3661:
3737:
3588:
3517:
38:
1533:
1441:
2319:. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since
2897:
2717:
3071:
2072:
3288:
2446:
2544:
3293:
Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since by using
1766:
1147:
691:
3729:
1705:
1523:
1904:
1612:
1079:
960:
909:
3794:
2732:
2552:
3580:
3653:
3660:
3869:
288:
3132:
3186:
2309:
600:
562:
1028:
1798:
2247:
1827:
1248:
524:
499:
477:
3938:
2162:
1333:
2197:
854:
804:
1215:
1186:
452:
2908:
1942:
757:
824:
777:
734:
714:
423:
403:
375:
355:
331:
311:
221:
3736:
1954:
3587:
1392:
of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator.
3516:
3194:
2356:
4106:. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials.
3329:
the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any
2457:
4176:
4121:
in analytical chemistry, enzyme kinetics in biochemistry, electronic circuitry, aerodynamics, medicine concentrations in vivo,
4032:
3975:
1716:
1411:
1084:
4395:
4271:
2260:
can be written as a rational function while the converse is not necessarily true, i.e., the ring of
Laurent polynomials is a
608:
3668:
3981:
2343:, which can be found by equating the rational function to a Taylor series with indeterminate coefficients, and collecting
1626:
2892:{\displaystyle 1=\sum _{k=2}^{\infty }a_{k-2}x^{k}-\sum _{k=1}^{\infty }a_{k-1}x^{k}+2\sum _{k=0}^{\infty }a_{k}x^{k}.}
2712:{\displaystyle 1=\sum _{k=0}^{\infty }a_{k}x^{k+2}-\sum _{k=0}^{\infty }a_{k}x^{k+1}+2\sum _{k=0}^{\infty }a_{k}x^{k}.}
502:
1453:
4306:
1837:
1545:
1033:
914:
863:
81:
59:
3744:
1414:, a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a
52:
4171:
3524:
3595:
4109:
Rational functions are used to approximate or model more complex equations in science and engineering including
4435:
4331:
4233:
4214:
4166:
3810:
1539:
1447:
1385:
229:
4430:
4374:
4425:
4145:
3094:
3151:
2270:
4440:
4369:
3425:
2095:
is a rational function since constants are polynomials. The function itself is rational, even though the
567:
529:
164:
of the function is the set of the values of the variables for which the denominator is not zero, and the
4364:
4031:
An extended version of the abstract idea of rational function is used in algebraic geometry. There the
3940:. Every rational function can be naturally extended to a function whose domain and range are the whole
965:
24:
4125:
for atoms and molecules, optics and photography to improve image resolution, and acoustics and sound.
4149:
3963:
1774:
2202:
1803:
1403:
of a rational function is the difference between the degrees of the numerator and the denominator.
1228:
46:
1377:
507:
482:
460:
4253:
4099:
3945:
1222:
20:
3908:
4410:
3476:
3457:
2123:
1265:
1156:
149:
103:
63:
4062:
are considered as regular functions in the sense of algebraic geometry on non-empty open sets
1302:
3081:
2316:
2167:
1365:
4091:
3460:
of integers can always be written uniquely in lowest terms by canceling out common factors.
3066:{\displaystyle 1=2a_{0}+(2a_{1}-a_{0})x+\sum _{k=2}^{\infty }(a_{k-2}-a_{k-1}+2a_{k})x^{k}.}
829:
4348:
3952:
3429:
3314:
782:
382:
1191:
1162:
428:
8:
4184: – a generalization of rational functions that allows taking integer roots
3330:
2340:
2118:
2096:
1396:
1389:
334:
188:
135:
1924:
739:
4181:
4161:
4079:
3358:
3350:
2257:
809:
762:
719:
699:
408:
388:
360:
340:
316:
296:
206:
184:
180:
111:
4391:
4327:
4302:
4267:
4229:
4210:
4133:
4129:
2081:
1407:
1361:
857:
161:
2067:{\displaystyle f(i)={\frac {i^{2}+2}{i^{2}+1}}={\frac {-1+2}{-1+1}}={\frac {1}{0}},}
4383:
4294:
4259:
4141:
4047:
3801:
3453:
3326:
3310:
3294:
1257:
There are several non equivalent definitions of the degree of a rational function.
