the series computes the inverse function
1406:can be written using the index notation as
1117:infinite compositions of analytic functions
8836:"Finding f such that f(f(x))=g(x) given g"
8687:: CS1 maint: location missing publisher (
6565:Definitions in terms of iterated functions
4910:{\displaystyle \alpha ={\frac {2ax+b}{2}}}
2938:, so the above formula terminates to just
1547:is the inverse composed with itself, i.e.
9223:
9202:
9008:
8933:
8892:
8866:
8729:Functional equations in a single variable
8219:
7950:
7925:
7912:
7906:
7856:
7807:
7748:
7741:
7735:
7713:{\displaystyle f^{n}(x)=h^{-1}(h(x)+n)~.}
7671:
7649:
7643:
7612:
7591:
7571:
7514:
7493:
7341:
7292:
7251:
7221:
7211:
7190:
7102:
7048:
7038:
7014:
6961:
6951:
6949:
6886:
6798:
6787:
6764:
6710:
6619:
6608:
6585:
6455:
6430:
6424:
6402:{\displaystyle T_{m}(x)=\cos(m\arccos x)}
6357:
6351:
6302:
6295:
6286:
6276:
6270:
6268:
6238:
6229:
6227:
6203:
6202:
6173:
6166:
6145:
6135:
6134:
6125:
6119:
6067:
6066:
6036:
6035:
6026:
6020:
5985:
5979:
5977:
5947:
5941:
5939:
5915:
5914:
5887:
5886:
5877:
5871:
5844:
5843:
5819:
5818:
5809:
5803:
5779:
5778:
5772:
5771:
5753:
5752:
5746:
5736:
5735:
5726:
5720:
5698:
5697:
5691:
5690:
5672:
5671:
5662:
5661:
5652:
5646:
5600:
5582:
5567:
5559:
5517:
5499:
5484:
5476:
5443:
5406:
5364:
5321:
5290:
5262:
5249:
5247:
5194:
5192:
5152:
5128:
5107:
5099:
5054:
5046:
5028:
5023:
5013:
5011:
4961:
4954:
4936:
4927:
4883:
4875:
4830:
4825:
4815:
4813:
4769:
4762:
4744:
4735:
4709:
4704:
4673:
4666:
4660:
4620:
4611:
4564:
4557:
4545:
4539:
4491:
4459:
4432:
4399:
4393:
4363:
3727:
3696:
3674:
3661:
3642:
3633:
3602:
3589:
3575:
3551:
3523:
3517:
3366:
3335:
3316:
3307:
3267:
3250:
3243:
3241:
3234:
3232:
3225:
3222:
3144:
3137:
3135:
3128:
3126:
3119:
3116:
3064:
3051:
3039:
3026:
2999:
2967:
2949:
2943:
2840:
2794:
2777:
2758:
2717:
2698:
2674:
2668:
2600:
2579:
2564:
2514:
2495:
2486:
2429:
2423:
2379:
2309:
2259:
2240:
2231:
2174:
2168:
2103:
2067:
1976:
1954:
1948:
1913:
1893:
1880:
1866:
1860:
1843:
1824:
1809:
1789:
1770:
1732:
1710:
1704:
1261:must be used with care when the equation
679:
666:
653:
640:
627:
621:
426:
375:
347:
346:
341:
339:
338:
323:
317:
296:
274:
273:
268:
266:
265:
256:
250:
132:
75:
8785:Carleson, L.; Gamelin, T. D. W. (1993).
8439:
8278:
4114:) and ending with the 64th iterate. The
601:Abelian property and iteration sequences
20:
9121:
9119:
8302:(in French). Vol. IV. p. 229.
8185:
8150:, introduced by Hans Maurer (1901) and
8085:
7379:{\displaystyle g(f(x))=\exp \leftg(x).}
7145:
467:. This notation has been traced to and
8763:
8680:
8649:(3rd ed.). CRC Press. p. 2.
3997:locally conjugate to a mere dilation,
1634:Some formulas for fractional iteration
1067:, and produces quadratic convergence.
9129:(1870). "Ueber iterirte Functionen".
9085:. This solution is also the infinite
8506:for log. log. log.
7389:For example, for rigid advection, if
3755:which is simply the Taylor series of
1111:, according to the behavior of small
1007:(that is, the period of the orbit of
503:), some mathematicians choose to use
7:
8435:Philosophical Transactions of London
8432:, etc., was published by him in the
8277:(NB. Inhere, Herschel refers to his
7553:{\displaystyle f(x)=h^{-1}(h(x)+1),}
5229:{\displaystyle {\frac {ax+b}{cx+d}}}
1186:is a trivial functional 5th root of
9172:, September 1955, 489–492.
8815:Fixed Point Theory, An Introduction
8353:We note here the symbolism used by
8331:. pp. 108, 176–179, 336, 346.
8324:A History of Mathematical Notations
973:problem in computer science is the
8646:Encounters with Chaos and Fractals
8643:Gulick, Denny; Ford, Jeff (2024).
8568:Powers of trigonometric functions.
8422:'s notation for inverse functions,
7762:
7751:
7347:
7343:
7238:
7214:
6255:{\displaystyle {\sqrt {ax^{2}+b}}}
6005:{\displaystyle {\sqrt {x^{2}+bn}}}
4349:, as well as non-negative integer
3837:is topologically conjugate to the
1428:is the function defined such that
354:
351:
348:
281:
278:
275:
164:Iterated functions are studied in
55:is a function that is obtained by
14:
9191:The Michigan Mathematical Journal
6488:Note: these two special cases of
5964:{\displaystyle {\sqrt {x^{2}+b}}}
4647:{\displaystyle ax^{b}\ (b\neq 1)}
1335:are solutions; so the expression
1051:There are several techniques for
8252:Herschel, John Frederick William
8194:Herschel, John Frederick William
7635:This is evident by noting that
6557:, or narrower ones, such as the
5238:fractional linear transformation
499:(the latter is commonly used in
9185:Berkson, E.; Porta, H. (1978).
8602:; second, sin (sin
8494:, we ought to write sin.
8266:from the original on 2020-08-04
7896:translation functional equation
7723:For continuous iteration index
4526:{\displaystyle ax+b\ (a\neq 1)}
4098:Iterates of the sine function (
740:translation functional equation
469:John Frederick William Herschel
245:is a non-negative integer, by:
9242:10.1088/1751-8113/43/44/445101
9027:10.1088/1751-8113/42/48/485208
8750:. Cambridge University Press.
8746:Iterative Functional Equations
8438:, for the year 1813. He says (
7968:
7962:
7940:
7937:
7931:
7918:
7871:
7868:
7862:
7849:
7840:
7837:
7828:
7822:
7816:
7800:
7791:
7785:
7774:
7768:
7701:
7692:
7686:
7680:
7661:
7655:
7606:
7600:
7582:
7576:
7544:
7535:
7529:
7523:
7504:
7498:
7459:Conversely, one may specify
7370:
7364:
7338:
7332:
7312:
7309:
7303:
7297:
7233:
7227:
7201:
7195:
7129:
7120:
7114:
7095:
7083:
7077:
7066:
7060:
7035:
7032:
7026:
7007:
6990:
6984:
6973:
6967:
6875:
6869:
6845:
6813:
6807:
6699:
6693:
6666:
6634:
6628:
6470:
6448:
6396:
6381:
6369:
6363:
6160:
6154:
6100:
6076:
6057:
6051:
5902:
5896:
5834:
5828:
5768:
5762:
5687:
5681:
5597:
5584:
5514:
5501:
5436:
5415:
5399:
5378:
5357:
5336:
5314:
5293:
5149:
5130:
4641:
4629:
4520:
4508:
4411:
4405:
4374:
4368:
3986:for a function Ψ, which makes
3966:, a form known as the
3724:
3711:
3708:
3689:
3686:
3667:
3630:
3617:
3614:
3595:
3569:
3557:
3535:
3529:
3458:causes the series to diverge.
3447:. Using the other fixed point
3404:
3386:
3378:
3363:
3350:
3347:
3332:
3319:
3304:
3291:
3279:
3273:
2961:
2955:
2866:
2860:
2837:
2830:
2819:
2807:
2774:
2761:
2752:
2749:
2743:
2729:
2714:
2701:
2686:
2680:
2628:
2622:
2597:
2590:
2576:
2561:
2554:
2543:
2537:
2526:
2511:
2498:
2483:
2476:
2465:
2453:
2441:
2435:
2376:
2369:
2349:
2343:
2321:
2306:
2299:
2288:
2282:
2271:
2256:
2243:
2228:
2221:
2210:
2198:
2186:
2180:
2124:
2121:
2115:
2096:
2082:
2079:
2073:
2060:
2049:
2046:
2040:
2034:
2023:
2017:
2006:
1994:
1988:
1982:
1966:
1960:
1905:
1899:
1840:
1827:
1801:
1795:
1762:
1750:
1744:
1738:
1722:
1716:
1604:, or, equivalently, such that
1515:
738:, this relation is called the
492:exponentiation of the function
144:
138:
122:
116:
92:
86:
1:
9166:American Mathematical Monthly
8442:): "This notation cos.
8329:Open court publishing company
8168:pre-superscript notation for
7146:Lie's data transport equation
3982:(0) = 0, one may often solve
2656:There is a special case when
1090:of the orbit is known as the
1055:of the sequences produced by
949:. The smallest such value of
9295:. Colorado State University.
9099:) − 2)/(ln 2)
8813:Istratescu, Vasile (1981).
8675:Curves, Functions and Forces
6756:and the equivalent product:
4087:) has generalized to a full
4083:of the Picard sequence (cf.
4075:Here, however, the exponent
3909:, for any function
3841:. As a special case, taking
3172:is not an integer). We have
522:-th iterate of the function
4180:, such as the chaotic case
3856:, one has the iteration of
3103:which is trivial to check.
1466:may be defined as equal to
1149:can both be interpreted as
1046:Brouwer fixed point theorem
869:The sequence of functions
471:in 1813. Herschel credited
9351:
9325:Fixed points (mathematics)
9079:) = ln 2 Ψ(
8498:for sin. sin.
8137:must not be confused with
7890:The initial flow velocity
7149:
1350:can always be obtained if
1115:under iteration. Also see
1042:Banach fixed point theorem
1036:. There exist a number of
42:, which both have order 3.
8618:and log (log
6531:Artin–Mazur zeta function
4168:iterated function systems
1532:is the normal inverse of
8817:, D. Reidel, Holland.
8160:David Patterson Ellerman
8139:Rudolf von Bitter Rucker
8003:Iterated function system
7280:iterate of the function
4417:{\displaystyle f^{n}(x)}
4085:transformation semigroup
3917:Making the substitution
1159:subshifts of finite type
1053:convergence acceleration
945:, the orbit is called a
563:. For the same purpose,
8300:Formulaire mathématique
8212:Royal Society of London
7180:beta function (physics)
4339:closed-form expressions
3810:topologically conjugate
1691:around the fixed point
9330:Functions and mappings
9204:10.1307/mmj/1029002009
8935:10.1006/jmaa.1998.5986
8221:10.1098/rstl.1813.0005
8152:Reuben Louis Goodstein
8038:Functional square root
7981:
7881:
7714:
7626:
7554:
7481:, through the generic
7380:
7267:
7136:
6923:
6803:
6747:
6624:
6561:of computer programs.