4110:
4067:
4039:
3362:
3283:{\displaystyle a_{k}={\frac {1}{2}}(a_{k-1}-a_{k-2})\quad {\text{for}}\ k\geq 2.}
1218:
131:
3313:, giving an explicit formula for the Taylor coefficients; this is the method of
3959:
3941:
1945:
1914:
1372:
4419:
4122:
4087:
4083:
3368:. Any rational expression can be written as the quotient of two polynomials
3085:
2336:
378:
4095:
4118:
4051:
1281:
4263:
3297:
we can write any proper rational function as a sum of factors of the form
1360:, where two or more solutions coincide or where some solution is rejected
141:. In this case, one speaks of a rational function and a rational fraction
4137:
3142:
1948:
or its negative), then formal evaluation would lead to division by zero:
1910:
127:
119:
95:
3969:
1532:
4382:
Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007),
2441:{\displaystyle {\frac {1}{x^{2}-x+2}}=\sum _{k=0}^{\infty }a_{k}x^{k}.}
2344:
123:
3088:
on the left must equal the constant term on the right it follows that
3508:
3084:
of the original Taylor series, we can compute as follows. Since the
115:
4098:. Approximations in terms of rational functions are well suited for
4103:
2249:
is not a rational function. However, the adjective "irrational" is
165:
2722:
After adjusting the indices of the sums to get the same powers of
1440:
3874:
is the ratio of two polynomials with complex coefficients, where
2261:
1364:(that is, when the degree of the equation decreases after having
4384:"Section 3.4. Rational Function Interpolation and Extrapolation"
4381:
223:
is called a rational function if it can be written in the form
4114:
2539:{\displaystyle 1=(x^{2}-x+2)\sum _{k=0}^{\infty }a_{k}x^{k}}
4411:
Dynamic visualization of rational functions with JSXGraph
2451:
Multiplying through by the denominator and distributing,
1761:{\displaystyle x^{2}=5\Leftrightarrow x=\pm {\sqrt {5}}.}
3471:). This field is said to be generated (as a field) over
3320:
2199:
A function that cannot be written in this form, such as
1142:{\displaystyle \textstyle {\frac {P_{1}(x)}{Q_{1}(x)}}.}
3491:) does not contain any proper subfield containing both
1375:
coefficients, a rational function with degree one is a
4226:
3145:
on the right must be zero, from which it follows that
2290:
1779:
1088:
1037:
918:
867:
686:{\displaystyle f_{1}(x)={\frac {P_{1}(x)}{Q_{1}(x)}},}
571:
533:
511:
486:
464:
3970:
Notion of a rational function on an algebraic variety
3911:
3813:
3747:
3724:{\displaystyle {\frac {z^{2}-0.2+0.7i}{z^{2}+0.917}}}
3671:
3598:
3527:
3197:
3154:
3097:
2911:
2735:
2555:
2460:
2359:
2273:
2205:
2170:
2126:
1957:
1927:
1840:
1806:
1777:
1719:
1629:
1548:
1456:
1305:
1231:
1194:
1165:
1087:
1036:
968:
917:
866:
832:
812:
785:
765:
742:
722:
702:
611:
570:
532:
510:
485:
463:
431:
411:
391:
363:
343:
319:
299:
232:
209:
1700:{\displaystyle f(x)={\frac {x^{3}-2x}{2(x^{2}-5)}}}
4388:Numerical Recipes: The Art of Scientific Computing
3984:, rational expressions can also be generalized to
3951:Rational functions are representative examples of
3932:
3863:
3788:
3723:
3647:
3574:
3282:
3180:
3126:
3065:
2891:
2711:
2538:
2440:
2303:
2241:
2191:
2156:
2066:
1936:
1898:
1821:
1792:
1760:
1699:
1606:
1517:
1327:
1242:
1209:
1180:
1141:
1073:
1022:
954:
903:
848:
818:
806:, that is to extend "by continuity" the domain of
798:
771:
751:
728:
708:
685:
594:
556:
518:
493:
471:
446:
417:
397:
369:
349:
325:
305:
282:
215:
3380:≠0, although this representation isn't unique.