6559:denotational semantics
6477:
6403:
6337:
6256:
6213:
6107:
6006:
5965:
5925:
5854:
5789:
5708:
5630:
5547:
5460:
5230:
5176:
5083:
4997:
4911:
4859:
4799:
4721:
4648:
4597:
4527:
4477:
4447:
4418:
4381:
4123:
3749:
3430:is positive. Also see
3420:
3158:
3097:
2923:gives the fixed point
2884:
2650:
2406:
2137:
1935:
1252:
1077:attractive fixed point
701:
533:, as in, for example,
435:
434:{\displaystyle \circ }
388:
306:
154:
44:
8574:, namely, (sin
8283:Hans Heinrich Bürmann
8135:function compositions
7982:
7882:
7715:
7627:
7555:
7381:
7268:
7137:
6939:functional derivative
6933:Functional derivative
6924:
6783:
6748:
6604:
6478:
6404:
6338:
6257:
6214:
6108:
6007:
5966:
5926:
5855:
5790:
5709:
5631:
5548:
5461:
5231:
5177:
5084:
4998:
4912:
4860:
4800:
4722:
4649:
4598:
4528:
4478:
4448:
4419:
4382:
4097:
3750:
3421:
3159:
3098:
2905:For example, setting
2885:
2651:
2416:geometric progression
2407:
2138:
1936:
1581:is defined such that
1213:. The computation of
1172:
1057:fixed point iteration
756:Chebyshev polynomials
702:
478:Because the notation
473:Hans Heinrich Bürmann
436:
389:
307:
178:renormalization group
155:
38:Below that are their
24:
9335:Functional equations
9320:Sequences and series
9056:) = Ψ((ln 2) Ψ(
8997:Journal of Physics A
8018:Sarkovskii's theorem
7905:
7734:
7642:
7570:
7492:
7452:, action by a plain
7291:
7189:
6948:
6763:
6584:
6423:
6411:Chebyshev polynomial
6350:
6267:
6226:
6118:
6019:
5976:
5938:
5870:
5802:
5719:
5645:
5558:
5475:
5246:
5191:
5098:
5010:
4926:
4874:
4812:
4734:
4659:
4610:
4538:
4490:
4476:{\displaystyle x+nb}
4458:
4431:
4392:
4380:{\displaystyle f(x)}
4362:
3516:
3221:
3115:
2942:
2667:
2422:
2167:
1947:
1703:
1501:continuous parameter
1081:unstable fixed point
1038:fixed-point theorems
881:Charles Émile Picard
620:
465:function composition
425:
316:
249:
74:
35:with infinite order.
9234:2010JPhA...43R5101C
9068:Schröder's equation
9019:2009JPhA...42V5208C
8965:Funkcialaj Ekvacioj
8922:J. Math. Anal. Appl
8877:1998JMP....39.5324A
8510:. Just as we write
8351:Iterated logarithms
8122:'s (1907) notation
8073:Functional equation
8053:Böttcher's equation
8048:Schröder's equation
8033:Schröder's equation
8028:Recurrence relation
8023:Fractional calculus
7998:Irrational rotation
7470:given an arbitrary
6551:recursive functions
6541:In computer science
4446:{\displaystyle x+b}
3984:Schröder's equation
3759:expanded around 1.
3189:. A fixed point is
3164:where this is done
2418:to simplify terms,
1088:accumulation points
1065:Steffensen's method
1059:. For example, the
965:itself is called a
959:period of the orbit
744:Schröder's equation
9262:Aczel, J. (2006),
9143:10.1007/BF01443992
8971:2012-04-26 at the
8703:Pringsheim, Alfred
8614:⋅ log
8610:, where log
8598:⋅ sin
8210:(Part 1). London:
8063:Flow (mathematics)
7977:
7877:
7710:
7622:
7550:
7421:. Consequently,
7376:
7263:
7176:iteration velocity
7132:
6919:
6743:
6535:transfer operators
6473:
6399:
6333:
6252:
6209:
6103:
6002:
5961:
5921:
5850:
5785:
5704:
5626:
5543:
5456:
5226:
5172:
5079:
4993:
4907:
4855:
4795:
4717:
4644:
4593:
4523:
4473:
4443:
4414:
4377:
4124:
3745:
3498:With the function
3416:
3154:
3111:Find the value of
3093:
2880:
2646:
2402:
2145:Substitute in for
2133:
1931:
1282:Babbage's equation
1253:
1071:Limiting behaviour
697:
431:
384:
302:
176:, mathematics and
150:
45:
9310:Dynamical systems
9110:Curtright, T. L.
8518:=arc (sin.=
8338:978-1-60206-714-1
8116:Alfred Pringsheim
8098:is taken for the
7973:
7778:
7759:
7756:
7747:
7706:
7610:
7485:discussed above,
7354:
7245:
7087:
6994:
6486:
6485:
6331:
6326:
6250:
6197:
6075:
6047:
6000:
5959:
5624:
5618:
5541:
5535:
5450:
5284:
5257:
5224:
5170:
5164:
5077:
4991:
4905:
4853:
4793:
4697:
4628:
4588:
4507:
4225:) = sin(2 arcsin(
4160:many chaotic maps
4144:stochastic matrix
3741:
3655:
3583:
3408:
3248:
3239:
3230:
3142:
3133:
3124:
3089:
3082:
3015:
2983:
2802:
2787:
2727:
2638:
2524:
2269:
1887:
1853:
1783:
1140:transfer operator
1136:invariant measure
1130:Invariant measure
941:for some integer
693:
580:Alfred Pringsheim
407:identity function
364:
359:
337:
291:
286:
264:
174:dynamical systems
100:
53:iterated function
43:
9342:
9296:
9291:(January 2017).
9275:
9260:
9254:
9253:
9227:
9208:
9206:
9182:
9176:
9161:
9155:
9154:
9123:
9114:
9108:
9102:
9100:
9084:
9078:
9077:
9061:
9045:
9039:
9038:
9012:
8989:Curtright, T. L.
8985:
8979:
8960:
8954:
8953:
8946:
8940:
8939:
8937:
8913:
8907:
8906:
8896:
8885:10.1063/1.532574
8870:
8850:
8844:
8843:
8832:
8826:
8811:
8805:
8804:
8792:
8789:Complex dynamics
8782:
8776:
8775:
8769:
8761:
8749:
8739:
8733:
8732:
8721:
8715:
8714:
8699:
8693:
8692:
8686:
8678:
8671:Peirce, Benjamin
8667:
8661:
8660:
8640:
8634:
8632:
8458:for (cos.
8346:
8345:
8315:
8304:
8303:
8292:
8286:
8274:
8272:
8271:
8248:
8242:
8241:
8223:
8190:
8173:
8167:
8149:
8132:
8113:
8107:
8104:
8097:
8090:
8008:Iterative method
7986:
7984:
7983:
7978:
7971:
7961:
7960:
7930:
7929:
7917:
7916:
7893:
7886:
7884:
7883:
7878:
7861:
7860:
7815:
7814:
7781:
7780:
7779:
7777:
7760:
7757:
7754:
7749:
7745:
7726:
7719:
7717:
7716:
7711:
7704:
7679:
7678:
7654:
7653:
7631:
7629:
7628:
7623:
7611:
7609:
7592:
7559:
7557:
7556:
7551:
7522:
7521:
7480:
7469:
7451:
7420:
7406:
7385:
7383:
7382:
7377:
7360:
7356:
7355:
7353:
7342:
7283:
7279:
7272:
7270:
7269:
7264:
7262:
7261:
7250:
7246:
7244:
7236:
7226:
7225:
7212:
7170:
7141:
7139:
7138:
7133:
7113:
7112:
7088:
7086:
7069:
7059:
7058:
7039:
7025:
7024:
7006:
6995:
6993:
6976:
6966:
6965:
6952:
6928:
6926:
6925:
6920:
6903:
6902:
6885:
6881:
6820:
6816:
6802:
6797:
6752:
6750:
6749:
6744:
6727:
6726:
6709:
6705:
6641:
6637:
6623:
6618:
6547:computer science
6501:
6482:
6480:
6479:
6474:
6460:
6459:
6438:
6437:
6408:
6406:
6405:
6400:
6362:
6361:
6342:
6340:
6339:
6334:
6332:
6327:
6325:
6314:
6307:
6306:
6296:
6291:
6290:
6281:
6280:
6271:
6261:
6259:
6258:
6253:
6251:
6243:
6242:
6230:
6218:
6216:
6215:
6210:
6208:
6207:
6198:
6196:
6185:
6178:
6177:
6167:
6150:
6149:
6140:
6139:
6133:
6132:
6112:
6110:
6109:
6104:
6073:
6072:
6071:
6045:
6041:
6040:
6034:
6033:
6011:
6009:
6008:
6003:
6001:
5990:
5989:
5980:
5970:
5968:
5967:
5962:
5960:
5952:
5951:
5942:
5930:
5928:
5927:
5922:
5920:
5919:
5892:
5891:
5885:
5884:
5860: (generic
5859:
5857:
5856:
5851:
5849:
5848:
5824:
5823:
5817:
5816:
5794:
5792:
5791:
5786:
5784:
5783:
5777:
5776:
5758:
5757:
5751:
5750:
5741:
5740:
5734:
5733:
5713:
5711:
5710:
5705:
5703:
5702:
5696:
5695:
5677:
5676:
5667:
5666:
5660:
5659:
5635:
5633:
5632:
5627:
5625:
5620:
5619:
5605:
5604:
5583:
5568:
5552:
5550:
5549:
5544:
5542:
5537:
5536:
5522:
5521:
5500:
5485:
5465:
5463:
5462:
5457:
5455:
5451:
5449:
5448:
5447:
5411:
5410:
5376:
5375:
5374:
5332:
5331:
5291:
5285:
5280:
5263:
5258:
5250:
5235:
5233:
5232:
5227:
5225:
5223:
5209:
5195:
5181:
5179:
5178:
5173:
5171:
5166:
5165:
5157:
5156:
5129:
5108:
5088:
5086:
5085:
5080:
5078:
5076:
5068:
5061:
5060:
5059:
5058:
5035:
5034:
5033:
5032:
5014:
5002:
5000:
4999:
4994:
4992:
4990:
4982:
4966:
4965:
4955:
4941:
4940:
4916:
4914:
4913:
4908:
4906:
4901:
4884:
4864:
4862:
4861:
4856:
4854:
4852:
4844:
4837:
4836:
4835:
4834:
4816:
4804:
4802:
4801:
4796:
4794:
4792:
4784:
4774:
4773:
4763:
4749:
4748:
4726:
4724:
4723:
4718:
4716:
4715:
4714:
4713:
4699:
4698:
4696:
4685:
4678:
4677:
4667:
4653:
4651:
4650:
4645:
4626:
4625:
4624:
4602:
4600:
4599:
4594:
4589:
4587:
4576:
4569:
4568:
4558:
4550:
4549:
4532:
4530:
4529:
4524:
4505:
4482:
4480:
4479:
4474:
4452:
4450:
4449:
4444:
4423:
4421:
4420:
4415:
4404:
4403:
4386:
4384:
4383:
4378:
4356:
4325:
4316:
4310:
4308:
4307:
4304:
4301:
4285:
4279:
4277:
4276:
4273:
4270:
4257:
4235:
4233:
4232:
4215:
4213:
4212:
4198:
4117:
4113:
4109:
4105:
4101:
4089:continuous group
4078:
4070:
4062:
4047:
4036:
4014:
3996:
3981:
3977:
3965:
3939:
3912:
3908:
3878:
3877:) + 1)
3855:
3832:
3807:
3803:
3799:
3781:
3774:
3770:
3754:
3752:
3751:
3746:
3739:
3732:
3731:
3701:
3700:
3679:
3678:
3666:
3665:
3656:
3654:
3643:
3638:
3637:
3607:
3606:
3594:
3593:
3584:
3576:
3556:
3555:
3528:
3527:
3511:
3489:
3487:
3486:
3483:
3480:
3473:
3467:
3457:
3446:
3445:
3444:
3425:
3423:
3422:
3417:
3409:
3407:
3381:
3371:
3370:
3346:
3345:
3317:
3312:
3311:
3272:
3271:
3259:
3258:
3257:
3256:
3255:
3254:
3249:
3244:
3240:
3235:
3231:
3226:
3216:
3209:
3199:
3188:
3187:
3186:
3163:
3161:
3160:
3155:
3153:
3152:
3151:
3150:
3149:
3148:
3143:
3138:
3134:
3129:
3125:
3120:
3102:
3100:
3099:
3094:
3087:
3083:
3081:
3070:
3069:
3068:
3052:
3044:
3043:
3031:
3030:
3021:
3017:
3016:
3014:
3000:
2984:
2982:
2968:
2954:
2953:
2937:
2922:
2889:
2887:
2886:
2881:
2873:
2869:
2859:
2845:
2844:
2829:
2803:
2795:
2788:
2783:
2782:
2781:
2759:
2742:
2728:
2723:
2722:
2721:
2699:
2679:
2678:
2662:
2655:
2653:
2652:
2647:
2639:
2637:
2621:
2612:
2605:
2604:
2589:
2580:
2575:
2574:
2553:
2536:
2525:
2520:
2519:
2518:
2496:
2491:
2490:
2475:
2434:
2433:
2414:Make use of the
2411:
2409:
2408:
2403:
2395:
2391:
2390:
2389:
2368:
2342:
2320:
2319:
2298:
2281:
2270:
2265:
2264:
2263:
2241:
2236:
2235:
2220:
2179:
2178:
2158:
2142:
2140:
2139:
2134:
2114:
2113:
2095:
2072:
2071:
2059:
2033:
2016:
1981:
1980:
1959:
1958:
1940:
1938:
1937:
1932:
1924:
1923:
1912:
1908:
1898:
1897:
1888:
1886:
1885:
1884:
1871:
1870:
1861:
1854:
1849:
1848:
1847:
1825:
1820:
1819:
1808:
1804:
1794:
1793:
1784:
1782:
1771:
1737:
1736:
1715:
1714:
1690:
1672:
1655:
1629:
1603:
1580:
1569:
1546:
1535:
1531:
1514:(cf. section on
1498:
1494:
1480:
1465:
1454:
1427:
1416:
1405:
1395:. This function
1394:
1371:
1367:
1356:
1345:
1334:
1319:
1305:
1290:
1279:
1260:
1250:
1249:
1245:
1236:
1235:
1231:
1226:
1225:
1221:
1212:
1185:
1157:. The theory of
1147:Koopman operator
1124:wandering points
1035:
1018:
1014:
1010:
1006:
1002:
998:
964:
956:
952:
944:
940:
918:
908:
893:
889:
874:
865:
850:
819:
795:
752:nesting property
737:
733:
726:
706:
704:
703:
698:
691:
690:
689:
671:
670:
658:
657:
645:
644:
632:
631:
612:
608:
596:
573:
562:
543:
532:
521:
517:
506:
498:
489:
483:
462:
441:
440:
438:
437:
432:
414:
404:
393:
391:
390:
385:
380:
379:
362:
361:
360:
358:
357:
345:
340:
335:
334:
333:
311:
309:
308:
303:
301:
300:
289:
288:
287:
285:
284:
272:
267:
262:
261:
260:
240:
230:
217:
203:
166:computer science
160:
159:
157:
156:
151:
137:
136:
98:
37:
9350:
9349:
9345:
9344:
9343:
9341:
9340:
9339:
9300:
9299:
9287:
9284:
9279:
9278:
9261:
9257:
9209:
9184:
9183:
9179:
9162:
9158:
9127:Schröder, Ernst
9125:
9124:
9117:
9109:
9105:
9090:
9075:
9073:
9071:
9048:
9046:
9042:
8987:
8986:
8982:
8973:Wayback Machine
8961:
8957:
8950:"Tetration.org"
8948:
8947:
8943:
8915:
8914:
8910:
8868:physics/9712026
8852:
8851:
8847:
8834:
8833:
8829:
8812:
8808:
8801:
8784:
8783:
8779:
8762:
8758:
8741:
8740:
8736:
8723:
8722:
8718:
8701:
8700:
8696:
8679:
8669:
8668:
8664:
8657:
8642:
8641:
8637:
8626:for (sin
8556:Benjamin Peirce
8412:
8406:
8396:
8386:
8380:
8370:
8361:in their joint
8343:
8341:
8339:
8319:Cajori, Florian
8317:
8316:
8307:
8296:Peano, Giuseppe
8294:
8293:
8289:
8285:'s older work.)
8269:
8267:
8250:
8249:
8245:
8192:
8191:
8187:
8182:
8177:
8176:
8163:
8145:
8123:
8114:
8110:
8100:
8093:
8091:
8087:
8082:
8077:
8013:Rotation number
7993:
7946:
7921:
7908:
7903:
7902:
7891:
7852:
7803:
7761:
7750:
7737:
7732:
7731:
7724:
7667:
7645:
7640:
7639:
7596:
7568:
7567:
7510:
7490:
7489:
7471:
7460:
7422:
7408:
7390:
7346:
7328:
7324:
7289:
7288:
7281:
7277:
7237:
7217:
7213:
7208:
7207:
7187:
7186:
7157:
7154:
7148:
7098:
7070:
7044:
7040:
7010:
6999:
6977:
6957:
6953:
6946:
6945:
6935:
6829:
6825:
6824:
6770:
6766:
6761:
6760:
6650:
6646:
6645:
6591:
6587:
6582:
6581:
6567:
6555:lambda calculus
6543:
6527:
6489:
6451:
6426:
6421:
6420:
6353:
6348:
6347:
6315:
6298:
6297:
6282:
6272:
6265:
6264:
6234:
6224:
6223:
6186:
6169:
6168:
6141:
6121:
6116:
6115:
6022:
6017:
6016:
5981:
5974:
5973:
5943:
5936:
5935:
5873:
5868:
5867:
5805:
5800:
5799:
5742:
5722:
5717:
5716:
5648:
5643:
5642:
5596:
5569:
5556:
5555:
5513:
5486:
5473:
5472:
5466:
5439:
5402:
5377:
5360:
5317:
5292:
5286:
5264:
5244:
5243:
5210:
5196:
5189:
5188:
5148:
5109:
5096:
5095:
5089:
5069:
5050:
5042:
5024:
5019:
5015:
5008:
5007:
5004:
4983:
4957:
4956:
4932:
4924:
4923:
4885:
4872:
4871:
4865:
4845:
4826:
4821:
4817:
4810:
4809:
4806:
4785:
4765:
4764:
4740:
4732:
4731:
4705:
4700:
4686:
4669:
4668:
4662:
4657:
4656:
4616:
4608:
4607:
4577:
4560:
4559:
4541:
4536:
4535:
4488:
4487:
4456:
4455:
4429:
4428:
4395:
4390:
4389:
4360:
4359:
4321:
4305:
4302:
4299:
4298:
4296:
4287:
4274:
4271:
4268:
4267:
4265:
4259:
4240:
4228:
4226:
4217:
4208:
4206:
4200:
4181:
4156:
4140:
4132:Carleman matrix
4115:
4111:
4107:
4103:
4099:
4076:
4068:
4052:
4042:
4019:
3998:
3987:
3979:
3975:
3944:
3918:
3910:
3883:
3857:
3842:
3816:
3808:are said to be
3805:
3801:
3783:
3779:
3772:
3768:
3765:
3723:
3692:
3670:
3657:
3647:
3629:
3598:
3585:
3547:
3519:
3514:
3513:
3499:
3496:
3484:
3481:
3475:
3474:
3471:
3469:
3462:
3448:
3442:
3440:
3435:
3382:
3362:
3331:
3318:
3303:
3263:
3242:
3233:
3224:
3219:
3218:
3211:
3204:
3190:
3184:
3182:
3173:
3136:
3127:
3118:
3113:
3112:
3109:
3071:
3060:
3053:
3035:
3022:
3004:
2992:
2988:
2972:
2945:
2940:
2939:
2924:
2906:
2903:
2852:
2836:
2822:
2793:
2789:
2773:
2760:
2735:
2713:
2700:
2670:
2665:
2664:
2657:
2614:
2613:
2596:
2582:
2581:
2560:
2546:
2529:
2510:
2497:
2482:
2468:
2425:
2420:
2419:
2375:
2361:
2335:
2328:
2324:
2305:
2291:
2274:
2255:
2242:
2227:
2213:
2170:
2165:
2164:
2146:
2099:
2088:
2063:
2052:
2026:
2009:
1972:
1950:
1945:
1944:
1889:
1876:
1872:
1862:
1859:
1856:
1855:
1839:
1826:
1785:
1775:
1769:
1766:
1765:
1728:
1706:
1701:
1700:
1681:
1660:
1643:
1636:
1605:
1582:
1571:
1548:
1537:
1533:
1522:
1496:
1482:
1467:
1456:
1429:
1418:
1407:
1396:
1373:
1369:
1365:
1354:
1336:
1321:
1307:
1292:
1285:
1262:
1256:
1247:
1243:
1242:
1233:
1229:
1228:
1223:
1219:
1218:
1187:
1173:
1167:
1151:shift operators
1132:
1073:
1026:
1016:
1012:
1008:
1004:
1000:
986:
983:
971:cycle detection
962:
954:
950:
942:
923:
916:
899:
891:
887:
877:Picard sequence
870:
852:
824:
805:
797:
789:
776:
767:
759:
735:
731:
715:
675:
662:
649:
636:
623:
618:
617:
610:
606:
603:
587:
576:Benjamin Peirce
564:
545:
534:
523:
519:
508:
504:
494:
485:
479:
423:
422:
421:
416:
410:
403:
397:
371:
319:
314:
313:
292:
252:
247:
246:
236:
235:-th iterate of
226:
205:
199:
186:
128:
72:
71:
70:
36:
26:
17:
12:
11:
5:
9348:
9346:
9338:
9337:
9332:
9327:
9322:
9317:
9312:
9302:
9301:
9298:
9297:
9283:
9282:External links
9280:
9277:
9276:
9272:978-0486445236
9255:
9218:(44): 445101.