860:of fractions of polynomials, where two fractions
856:Indeed, one can define a rational fraction as an
4417:
1518:{\displaystyle y={\frac {x^{3}-2x}{2(x^{2}-5)}}}
1899:{\displaystyle f(x)={\frac {x^{2}+2}{x^{2}+1}}}
1607:{\displaystyle y={\frac {x^{2}-3x-2}{x^{2}-4}}}
1538:Rational function of degree 2, with a graph of
1446:Rational function of degree 3, with a graph of
1074:{\displaystyle \textstyle {\frac {P(x)}{Q(x)}}}
955:{\displaystyle \textstyle {\frac {C(x)}{D(x)}}}
904:{\displaystyle \textstyle {\frac {A(x)}{B(x)}}}
3958:Iteration of rational functions (maps) on the
3789:{\displaystyle {\frac {z^{2}}{z^{9}-z+0.025}}}
4204:
4152:are rational functions over complex numbers.
3502:
3463:The field of rational expressions is denoted
1264:of a rational function is the maximum of the
4390:(3rd ed.), Cambridge University Press,
3575:{\displaystyle {\frac {1}{az^{5}+z^{3}+bz}}}
4324:Introduction to Circuit Analysis and Design
4066:, and also may be seen as morphisms to the
4038:is formed as the field of fractions of the
3648:{\displaystyle {\frac {1}{z^{3}+z(-3-3i)}}}
175:The set of rational functions over a field
3864:{\displaystyle f(z)={\frac {P(z)}{Q(z)}}}
1324:
1233:
283:{\displaystyle f(x)={\frac {P(x)}{Q(x)}}}
82:Learn how and when to remove this message
4293:
4205:Corless, Martin J.; Frazho, Art (2003).
3905:is the set of complex numbers such that
45:This article includes a list of general
4321:
4251:
4223:
4418:
4177:Function field of an algebraic variety
4033:function field of an algebraic variety
4004:, by taking the field of fractions of
3976:Function field of an algebraic variety
3898:taking the indeterminate value 0/0).
3321:Abstract algebra and geometric notion
3127:{\displaystyle a_{0}={\frac {1}{2}}.}
4346:
3181:{\displaystyle a_{1}={\frac {1}{4}}}
2304:{\displaystyle f(x)={\tfrac {x}{x}}}
1155:is a rational function in which the
696:which may have a larger domain than
31:
23:. For the use in monoid theory, see
19:For the use in automata theory, see
4140:(for discrete-time systems) of the
3892:have no common factor (this avoids
3137:Then, since there are no powers of
2339:of any rational function satisfy a
595:{\displaystyle \textstyle Q=Q_{1}R}
557:{\displaystyle \textstyle P=P_{1}R}
13:
3333:. In this setting, given a field
2988:
2861:
2808:
2758:
2681:
2628:
2578:
2511:
2410:
1813:
1280:, when the fraction is reduced to
503:polynomial greatest common divisor
51:it lacks sufficient corresponding
14:
4452:
4404:
4349:"Iteration of Rational Functions"
3448:polynomials of lowest degree and
1023:{\displaystyle A(x)D(x)=B(x)C(x)}
759:It is a common usage to identify
4172:Partial fractions in integration
4136:(for continuous systems) or the
3735:
3659:
3586:
3515:
2347:after clearing the denominator.
2330:
1531:
1439:
36:
4078:Rational functions are used in
4073:
3880:is not the zero polynomial and
3262:
1793:{\displaystyle {\tfrac {x}{2}}}
1268:of its constituent polynomials
130:of the polynomials need not be
4340:
4315:
4287:
4245:
4228:. Prentice-Hall. p. 203.
4195:
4167:Partial fraction decomposition
4090:of functions, for example the
3921:
3915:
3855:
3849:
3841:
3835:
3823:
3817:
3639:
3621:
3295:partial fraction decomposition
3259:
3221:
3076:Since this holds true for all
3047:
2993:
2963:
2934:
2492:
2467:
2283:
2277:
2253:generally used for functions.