9177:
9156:
9137:(2): 296–322.
9115:
9103:
9040:
9003:(48): 485208.
8980:
8955:
8941:
8908:
8845:
8827:
8806:
8799:
8777:
8756:
8734:
8716:
8694:
8662:
8655:
8635:
8562:," "log
8542:), sin.
8522:), log.
8408:
8402:
8392:
8382:
8376:
8366:
8337:
8305:
8287:
8243:
8184:
8183:
8181:
8178:
8175:
8174:
8108:
8084:
8083:
8081:
8078:
8076:
8075:
8070:
8065:
8060:
8055:
8050:
8045:
8040:
8035:
8030:
8025:
8020:
8015:
8010:
8005:
8000:
7994:
7992:
7989:
7988:
7987:
7976:
7970:
7967:
7964:
7959:
7956:
7953:
7949:
7945:
7942:
7939:
7936:
7933:
7928:
7924:
7920:
7915:
7911:
7888:
7887:
7876:
7873:
7870:
7867:
7864:
7859:
7855:
7851:
7848:
7845:
7842:
7839:
7836:
7833:
7830:
7827:
7824:
7821:
7818:
7813:
7810:
7806:
7802:
7799:
7796:
7793:
7790:
7787:
7784:
7776:
7773:
7770:
7767:
7764:
7753:
7744:
7740:
7721:
7720:
7709:
7703:
7700:
7697:
7694:
7691:
7688:
7685:
7682:
7677:
7674:
7670:
7666:
7663:
7660:
7657:
7652:
7648:
7633:
7632:
7621:
7618:
7615:
7608:
7605:
7602:
7599:
7595:
7590:
7587:
7584:
7581:
7578:
7575:
7561:
7560:
7549:
7546:
7543:
7540:
7537:
7534:
7531:
7528:
7525:
7520:
7517:
7513:
7509:
7506:
7503:
7500:
7497:
7454:shift operator
7387:
7386:
7375:
7372:
7369:
7366:
7363:
7359:
7352:
7349:
7345:
7340:
7337:
7334:
7331:
7327:
7323:
7320:
7317:
7314:
7311:
7308:
7305:
7302:
7299:
7296:
7274:
7273:
7260:
7257:
7254:
7249:
7243:
7240:
7235:
7232:
7229:
7224:
7220:
7216:
7210:
7206:
7203:
7200:
7197:
7194:
7147:
7144:
7143:
7142:
7131:
7128:
7125:
7122:
7119:
7116:
7111:
7108:
7105:
7101:
7097:
7094:
7091:
7085:
7082:
7079:
7076:
7073:
7068:
7065:
7062:
7057:
7054:
7051:
7047:
7043:
7037:
7034:
7031:
7028:
7023:
7020:
7017:
7013:
7009:
7005:
7002:
6998:
6992:
6989:
6986:
6983:
6980:
6975:
6972:
6969:
6964:
6960:
6956:
6934:
6931:
6930:
6929:
6918:
6915:
6912:
6909:
6906:
6901:
6898:
6895:
6892:
6889:
6884:
6880:
6877:
6874:
6871:
6868:
6865:
6862:
6859:
6856:
6853:
6850:
6847:
6844:
6841:
6838:
6835:
6832:
6828:
6823:
6819:
6815:
6812:
6809:
6806:
6801:
6796:
6793:
6790:
6786:
6782:
6779:
6776:
6773:
6769:
6754:
6753:
6742:
6739:
6736:
6733:
6730:
6725:
6722:
6719:
6716:
6713:
6708:
6704:
6701:
6698:
6695:
6692:
6689:
6686:
6683:
6680:
6677:
6674:
6671:
6668:
6665:
6662:
6659:
6656:
6653:
6649:
6644:
6640:
6636:
6633:
6630:
6627:
6622:
6617:
6614:
6611:
6607:
6603:
6600:
6597:
6594:
6590:
6569:Two important
6566:
6563:
6542:
6539:
6526:
6525:Means of study
6523:
6484:
6483:
6472:
6469:
6466:
6463:
6458:
6454:
6450:
6447:
6444:
6441:
6436:
6433:
6429:
6418:
6398:
6395:
6392:
6389:
6386:
6383:
6380:
6377:
6374:
6371:
6368:
6365:
6360:
6356:
6344:
6343:
6330:
6324:
6321:
6318:
6313:
6310:
6305:
6301:
6294:
6289:
6285:
6279:
6275:
6262:
6249:
6246:
6241:
6237:
6233:
6220:
6219:
6206:
6201:
6195:
6192:
6189:
6184:
6181:
6176:
6172:
6165:
6162:
6159:
6156:
6153:
6148:
6144:
6138:
6131:
6128:
6124:
6113:
6102:
6099:
6096:
6093:
6090:
6087:
6084:
6081:
6078:
6070:
6065:
6062:
6059:
6056:
6053:
6050:
6044:
6039:
6032:
6029:
6025:
6013:
6012:
5999:
5996:
5993:
5988:
5984:
5971:
5958:
5955:
5950:
5946:
5932:
5931:
5918:
5913:
5910:
5907:
5904:
5901:
5898:
5895:
5890:
5883:
5880:
5876:
5865:
5847:
5842:
5839:
5836:
5833:
5830:
5827:
5822:
5815:
5812:
5808:
5796:
5795:
5782:
5775:
5770:
5767:
5764:
5761:
5756:
5749:
5745:
5739:
5732:
5729:
5725:
5714:
5701:
5694:
5689:
5686:
5683:
5680:
5675:
5670:
5665:
5658:
5655:
5651:
5639:
5638:
5637:
5636:
5623:
5617:
5614:
5611:
5608:
5603:
5599:
5595:
5592:
5589:
5586:
5581:
5578:
5575:
5572:
5566:
5563:
5553:
5540:
5534:
5531:
5528:
5525:
5520:
5516:
5512:
5509:
5506:
5503:
5498:
5495:
5492:
5489:
5483:
5480:
5454:
5446:
5442:
5438:
5435:
5432:
5429:
5426:
5423:
5420:
5417:
5414:
5409:
5405:
5401:
5398:
5395:
5392:
5389:
5386:
5383:
5380:
5373:
5370:
5367:
5363:
5359:
5356:
5353:
5350:
5347:
5344:
5341:
5338:
5335:
5330:
5327:
5324:
5320:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5289:
5283:
5279:
5276:
5273:
5270:
5267:
5261:
5256:
5253:
5241:
5222:
5219:
5216:
5213:
5208:
5205:
5202:
5199:
5185:
5184:
5183:
5182:
5169:
5163:
5160:
5155:
5151:
5147:
5144:
5141:
5138:
5135:
5132:
5127:
5124:
5121:
5118:
5115:
5112:
5106:
5103:
5075:
5072:
5067:
5064:
5057:
5053:
5049:
5045:
5041:
5038:
5031:
5027:
5022:
5018:
5005:
4989:
4986:
4981:
4978:
4975:
4972:
4969:
4964:
4960:
4953:
4950:
4947:
4944:
4939:
4935:
4931:
4920:
4919:
4918:
4917:
4904:
4900:
4897:
4894:
4891:
4888:
4882:
4879:
4851:
4848:
4843:
4840:
4833:
4829:
4824:
4820:
4807:
4791:
4788:
4783:
4780:
4777:
4772:
4768:
4761:
4758:
4755:
4752:
4747:
4743:
4739:
4728:
4727:
4712:
4708:
4703:
4695:
4692:
4689:
4684:
4681:
4676:
4672:
4665:
4654:
4643:
4640:
4637:
4634:
4631:
4623:
4619:
4615:
4604:
4603:
4592:
4586:
4583:
4580:
4575:
4572:
4567:
4563:
4556:
4553:
4548:
4544:
4533:
4522:
4519:
4516:
4513:
4510:
4504:
4501:
4498:
4495:
4484:
4483:
4472:
4469:
4466:
4463:
4453:
4442:
4439:
4436:
4425:
4424:
4413:
4410:
4407:
4402:
4398:
4387:
4376:
4373:
4370:
4367:
4174:Ernst Schröder
4164:Mandelbrot set
4155:
4152:
4139:
4136:
4065:
4064:
4039:
4038:
3972:
3971:
3915:
3914:
3903:) +
3854: + 1
3764:
3761:
3744:
3738:
3735:
3730:
3726:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3699:
3695:
3691:
3688:
3685:
3682:
3677:
3673:
3669:
3664:
3660:
3653:
3650:
3646:
3641:
3636:
3632:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3605:
3601:
3597:
3592:
3588:
3582:
3579:
3574:
3571:
3568:
3565:
3562:
3559:
3554:
3550:
3546:
3543:
3540:
3537:
3534:
3531:
3526:
3522:
3495:
3492:
3415:
3412:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3380:
3377:
3374:
3369:
3365:
3361:
3358:
3355:
3352:
3349:
3344:
3341:
3338:
3334:
3330:
3327:
3324:
3321:
3315:
3310:
3306:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3270:
3266:
3262:
3253:
3247:
3238:
3229:
3147:
3141:
3132:
3123:
3108:
3105:
3092:
3086:
3080:
3077:
3074:
3067:
3063:
3059:
3056:
3050:
3047:
3042:
3038:
3034:
3029:
3025:
3020:
3013:
3010:
3007:
3003:
2998:
2995:
2991:
2987:
2981:
2978:
2975:
2971:
2966:
2963:
2960:
2957:
2952:
2948:
2902:
2899:
2891:
2890:
2879:
2876:
2872:
2868:
2865:
2862:
2858:
2855:
2851:
2848:
2843:
2839:
2835:
2832:
2828:
2825:
2821:
2818:
2815:
2812:
2809:
2806:
2801:
2798:
2792:
2786:
2780:
2776:
2772:
2769:
2766:
2763:
2757:
2754:
2751:
2748:
2745:
2741:
2738:
2734:
2731:
2726:
2720:
2716:
2712:
2709:
2706:
2703:
2697:
2694:
2691:
2688:
2685:
2682:
2677:
2673:
2645:
2642:
2636:
2633:
2630:
2627:
2624:
2620:
2617:
2611:
2608:
2603:
2599:
2595:
2592:
2588:
2585:
2578:
2573:
2570:
2567:
2563:
2559:
2556:
2552:
2549:
2545:
2542:
2539:
2535:
2532:
2528:
2523:
2517:
2513:
2509:
2506:
2503:
2500:
2494:
2489:
2485:
2481:
2478:
2474:
2471:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2432:
2428:
2412:
2401:
2398:
2394:
2388:
2385:
2382:
2378:
2374:
2371:
2367:
2364:
2360:
2357:
2354:
2351:
2348:
2345:
2341:
2338:
2334:
2331:
2327:
2323:
2318:
2315:
2312:
2308:
2304:
2301:
2297:
2294:
2290:
2287:
2284:
2280:
2277:
2273:
2268:
2262:
2258:
2254:
2251:
2248:
2245:
2239:
2234:
2230:
2226:
2223:
2219:
2216:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2177:
2173:
2143:
2132:
2129:
2126:
2123:
2120:
2117:
2112:
2109:
2106:
2102:
2098:
2094:
2091:
2087:
2084:
2081:
2078:
2075:
2070:
2066:
2062:
2058:
2055:
2051:
2048:
2045:
2042:
2039:
2036:
2032:
2029:
2025:
2022:
2019:
2015:
2012:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1979:
1975:
1971:
1968:
1965:
1962:
1957:
1953:
1941:
1930:
1927:
1922:
1919:
1916:
1911:
1907:
1904:
1901:
1896:
1892:
1883:
1879:
1875:
1869:
1865:
1858:
1852:
1846:
1842:
1838:
1835:
1832:
1829:
1823:
1818:
1815:
1812:
1807:
1803:
1800:
1797:
1792:
1788:
1781:
1778:
1774:
1768:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1735:
1731:
1727:
1724:
1721:
1718:
1713:
1709:
1678:
1657:
1635:
1632:
1368:is a function
1364:of a function
1166:
1163:
1131:
1128:
1072:
1069:
982:
979:
967:periodic point
957:is called the
947:periodic orbit
909:is called the
879:, named after
823:The relation
801:
785:
772:
763:
712:exponentiation
708:
707:
696:
688:
685:
682:
678:
674:
669:
665:
661:
656:
652:
648:
643:
639:
635:
630:
626:
602:
599:
430:
399:
383:
378:
374:
370:
367:
356:
353:
350:
344:
332:
329:
326:
322:
299:
295:
283:
280:
277:
271:
259:
255:
185:
182:
162:
161:
149:
146:
143:
140:
135:
131:
127:
124:
121:
118:
115:
112:
109:
106:
103:
97:
94:
91:
88:
85:
82:
79:
15:
13:
10:
9:
6:
4:
3:
2:
9347:
9336:
9333:
9331:
9328:
9326:
9323:
9321:
9318:
9316:
9313:
9311:
9308:
9307:
9305:
9294:
9290:
9286:
9285:
9281:
9273:
9269:
9265:
9259:
9256:
9251:
9247:
9243:
9239:
9235:
9231:
9226:
9221:
9217:
9213:
9205:
9200:
9196:
9192:
9188:
9181:
9178:
9175:
9171:
9168:
9167:
9160:
9157:
9152:
9148:
9144:
9140:
9136:
9132:
9128:
9122:
9120:
9116:
9113:
9107:
9104:
9098:
9094:
9088:
9082:
9069:
9065:
9059:
9055:
9051:
9044:
9041:
9036:
9032:
9028:
9024:
9020:
9016:
9011:
9006:
9002:
8998:
8994:
8993:Zachos, C. K.
8990:
8984:
8981:
8977:
8974:
8970:
8967:
8966:
8959:
8956:
8951:
8945:
8942:
8936:
8931:
8927:
8923:
8919:
8912:
8909:
8904:
8900:
8895:
8890:
8886:
8882:
8878:
8874:
8869:
8864:
8860:
8856:
8855:J. Math. Phys
8849:
8846:
8841:
8837:
8831:
8828:
8824:
8823:90-277-1224-7
8820:
8816:
8810:
8807:
8802:
8800:0-387-97942-5
8796:
8791:
8790:
8781:
8778:
8773:
8767:
8759:
8757:0-521-35561-3
8753:
8748:
8747:
8738:
8735:
8730:
8726:
8725:Kuczma, Marek
8720:
8717:
8712:
8708:
8704:
8698:
8695:
8690:
8684:
8676:
8672:
8666:
8663:
8658:
8656:9781003835776
8652:
8648:
8647:
8639:
8636:
8631:
8629:
8625:
8621:
8617:
8613:
8609:
8605:
8601:
8597:
8593:
8589:
8585:
8581:
8578:), sin
8577:
8573:
8569:
8565:
8561:
8557:
8553:
8549:
8545:
8541:
8537:
8533:
8529:
8525:
8521:
8517:
8513:
8509:
8505:
8502:, log.
8501:
8497:
8493:
8489:
8485:
8481:
8477:
8473:
8469:
8465:
8461:
8457:
8453:
8449:
8445:
8441:
8437:
8436:
8431:
8427:
8423:
8421:
8420:John Herschel
8416:
8411:
8405:
8400:
8395:
8390:
8385:
8379:
8374:
8369:
8365:article: "log
8364:
8360:
8356:
8352:
8340:
8334:
8330:
8326:
8325:
8320:
8314:
8312:
8310:
8306:
8301:
8297:
8291:
8288:
8284:
8281:and mentions
8280:
8276:
8265:
8261:
8257:
8253:
8247:
8244:
8239:
8235:
8231:
8227:
8222:
8217:
8213:
8209:
8205:
8204:
8199:
8195:
8189:
8186:
8179:
8171:
8166:
8161:
8157:
8153:
8148:
8144:
8140:
8136:
8130:
8126:
8121:
8117:
8112:
8109:
8106:
8105:th derivative
8103:
8096:
8089:
8086:
8079:
8074:
8071:
8069:
8066:
8064:
8061:
8059:
8056:
8054:
8051:
8049:
8046:
8044:
8043:Abel function
8041:
8039:
8036:
8034:
8031:
8029:
8026:
8024:
8021:
8019:
8016:
8014:
8011:
8009:
8006:
8004:
8001:
7999:
7996:
7995:
7990:
7974:
7965:
7957:
7954:
7951:
7947:
7943:
7934:
7926:
7922:
7913:
7909:
7901:
7900:
7899:
7897:
7874:
7865:
7857:
7853:
7846:
7843:
7834:
7831:
7825:
7819:
7811:
7808:
7804:
7797:
7794:
7788:
7782:
7771:
7765:
7742:
7738:
7730:
7729:
7728:
7707:
7698:
7695:
7689:
7683:
7675:
7672:
7668:
7664:
7658:
7650:
7646:
7638:
7637:
7636:
7619:
7616:
7613:
7603:
7597:
7593:
7588:
7585:
7579:
7573:
7566:
7565:
7564:
7547:
7541:
7538:
7532:
7526:
7518:
7515:
7511:
7507:
7501:
7495:
7488:
7487:
7486:
7484:
7483:Abel equation
7478:
7474:
7467:
7463:
7457:
7455:
7449:
7445:
7441:
7437:
7433:
7429:
7425:
7419:
7415:
7411:
7405:
7401:
7397:
7393:
7373:
7367:
7361:
7357:
7350:
7335:
7329:
7325:
7321:
7318:
7315:
7306:
7300:
7294:
7287:
7286:
7285:
7258:
7255:
7252:
7247:
7241:
7230:
7222:
7218:
7204:
7198:
7192:
7185:
7184:
7183:
7181:
7177:
7172:
7168:
7164:
7160:
7153:
7126:
7123:
7117:
7109:
7106:
7103:
7099:
7092:
7089:
7080:
7074:
7071:
7063:
7055:
7052:
7049:
7045:
7041:
7029:
7021:
7018:
7015:
7011:
7003:
7000:
6996:
6987:
6981:
6978:
6970:
6962:
6958:
6954:
6944:
6943:
6942:
6940:
6932:
6913:
6910:
6907:
6899:
6896:
6893:
6890:
6887:
6882:
6872:
6866:
6863:
6860:
6857:
6854:
6851:
6839:
6836:
6833:
6826:
6821:
6817:
6810:
6804:
6799:
6794:
6791:
6788:
6784:
6780:
6777:
6774:
6771:
6767:
6759:
6758:
6757:
6737:
6734:
6731:
6723:
6720:
6717:
6714:
6711:
6706:
6696:
6690:
6687:
6684:
6681:
6678:
6675:
6672:
6660:
6657:
6654:
6647:
6642:
6638:
6631:
6625:
6620:
6615:
6612:
6609:
6605:
6601:
6598:
6595:
6592:
6588:
6580:
6579:
6578:
6576:
6572:
6564:
6562:
6560:
6556:
6552:
6548:
6540:
6538:
6536:
6532:
6524:
6522:
6519:
6517:
6513:
6509:
6505:
6500:
6496:
6492:
6467:
6464:
6461:
6456:
6452:
6445:
6442:
6439:
6434:
6431:
6427:
6419:
6416:
6412:
6393:
6390:
6387:
6384:
6378:
6375:
6372:
6366:
6358:
6354:
6346:
6345:
6328:
6322:
6319:
6316:
6311:
6308:
6303:
6299:
6292:
6287:
6283:
6277:
6273:
6263:
6247:
6244:
6239:
6235:
6231:
6222:
6221:
6199:
6193:
6190:
6187:
6182:
6179:
6174:
6170:
6163:
6157:
6151:
6146:
6142:
6129:
6126:
6122:
6114:
6097:
6094:
6091:
6088:
6085:
6082:
6079:
6063:
6060:
6054:
6048:
6042:
6030:
6027:
6023:
6015:
6014:
5997:
5994:
5991:
5986:
5982:
5972:
5956:
5953:
5948:
5944:
5934:
5933:
5911:
5908:
5905:
5899:
5893:
5881:
5878:
5874:
5866:
5863:
5862:Abel equation
5840:
5837:
5831:
5825:
5813:
5810:
5806:
5798:
5797:
5765:
5759:
5747:
5743:
5730:
5727:
5723:
5715:
5684:
5678:
5668:
5656:
5653:
5649:
5641:
5640:
5621:
5615:
5612:
5609:
5606:
5601:
5593:
5590:
5587:
5579:
5576:
5573:
5570:
5564:
5561:
5554:
5538:
5532:
5529:
5526:
5523:
5518:
5510:
5507:
5504:
5496:
5493:
5490:
5487:
5481:
5478:
5471:
5470:
5469:
5452:
5444:
5440:
5433:
5430:
5427:
5424:
5421:
5418:
5412:
5407:
5403:
5396:
5393:
5390:
5387:
5384:
5381:
5371:
5368:
5365:
5361:
5354:
5351:
5348:
5345:
5342:
5339:
5333:
5328:
5325:
5322:
5318:
5311:
5308:
5305:
5302:
5299:
5296:
5287:
5281:
5277:
5274:
5271:
5268:
5265:
5259:
5254:
5251:
5242:
5239:
5220:
5217:
5214:
5211:
5206:
5203:
5200:
5197:
5187:
5186:
5167:
5161:
5158:
5153:
5145:
5142:
5139:
5136:
5133:
5125:
5122:
5119:
5116:
5113:
5110:
5104:
5101:
5094:
5093:
5092:
5073:
5070:
5065:
5062:
5055:
5051:
5047:
5043:
5039:
5036:
5029:
5025:
5020:
5016:
5006:
4987:
4984:
4979:
4976:
4973:
4970:
4967:
4962:
4958:
4951:
4948:
4945:
4942:
4937:
4933:
4929:
4922:
4921:
4902:
4898:
4895:
4892:
4889:
4886:
4880:
4877:
4870:
4869:
4868:
4849:
4846:
4841:
4838:
4831:
4827:
4822:
4818:
4808:
4789:
4786:
4781:
4778:
4775:
4770:
4766:
4759:
4756:
4753:
4750:
4745:
4741:
4737:
4730:
4729:
4710:
4706:
4701:
4693:
4690:
4687:
4682:
4679:
4674:
4670:
4663:
4655:
4638:
4635:
4632:
4621:
4617:
4613:
4606:
4605:
4590:
4584:
4581:
4578:
4573:
4570:
4565:
4561:
4554:
4551:
4546:
4542:
4534:
4517:
4514:
4511:
4502:
4499:
4496:
4493:
4486:
4485:
4470:
4467:
4464:
4461:
4454:
4440:
4437:
4434:
4427:
4426:
4408:
4400:
4396:
4388:
4371:
4365:
4358:
4357:
4354:
4352:
4348:
4344:
4340:
4335:
4333:
4329:
4324:
4318:
4314:
4294:
4290:
4286:, and hence
4283:
4263:
4255:
4251:
4247:
4243:
4237:
4231:
4224:
4220:
4211:
4204:
4196:
4192:
4188:
4184:
4179:
4175:
4171:
4169:
4165:
4161:
4153:
4151:
4149:
4145:
4138:Markov chains
4137:
4135:
4133:
4129:
4128:eigenfunction
4121:
4096:
4092:
4090:
4086:
4082:
4074:
4060:
4056:
4051:
4050:
4049:
4045:
4034:
4030:
4026:
4022:
4018:
4017:
4016:
4013:
4009:
4005:
4001:
3994:
3990:
3985:
3969:
3968:Abel equation
3963:
3959:
3955:
3951:
3947:
3943:
3942:
3941:
3937:
3933:
3929:
3925:
3921:
3906:
3902:
3898:
3894:
3890:
3886:
3882:
3881:
3880:
3876:
3872:
3868:
3864:
3860:
3853:
3849:
3845:
3840:
3836:
3831:
3827:
3824: ○
3823:
3820: =
3819:
3813:
3811:
3798:
3794:
3790:
3786:
3778:
3777:homeomorphism
3762:
3760:
3758:
3742:
3736:
3733:
3728:
3720:
3717:
3714:
3705:
3702:
3697:
3693:
3683:
3680:
3675:
3671:
3662:
3658:
3651:
3648:
3644:
3639:
3634:
3626:
3623:
3620:
3611:
3608:
3603:
3599:
3590:
3586:
3580:
3577:
3572:
3566:
3563:
3560:
3552:
3548:
3544:
3541:
3538:
3532:
3524:
3520:
3510:
3506:
3502:
3493:
3491:
3479:
3465:
3459:
3455:
3451:
3438:
3433:
3429:
3413:
3410:
3401:
3398:
3395:
3392:
3389:
3383:
3375:
3372:
3367:
3359:
3356:
3353:
3342:
3339:
3336:
3328:
3325:
3322:
3313:
3308:
3300:
3297:
3294:
3288:
3285:
3282:
3276:
3268:
3264:
3260:
3251:
3245:
3236:
3227:
3214:
3207:
3201:
3197:
3193:
3180:
3176:
3171:
3167:
3145:
3139:
3130:
3121:
3106:
3104:
3090:
3084:
3078:
3075:
3072:
3065:
3061:
3057:
3054:
3048:
3045:
3040:
3036:
3032:
3027:
3023:
3018:
3011:
3008:
3005:
3001:
2996:
2993:
2989:
2985:
2979:
2976:
2973:
2969:
2964:
2958:
2950:
2946:
2935:
2931:
2927:
2921:
2917:
2913:
2909:
2900:
2898:
2896:
2877:
2874:
2870:
2863:
2856:
2853:
2849:
2846:
2841:
2833:
2826:
2823:
2816:
2813:
2810:
2804:
2799:
2796:
2790:
2784:
2778:
2770:
2767:
2764:
2755:
2746:
2739:
2736:
2732:
2724:
2718:
2710:
2707:
2704:
2695:
2692:
2689:
2683:
2675:
2671:
2660:
2643:
2640:
2634:
2631:
2625:
2618:
2615:
2609:
2606:
2601:
2593:
2586:
2583:
2571:
2568:
2565:
2557:
2550:
2547:
2540:
2533:
2530:
2521:
2515:
2507:
2504:
2501:
2492:
2487:
2479:
2472:
2469:
2462:
2459:
2456:
2450:
2447:
2444:
2438:
2430:
2426:
2417:
2413:
2399:
2396:
2392:
2386:
2383:
2380:
2372:
2365:
2362:
2358:
2355:
2352:
2346:
2339:
2336:
2332:
2329:
2325:
2316:
2313:
2310:
2302:
2295:
2292:
2285:
2278:
2275:
2266:
2260:
2252:
2249:
2246:
2237:
2232:
2224:
2217:
2214:
2207:
2204:
2201:
2195:
2192:
2189:
2183:
2175:
2171:
2162:
2157:
2153:
2149:
2144:
2130:
2127:
2118:
2110:
2107:
2104:
2100:
2092:
2089:
2085:
2076:
2068:
2064:
2056:
2053:
2043:
2037:
2030:
2027:
2020:
2013:
2010:
2003:
2000:
1997:
1991:
1985:
1977:
1973:
1969:
1963:
1955:
1951:
1942:
1928:
1925:
1920:
1917:
1914:
1909:
1902:
1894:
1890:
1881:
1877:
1873:
1867:
1863:
1850:
1844:
1836:
1833:
1830:
1821:
1816:
1813:
1810:
1805:
1798:
1790:
1786:
1779:
1776:
1772:
1759:
1756:
1753:
1747:
1741:
1733:
1729:
1725:
1719:
1711:
1707:
1698:
1697:Taylor series
1694:
1688:
1684:
1679:
1676:
1671:
1667:
1663:
1658:
1654:
1650:
1646:
1641:
1640:
1639:
1633:
1631:
1628:
1624:
1620:
1616:
1612:
1608:
1601:
1597:
1593:
1589:
1585:
1578:
1574:
1567:
1563:
1559:
1555:
1551:
1544:
1540:
1529:
1525:
1519:
1517:
1513:
1508:
1506:
1502:
1493:
1489:
1485:
1478:
1474:
1470:
1463:
1459:
1452:
1448:
1444:
1440:
1436:
1432:
1425:
1421:
1417:. Similarly,
1414:
1410:
1403:
1399:
1392:
1388:
1384:
1380:
1376:
1363:
1358:
1353:
1349:
1343:
1339:
1332:
1328:
1324:
1318:
1314:
1310:
1303:
1299:
1295:
1288:
1283:
1277:
1273:
1269:
1265:
1259:
1240:
1216:
1210:
1206:
1202:
1198:
1194:
1190:
1184:
1180:
1176:
1171:
1164:
1162:
1160:
1156:
1152:
1148:
1143:
1141:
1137:
1129:
1127:
1125:
1120:
1118:
1114:
1113:neighborhoods
1110:
1109:unstable sets
1106:
1101:
1099:
1095:
1094:
1089:
1084:
1082:
1078:
1070:
1068:
1066:
1062:
1061:Aitken method
1058:
1054:
1049:
1047:
1043:
1039:
1033:
1029:
1024:
1023:
996:
992:
988:
980:
978:
976:
972:
968:
960:
948:
938:
934:
930:
926:
920:
914:
913:
906:
902:
897:
884:
882:
878:
873:
867:
864:
860:
856:
848:
844:
840:
836:
832:
828:
821:
817:
813:
809:
804:
800:
793:
788:
784:
780:
775:
771:
766:
762:
757:
753:
749:
748:Abel equation
745:
741:
728:
725:
721:
718:
713:
694:
686:
683:
680:
676:
672:
667:
663:
659:
654:
650:
646:
641:
637:
633:
628:
624:
616:
615:
614:
600:
598:
594:
590:
585:
581:
577:
571:
567:
560:
556:
552:
548:
541:
537:
530:
526:
515:
511:
502:
497:
493:
488:
482:
476:
474:
470:
466:
460:
456:
452:
448:
444:
428:
420:
413:
408:
402:
394:
381:
376:
372:
368:
365:
342:
330:
327:
324:
320:
297:
293:
269:
257:
253:
244:
239:
234:
229:
223:
221:
216:
212:
208:
204:be a set and
202:
196:
194:
191:
183:
181:
179:
175:
171:
167:
147:
141:
133:
129:
125:
119:
113:
110:
107:
104:
101:
95:
89:
83:
80:
77:
69:
68:
67:
64:
62:
58:
54:
50:
41:
34:
33:shear mapping
30:
23:
19:
9263:
9258:
9215:
9211:
9194:
9190:
9180:
9169:
9164:
9159:
9134:
9130:
9106:
9096:
9092:
9086:
9080:
9063:
9057:
9053:
9049:
9043:
9000:
8996:
8983:
8975:
8964:
8958:
8944:
8925:
8921:
8911:
8861:(10): 5324.