2233:
2227:
2215:
2209:
2180:
2174:
2151:
2145:
2136:
2130:
1967:
1961:
1850:
1844:
1810:
1736:
1691:
1672:
1639:
1633:
1509:
1490:
1433:Examples of rational functions
1315:
1309:
1204:
1198:
1175:
1169:
1129:
1123:
1108:
1102:
1064:
1058:
1050:
1044:
1017:
1011:
1005:
999:
990:
984:
978:
972:
945:
939:
931:
925:
894:
888:
880:
874:
674:
668:
653:
647:
628:
622:
441:
435:
274:
268:
260:
254:
242:
236:
198:
1:
4301:. Springer. p. A.IV.20.
4188:
4146:linear time-invariant systems
2242:{\displaystyle f(x)=\sin(x),}
1822:{\displaystyle x\to \infty .}
1395:In some contexts, such as in
1350:except for certain values of
1243:{\displaystyle \mathbb {Q} .}
962:are considered equivalent if
602:produces a rational function
16:Ratio of polynomial functions
4224:Pownall, Malcolm W. (1983).
4046:(more accurately said, of a
3456:. This is similar to how a
3432:for any rational expression
2164:is a rational function with
519:{\displaystyle \textstyle R}
494:{\displaystyle \textstyle Q}
472:{\displaystyle \textstyle P}
405:is the set of all values of
7:
4370:Encyclopedia of Mathematics
4322:Glisson, Tildon H. (2011).
4155:
3426:unique factorization domain
2902:Combining like terms gives
2315:except 0, where there is a
2264:of the rational functions.
1425:
1188:is less than the degree of
134:; they may be taken in any
10:
4457:
4209:. CRC Press. p. 163.
4207:Linear Systems and Control
3973:
3964:discrete dynamical systems
3933:{\displaystyle Q(z)\neq 0}
3503:Complex rational functions
2341:linear recurrence relation
425:for which the denominator
152:may be taken in any field
25:Rational function (monoid)
18:
4150:infinite impulse response
2157:{\displaystyle f(x)=P(x)}
1252:
106:that can be defined by a
4347:Camarena, Omar AntolĂn.
4100:computer algebra systems
3357:) is any element of the
3141:on the left, all of the
1328:{\displaystyle f(z)=w\,}
1221:, named by analogy to a
1153:proper rational function
4252:Bourles, Henri (2010).
3946:complex projective line
3337:and some indeterminate
2192:{\displaystyle Q(x)=1.}
1366:cleared the denominator
66:more precise citations.
21:Finite-state transducer
4258:. Wiley. p. 515.
4008:, which is denoted by
3934:
3865:
3804:, a rational function
3790:
3725:
3649:
3576:
3477:transcendental element
3284:
3182:
3128:
3067:
2992:
2893:
2865:
2812:
2762:
2713:
2685:
2632:
2582:
2540:
2515:
2442:
2414:
2335:The coefficients of a
2327:is equivalent to 1/1.
2311:is equal to 1 for all
2305:
2267:The rational function
2243:
2193:
2158:
2110:is irrational for all
2068:
1938:
1921:were a square root of
1900:
1831:The rational function
1823:
1794:
1762:
1701:
1620:The rational function
1608:
1519:
1344:distinct solutions in
1329:
1244:
1211:
1182:
1143:
1075:
1024:
956:
905:
850:
849:{\displaystyle f_{1}.}
820:
800:
773:
753:
730:
710:
687:
596:
558:
520:
495:
473:
448:
419:
399:
371:
351:
327:
307:
284:
217:
4436:Meromorphic functions
4264:10.1002/9781118619988
3953:meromorphic functions
3935:
3866:
3791:
3726:
3650:
3577:
3430:unique representation
3285:
3183:
3129:
3082:radius of convergence
3068:
2972:
2894:
2845:
2792:
2742:
2714:
2665:
2612:
2562:
2541:
2495:
2443:
2394:
2317:removable singularity
2306:
2244:
2194:
2159:
2069:
1939:
1901:
1824:
1795:
1763:
1702:
1609:
1520:
1378:Möbius transformation
1330:
1245:
1212:
1183:
1144:
1076:
1025:
957:
906:
851:
821:
801:
799:{\displaystyle f_{1}}
774:
754:
731:
711:
688:
597:
559:
521:
496:
474:
449:
420:
400:
372:
352:
328:
308:
285:
218:
4431:Morphisms of schemes
4102:and other numerical
3909:
3811:
3745:
3669:
3596:
3525:
3315:generating functions
3309:and expand these as
3195:
3152:
3095:
2909:
2733:
2553:
2458:
2357:
2271:
2203:
2168:
2124:
2077:which is undefined.