8858:
8854:
8848:
8840:MathOverflow
8839:
8830:
8814:
8809:
8788:
8780:
8745:
8737:
8728:
8719:
8710:
8697:
8674:
8665:
8645:
8638:
8627:
8623:
8619:
8615:
8611:
8607:
8603:
8599:
8595:
8591:
8587:
8583:
8582:, sin
8579:
8575:
8571:
8567:
8563:
8559:
8551:
8543:
8539:
8535:
8531:
8527:
8523:
8519:
8515:
8511:
8507:
8503:
8499:
8495:
8491:
8487:
8486:, ΔΔΔ
8483:
8479:
8475:
8471:
8467:
8463:
8459:
8455:
8451:
8447:
8443:
8433:
8429:
8428:, tan
8425:
8418:
8414:
8409:
8403:
8398:
8393:
8388:
8383:
8377:
8372:
8367:
8363:Encyclopédie
8362:
8350:
8348:
8342:. Retrieved
8323:
8299:
8290:
8268:. Retrieved
8259:
8246:
8207:
8201:
8188:
8164:
8146:
8128:
8124:
8111:
8101:
8094:
8088:
7895:
7889:
7722:
7634:
7562:
7476:
7472:
7465:
7461:
7458:
7447:
7443:
7439:
7435:
7431:
7427:
7423:
7417:
7413:
7409:
7403:
7399:
7395:
7391:
7388:
7275:
7173:
7166:
7162:
7158:
7155:
6936:
6755:
6568:
6544:
6528:
6520:
6515:
6511:
6507:
6503:
6498:
6494:
6490:
6487:
6414:
6413:for integer
5467:
5090:
4866:
4350:
4346:
4342:
4336:
4322:
4319:
4312:
4292:
4288:
4281:
4261:
4253:
4249:
4245:
4241:
4238:
4229:
4222:
4218:
4209:
4202:
4194:
4190:
4186:
4182:
4178:logistic map
4172:
4157:
4148:Markov chain
4141:
4125:
4072:
4066:
4058:
4054:
4043:
4040:
4032:
4028:
4024:
4020:
4015:, that is
4011:
4007:
4003:
3999:
3992:
3988:
3973:
3961:
3957:
3953:
3949:
3945:
3935:
3931:
3927:
3923:
3919:
3916:
3904:
3900:
3896:
3892:
3888:
3884:
3874:
3870:
3866:
3862:
3858:
3851:
3847:
3843:
3839:logistic map
3829:
3825:
3821:
3817:
3814:
3796:
3792:
3788:
3784:
3766:
3756:
3508:
3504:
3500:
3497:
3477:
3463:
3460:
3453:
3449:
3436:
3427:
3212:
3205:
3202:
3195:
3191:
3178:
3174:
3169:
3165:
3110:
2933:
2929:
2925:
2919:
2915:
2911:
2907:
2904:
2894:
2892:
2658:
2160:
2155:
2151:
2147:
1692:
1686:
1682:
1674:
1669:
1665:
1661:
1652:
1648:
1644:
1637:
1626:
1622:
1618:
1614:
1610:
1606:
1599:
1595:
1591:
1587:
1583:
1576:
1572:
1565:
1561:
1557:
1553:
1549:
1542:
1538:
1527:
1523:
1520:
1509:
1500:
1491:
1487:
1483:
1476:
1472:
1468:
1461:
1457:
1450:
1446:
1442:
1438:
1434:
1430:
1423:
1419:
1412:
1408:
1401:
1397:
1390:
1386:
1382:
1378:
1374:
1362:half iterate
1359:
1351:
1347:
1341:
1337:
1330:
1326:
1322:
1316:
1312:
1308:
1301:
1297:
1293:
1286:
1275:
1271:
1267:
1263:
1257:
1254:
1238:
1214:
1208:
1204:
1200:
1196:
1192:
1188:
1182:
1178:
1174:
1153:action on a
1144:
1133:
1121:
1102:
1097:
1091:
1085:
1074:
1050:
1031:
1027:
1020:
1019:is called a
994:
990:
987:
984:
981:Fixed points
961:. The point
958:
953:for a given
946:
936:
932:
928:
924:
921:
910:
904:
900:
886:For a given
885:
876:
875:is called a
871:
868:
862:
858:
854:
846:
842:
838:
834:
830:
826:
822:
815:
811:
807:
802:
798:
791:
786:
782:
778:
773:
769:
764:
760:
751:
739:
729:
723:
719:
716:
709:
604:
592:
588:
574:was used by
569:
565:
558:
554:
550:
546:
539:
535:
528:
524:
513:
509:
501:trigonometry
495:
486:
480:
477:
458:
454:
450:
446:
442:
418:
411:
400:
395:
242:
237:
232:
227:
224:
214:
210:
206:
200:
197:
192:
187:
163:
65:
52:
46:
40:compositions
18:
9197:: 101–115.
8894:11449/65519
8707:Molk, Jules
8490:, ΣΣ
8417:)." §533.
8154:(1947) for
6571:functionals
4205:) = arcsin(
3782:such that
1943:Expand out
1255:The notion
1251:) is shown.
1155:shift space
1105:stable sets
1098:ω-limit set
1022:fixed point
975:algorithmic
898:of values
49:mathematics
9304:Categories
9289:Gill, John
8978:, 197-238.
8566:." §537.
8474:, Σ
8470:, Δ
8440:p. 10
8424:sin
8355:Pringsheim
8344:2016-01-18
8270:2020-08-04
8180:References
8162:'s (1995)
8158:, or with
8141:'s (1982)
8133:to denote
8120:Jules Molk
7284:, we have
7174:Given the
7150:See also:
5003:(see note)
4805:(see note)
4332:free group
4258:, yielded
4199:, so that
4158:There are
2159:, for any
1499:becomes a
1372:such that
586:suggested
584:Jules Molk
184:Definition
9250:115176169
9225:1002.0104
9151:116998358
9131:Math. Ann
9089:limit of
9035:115173476
9010:0909.2424
8928:: 81–90.
8903:119675869
8766:cite book
8683:cite book
8391:), …, log
8279:1813 work
8238:118124706
8196:(1813) .
8156:tetration
8068:Tetration
7958:τ
7927:τ
7809:−
7763:∂
7752:∂
7673:−
7589:∫
7516:−
7348:∂
7344:∂
7322:
7239:∂
7215:∂
7124:−
7107:−
7093:δ
7072:δ
7053:−
7042:δ
7019:−
6979:δ
6955:δ
6891:−
6846:→
6822:≡
6785:∏
6715:−
6667:→
6643:≡
6606:∑
6575:summation
6533:and with
6465:
6446:
6391:
6379:
6320:−
6309:−
6191:−
6180:−
6127:−
6089:∨
6083:≠
6028:−
5879:−
5811:−
5728:−
5654:−
5591:−
5580:−
5562:β
5508:−
5479:α
5441:β
5434:β
5425:−
5413:−
5404:α
5397:α
5388:−
5369:−
5362:β
5355:β
5346:−
5334:−
5326:−
5319:α
5312:α
5303:−
5272:−
5159:−
5126:±
5102:α
5063:−
5048:−
5044:α
5021:α
4977:−
4968:−
4878:α
4839:−
4823:α
4776:−
4691:−
4680:−
4636:≠
4582:−
4571:−
4515:≠
3763:Conjugacy
3737:⋯
3718:−
3703:−
3681:−
3624:−
3609:−
3564:−
3494:Example 3
3432:Tetration
3414:⋯
3411:−
3399:−
3393:
3373:−
3357:
3326:
3298:
3289:−
3252:⋯
3146:⋯
3107:Example 2
3076:−
3058:−
3009:−
2997:−
2977:−
2901:Example 1
2895:Conjugacy
2878:⋯
2814:−
2768:−
2708:−
2644:⋯
2632:−
2607:−
2569:−
2505:−
2460:−
2400:⋯
2384:−
2356:⋯
2314:−
2250:−
2205:−
2131:⋯
2108:−
2086:⋯
2001:−
1929:⋯
1834:−
1757:−
1516:conjugacy
1315:) = 6 − 2
1093:limit set
999:for some
796:, since
660:∘
634:∘
597:instead.
429:∘
369:∘
225:Defining
195:follows.
180:physics.
111:∘
61:iteration
57:composing
9315:Fractals
8969:Archived
8727:(1968).
8709:(1907).
8673:(1852).
8298:(1903).
8264:Archived
8254:(1820).
8143:notation
7991:See also
7434:) = exp(
7276:for the
7004:′
5236: (
4341:for the
4280:ln(1 − 2
4216:, hence
4154:Examples
4046:'(0) ≠ 1
3940:yields
3835:tent map
3800:, then
3470:2
2857:‴
2827:″
2740:″
2661:'(a) = 1
2619:′
2587:′
2551:′
2534:″
2473:′
2366:′
2340:′
2296:′
2279:″
2218:′
2093:′
2057:′
2031:′
2014:′
1673:for all
1536:, while
1518:below.)
1455:, while
1207:) = sin(
1044:and the
1015:), then
943:m > 0
896:sequence
810:) = cos(
578:whereas
544:meaning
518:for the
463:denotes
241:, where
220:function
170:fractals
9230:Bibcode
9074:√
9015:Bibcode
8873:Bibcode
8548:Burmann
8482:
8466:
8413:
8397:
8387:
8371:
8118:'s and
7407:, then
6514:= 4 = –
6506:= 2 = –
5468:where:
5091:where:
4867:where:
4326:is the
4311:((1 − 2
4309:
4297:
4278:
4266:
4227:√
4207:√
4130:Ψ, cf.
4057:'(0) Ψ(
4031:'(0) Ψ(
4027:) = Ψ(
3488:
3456:(4) = 4
3441:√
3203:So set
3198:(2) = 2
3183:√
1680:Expand
1659:Define
1306:, both
1246:⁄
1232:⁄
1222:⁄
1096:or the
814:arccos(
405:is the
231:as the
9270:
9248:
9174:online
9149:
9062:, for
9033:
8901:
8821:
8797:
8754:
8653:
8349:§473.
8335:
8236:
8230:107384
8228:
8092:while
7972:
7758:
7755:
7746:
7705:
7563:where
6462:arccos
6388:arccos
6074:
6046:
4627:
4506:
4328:action
4315:) − 1)
4104:orange
4081:monoid
4067:where
3740:
3439:(1) =
3088:
2932:/(1 −
1445:))) =
969:. The
894:, the
742:, cf.
692:
396:where
363:
336:
290:
263:
99:
9246:S2CID
9220:arXiv
9147:S2CID
9064:any n
9031:S2CID
9005:arXiv
8899:S2CID
8863:arXiv
8478:mean
8401:= log
8375:= log
8234:S2CID
8226:JSTOR
8170:roots
8080:Notes
7178:, or
4295:) = −
4264:) = −
4252:(1 −
4248:) = 2
4193:(1 −
4189:) = 4
4116:green
4010:'(0)
3978:= 0,
3956:)) =
1695:as a
1617:)) =
1594:)) =
1505:orbit
1385:)) =
1355:'
1329:) = 2
1300:) = 4
931:) =
912:orbit
857:) = (
833:) = (
781:)) =
714:that
218:be a
51:, an
29:order
9268:ISBN
8819:ISBN
8795:ISBN
8772:link
8752:ISBN
8689:link
8651:ISBN
8407:(log
8381:(log
8359:Molk
8357:and
8333:ISBN
7898:,
7416:) =
7398:) =
7182:,
6937:The
6510:and
4166:and
4108:blue
4100:blue
4006:) =
3930:) =
3891:) =
3879:as
3865:) =
3850:) =
3804:and
3771:and
3507:) =
3485:ln 2
3466:= −1
3461:For
3210:and
3181:) =
2914:) =
2154:) =
1668:) =
1651:) =
1625:) =
1556:) =
1512:flow
1320:and
1291:and
1270:) =
1227:) =
1107:and
922:If
861:) =
841:) =
746:and
734:and
609:and
582:and
449:) =
415:and
312:and
198:Let
9238:doi
9199:doi
9139:doi
9023:doi
8930:doi
8926:224
8889:hdl
8881:doi
8534:),
8216:doi
8208:103
7442:)
7438:∂/∂
7319:exp
6545:In
6443:cos
6376:cos
4320:If
4122:.)
4112:red
3964:+1)
3787:=
3767:If
3476:ln
3215:(1)
3208:= 1
1333:− 2
1304:− 6
1289:= 2
1028:Fix
1011:is
1003:in
991:= f
985:If
915:of
890:in
787:m n
754:of
561:)))
490:or
409:on
209::
190:set
47:In
9306::
9244:.
9236:.
9228:.
9216:43
9214:.
9195:25
9193:.
9189:.
9170:62
9145:.
9133:.
9118:^
9072:Ψ(
9070:,
9060:))
9029:.
9021:.
9013:.
9001:42
8999:.
8991:;
8976:14
8924:.
8920:.
8897:.
8887:.
8879:.
8871:.
8859:39
8857:.
8838:.
8768:}}
8764:{{
8705:;
8685:}}
8681:{{
8480:dd
8347:.
8308:^
8258:.
8232:.
8224:.
8206:.
8200:.
7456:.
7430:+
7402:+
7171:.
7169:))
6577::
6537:.
6497:+
6495:bx
6493:+
6491:ax
6417:)
5864:)
5240:)
5162:16
4353:.
4334:.
4317:.
4260:Ψ(
4236:.
4234:))
4201:Ψ(
4170:.
4150:.
4091:.
4061:))
4053:Ψ(
4035:))
3922:=
3828:○
3812:.
3795:○
3791:○
3490:.
3452:=
3434::
3390:ln
3354:ln
3323:ln
3295:ln
3200:.
3194:=
2928:=
2918:+
2916:Cx
2897:.
2663:,
2163:,
1699:,
1630:.
1568:))
1507:.
1490:=
1486:○
1479:))
1237:=
1199:,
1191::
1177::
1119:.
1100:.
1083:.
1048:.
919:.
883:.
866:.
837:)(
829:)(
820:.
818:))
758:,
727:.
722:=
613:,
461:))
445:)(
398:id
294:id
222:.
213:→
172:,
168:,
9274:.
9252:.
9240::
9232::
9222::
9207:.
9201::
9153:.
9141::
9135:3
9101:.
9097:x
9095:(
9093:f
9091:(
9087:m
9083:)
9081:x
9076:2
9058:x
9054:x
9052:(
9050:f
9037:.
9025::
9017::
9007::
8952:.
8938:.
8932::
8905:.
8891::
8883::
8875::
8865::
8842:.
8825:.
8803:.
8774:)
8760:.
8691:)
8659:.
8628:x
8624:x
8620:x
8616:x
8612:x
8608:x
8604:x
8600:x
8596:x
8592:x
8588:x
8584:x
8580:x
8576:x
8572:x
8564:x
8560:x
8544:x
8540:x
8538:(
8536:f
8532:x
8530:(
8528:f
8524:x
8520:x
8516:x
8512:d
8508:x
8504:x
8500:x
8496:x
8492:x
8488:x
8484:x
8476:x
8472:x
8468:x
8464:d
8460:A
8456:A
8452:e
8448:e
8444:e
8430:x
8426:x
8415:a
8410:b
8404:b
8399:a
8394:b
8389:a
8384:b
8378:b
8373:a
8368:b
8273:.
8240:.
8218::
8172:.
8165:x
8147:x
8131:)
8129:x
8127:(
8125:f
8102:n
8095:f
7975:.
7969:)
7966:x
7963:(
7955:+
7952:t
7948:f
7944:=
7941:)
7938:)
7935:x
7932:(
7923:f
7919:(
7914:t
7910:f
7892:v
7875:.
7872:)
7869:)
7866:x
7863:(
7858:t
7854:f
7850:(
7847:g
7844:=
7841:)
7838:)
7835:t
7832:+
7829:)
7826:x
7823:(
7820:h
7817:(
7812:1
7805:h
7801:(
7798:g
7795:=
7792:)
7789:x
7786:(
7783:g
7775:)
7772:x
7769:(
7766:h
7743:t
7739:e
7725:t
7708:.
7702:)
7699:n
7696:+
7693:)
7690:x
7687:(
7684:h
7681:(
7676:1
7669:h
7665:=
7662:)
7659:x
7656:(
7651:n
7647:f
7620:.
7617:x
7614:d
7607:)
7604:x
7601:(
7598:v
7594:1
7586:=
7583:)
7580:x
7577:(
7574:h
7548:,
7545:)
7542:1
7539:+
7536:)
7533:x
7530:(
7527:h
7524:(
7519:1
7512:h
7508:=
7505:)
7502:x
7499:(
7496:f
7479:)
7477:x
7475:(
7473:v
7468:)
7466:x
7464:(
7462:f
7450:)
7448:x
7446:(
7444:g
7440:x
7436:t
7432:t
7428:x
7426:(
7424:g
7418:t
7414:x
7412:(
7410:v
7404:t
7400:x
7396:x
7394:(
7392:f
7374:.
7371:)
7368:x
7365:(
7362:g
7358:]
7351:x
7339:)
7336:x
7333:(
7330:v
7326:[
7316:=
7313:)
7310:)
7307:x
7304:(
7301:f
7298:(
7295:g
7282:f
7278:n
7259:0
7256:=
7253:n
7248:|
7242:n
7234:)
7231:x
7228:(
7223:n
7219:f
7205:=
7202:)
7199:x
7196:(
7193:v
7167:x
7165:(
7163:f
7161:(
7159:g
7130:)
7127:y
7121:)
7118:x
7115:(
7110:1
7104:N
7100:f
7096:(
7090:+
7084:)
7081:y
7078:(
7075:f
7067:)
7064:x
7061:(
7056:1
7050:N
7046:f
7036:)
7033:)
7030:x
7027:(
7022:1
7016:N
7012:f
7008:(
7001:f
6997:=
6991:)
6988:y
6985:(
6982:f
6974:)
6971:x
6968:(
6963:N
6959:f
6917:}
6914:1
6911:,
6908:a
6905:{
6900:1
6897:+
6894:a
6888:b
6883:)
6879:}
6876:)
6873:i
6870:(
6867:g
6864:x
6861:,
6858:1
6855:+
6852:i
6849:{
6843:}
6840:x
6837:,
6834:i
6831:{
6827:(
6818:}
6814:)
6811:i
6808:(
6805:g
6800:b
6795:a
6792:=
6789:i
6781:,
6778:1
6775:+
6772:b
6768:{
6741:}
6738:0
6735:,
6732:a
6729:{
6724:1
6721:+
6718:a
6712:b
6707:)
6703:}
6700:)
6697:i
6694:(
6691:g
6688:+
6685:x
6682:,
6679:1
6676:+
6673:i
6670:{
6664:}
6661:x
6658:,
6655:i
6652:{
6648:(
6639:}
6635:)
6632:i
6629:(
6626:g
6621:b
6616:a
6613:=
6610:i
6602:,
6599:1
6596:+
6593:b
6589:{
6516:a
6512:b
6508:a
6504:b
6499:c
6471:)
6468:x
6457:n
6453:m
6449:(
6440:=
6435:n
6432:m
6428:T
6415:m
6409:(
6397:)
6394:x
6385:m
6382:(
6373:=
6370:)
6367:x
6364:(
6359:m
6355:T
6329:b
6323:1
6317:a
6312:1
6304:n
6300:a
6293:+
6288:2
6284:x
6278:n
6274:a
6248:b
6245:+
6240:2
6236:x
6232:a
6205:)
6200:b
6194:1
6188:a
6183:1
6175:n
6171:a
6164:+
6161:)
6158:x
6155:(
6152:g
6147:n
6143:a
6137:(
6130:1
6123:g
6101:)
6098:0
6095:=
6092:b
6086:1
6080:a
6077:(
6069:)
6064:b
6061:+
6058:)
6055:x
6052:(
6049:g
6043:a
6038:(
6031:1
6024:g
5998:n
5995:b
5992:+
5987:2
5983:x
5957:b
5954:+
5949:2
5945:x
5917:)
5912:b
5909:n
5906:+
5903:)
5900:x
5897:(
5894:g
5889:(
5882:1
5875:g
5846:)
5841:b
5838:+
5835:)
5832:x
5829:(
5826:g
5821:(
5814:1
5807:g
5781:)
5774:)
5769:)
5766:x
5763:(
5760:g
5755:(
5748:n
5744:h
5738:(
5731:1
5724:g
5700:)
5693:)
5688:)
5685:x
5682:(
5679:g
5674:(
5669:h
5664:(
5657:1
5650:g
5622:2
5616:c
5613:b
5610:4
5607:+
5602:2
5598:)
5594:d
5588:a
5585:(
5577:d
5574:+
5571:a
5565:=
5539:2
5533:c
5530:b
5527:4
5524:+
5519:2
5515:)
5511:d
5505:a
5502:(
5497:+
5494:d
5491:+
5488:a
5482:=
5453:]
5445:n
5437:)
5431:+
5428:a
5422:x
5419:c
5416:(
5408:n
5400:)
5394:+
5391:a
5385:x
5382:c
5379:(
5372:1
5366:n
5358:)
5352:+
5349:a
5343:x
5340:c
5337:(
5329:1
5323:n
5315:)
5309:+
5306:a
5300:x
5297:c
5294:(
5288:[
5282:c
5278:d
5275:a
5269:c
5266:b
5260:+
5255:c
5252:a
5221:d
5218:+
5215:x
5212:c
5207:b
5204:+
5201:x
5198:a
5168:4
5154:2
5150:)
5146:b
5143:+
5140:x
5137:a
5134:2
5131:(
5123:b
5120:+
5117:x
5114:a
5111:2
5105:=
5074:a
5071:2
5066:b
5056:n
5052:2
5040:2
5037:+
5030:n
5026:2
5017:2
4988:a
4985:4
4980:8
4974:b
4971:2
4963:2
4959:b
4952:+
4949:x
4946:b
4943:+
4938:2
4934:x
4930:a
4903:2
4899:b
4896:+
4893:x
4890:a
4887:2
4881:=
4850:a
4847:2
4842:b
4832:n
4828:2
4819:2
4790:a
4787:4
4782:b
4779:2
4771:2
4767:b
4760:+
4757:x
4754:b
4751:+
4746:2
4742:x
4738:a
4711:n
4707:b
4702:x
4694:1
4688:b
4683:1
4675:n
4671:b
4664:a
4642:)
4639:1
4633:b
4630:(
4622:b
4618:x
4614:a
4591:b
4585:1
4579:a
4574:1
4566:n
4562:a
4555:+
4552:x
4547:n
4543:a
4521:)
4518:1
4512:a
4509:(
4503:b
4500:+
4497:x
4494:a
4471:b
4468:n
4465:+
4462:x
4441:b
4438:+
4435:x
4412:)
4409:x
4406:(
4401:n
4397:f
4375:)
4372:x
4369:(
4366:f
4351:n
4347:n
4343:n
4323:f
4313:x
4306:2
4303:/
4300:1
4293:x
4291:(
4289:f
4284:)
4282:x
4275:2
4272:/
4269:1
4262:x
4256:)
4254:x
4250:x
4246:x
4244:(
4242:f
4230:x
4223:x
4221:(
4219:f
4214:)
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