1955:
1925:
1838:
1804:
1775:
1771:It is asymptotic to
1717:
1627:
1546:
1454:
1418:biquadratic function
1303:
1296:, then the equation
1229:
1210:{\displaystyle Q(x)}
1192:
1181:{\displaystyle P(x)}
1163:
1085:
1034:
966:
915:
864:
830:
810:
783:
763:
740:
720:
700:
609:
568:
530:
508:
501:have a non-constant
483:
461:
447:{\displaystyle Q(x)}
429:
409:
389:
361:
341:
335:polynomial functions
317:
297:
230:
207:
189:polynomial functions
148:. The values of the
4426:Algebraic varieties
4365:"Rational function"
4182:Algebraic fractions
4092:Padé approximations
4054:affine open set in
3343:rational expression
2119:polynomial function
1909:is defined for all
1397:asymptotic analysis
1284:. If the degree of
1260:Most commonly, the
114:such that both the
4441:Rational functions
4162:Field of fractions
4080:numerical analysis
3930:
3861:
3786:
3721:
3645:
3572:
3420:. However, since
3396:, for polynomials
3359:field of fractions
3351:algebraic geometry
3280:
3178:
3124:
3063:
2889:
2709:
2536:
2438:
2301:
2299:
2258:Laurent polynomial
2239:
2189:
2154:
2064:
1937:{\displaystyle -1}
1934:
1913:, but not for all
1896:
1819:
1790:
1788:
1758:
1710:is not defined at
1697:
1604:
1515:
1325:
1240:
1207:
1178:
1139:
1138:
1071:
1070:
1020:
952:
951:
901:
900:
846:
816:
796:
769:
752:{\displaystyle f.}
749:
726:
716:, and is equal to
706:
683:
592:
591:
554:
553:
516:
515:
491:
490:
469:
468:
444:
415:
395:
367:
347:
323:
303:
280:
213:
181:field of fractions
112:algebraic fraction
4397:978-0-521-88068-8
4273:978-1-84821-162-9
4144:of commonly-used
4134:Laplace transform
4130:signal processing
3859:
3784:
3719:
3643:
3570:
3511:for rational maps
3388:is equivalent to
3355:rational function
3347:rational fraction
3345:(also known as a
3270:
3266:
3219:
3176:
3119:
2389:
2298:
2082:constant function
2059:
2046:
2011:
1894:
1787:
1753:
1695:
1602:
1513:
1408:network synthesis
1133:
1081:is equivalent to
1068:
949:
898:
858:equivalence class
819:{\displaystyle f}
772:{\displaystyle f}
736:on the domain of
729:{\displaystyle f}
709:{\displaystyle f}
678:
418:{\displaystyle x}
398:{\displaystyle f}
370:{\displaystyle Q}
350:{\displaystyle x}
326:{\displaystyle Q}
306:{\displaystyle P}
278:
216:{\displaystyle f}
108:rational fraction
100:rational function
92:
91:
84:
4448:
4400:
4378:
4356:
4355:
4353:
4344:
4338:
4337:
4319:
4313:
4312:
4291:
4285:
4284:
4282:
4280:
4249:
4243:
4239:
4220:
4199:
4142:impulse response
4058:). Its elements
3939:
3937:
3936:
3931:
3904:
3897:
3891:
3885:
3879:
3870:
3868:
3867:
3862:
3860:
3858:
3844:
3830:
3802:complex analysis
3795:
3793:
3792:
3787:
3785:
3783:
3770:
3769:
3759:
3758:
3749:
3739:
3730:
3728:
3727:
3722:
3720:
3718:
3711:
3710:
3700:
3684:
3683:
3673:
3663:
3654:
3652:
3651:
3646:
3644:
3642:
3614:
3613:
3600:
3590:
3581:
3579:
3578:
3573:
3571:
3569:
3559:
3558:
3546:
3545:
3529:
3519:
3495:and the element
3327:abstract algebra
3311:geometric series
3308:
3289:
3287:
3286:
3281:
3268:
3267:
3264:
3258:
3257:
3239:
3238:
3220:
3212:
3207:
3206:
3187:
3185:
3184:
3179:
3177:
3169:
3164:
3163:
3133:
3131:
3130:
3125:
3120:
3112:
3107:
3106:
3072:
3070:
3069:
3064:
3059:
3058:
3046:
3045:
3030:
3029:
3011:
3010:
2991:
2986:
2962:
2961:
2949:
2948:
2930:
2929:
2898:
2896:
2895:
2890:
2885:
2884:
2875:
2874:
2864:
2859:
2838:
2837:
2828:
2827:
2811:
2806:
2788:
2787:
2778:
2777:
2761:
2756:
2718:
2716:
2715:
2710:
2705:
2704:
2695:
2694:
2684:
2679:
2658:
2657:
2642:
2641:
2631:
2626:
2608:
2607:
2592:
2591:
2581:
2576:
2545:
2543:
2542:
2537:
2535:
2534:
2525:
2524:
2514:
2509:
2479:
2478:
2447:
2445:
2444:
2439:
2434:
2433:
2424:
2423:
2413:
2408:
2390:
2388:
2375:
2374:
2361:
2310:
2308:
2307:
2302:
2300:
2291:
2248:
2246:
2245:
2240:
2198:
2196:
2195:
2190:
2163:
2161:
2160:
2155:
2113:
2109:
2094:
2073:
2071:
2070:
2065:
2060:
2052:
2047:
2045:
2031:
2017:
2012:
2010:
2003:
2002:
1992:
1985:
1984:
1974:
1943:
1941:
1940:
1935:
1905:
1903:
1902:
1897:
1895:
1893:
1886:
1885:
1875:
1868:
1867:
1857:
1828:
1826:
1825:
1820:
1799:
1797:
1796:
1791:
1789:
1780:
1767:
1765:
1764:
1759:
1754:
1749:
1729:
1728:
1706:
1704:
1703:
1698:
1696:
1694:
1684:
1683:
1667:
1657:
1656:
1646:
1613:
1611:
1610:
1605:
1603:
1601:
1594:
1593:
1583:
1567:
1566:
1556:
1535:
1524:
1522:
1521:
1516:
1514:
1512:
1502:
1501:
1485:
1475:
1474:
1464:
1443:
1420:
1419:
1412:network analysis
1355:
1349:
1343:
1334:
1332:
1331:
1326:
1295:
1289:
1279:
1273:
1249:
1247:
1246:
1241:
1236:
1219:real polynomials
1216:
1214:
1213:
1208:
1187:
1185:
1184:
1179:
1148:
1146:
1145:
1140:
1134:
1132:
1122:
1121:
1111:
1101:
1100:
1090:
1080:
1078:
1077:
1072:
1069:
1067:
1053:
1039:
1029:
1027:
1026:
1021:
961:
959:
958:
953:
950:
948:
934:
920:
910:
908:
907:
902:
899:
897:
883:
869:
855:
853:
852:
847:
842:
841:
825:
823:
822:
817:
805:
803:
802:
797:
795:
794:
778:
776:
775:
770:
758:
756:
755:
750:
735:
733:
732:
727:
715:
713:
712:
707:
692:
690:
689:
684:
679:
677:
667:
666:
656:
646:
645:
635:
621:
620:
601:
599:
598:
593:
587:
586:
563:
561:
560:
555:
549:
548:
525:
523:
522:
517:
500:
498:
497:
492:
478:
476:
475:
470:
453:
451:
450:
445:
424:
422:
421:
416:
404:
402:
401:
396:
376:
374:
373:
368:
356:
354:
353:
348:
332:
330:
329:
324:
312:
310:
309:
304:
289:
287:
286:
281:
279:
277:
263:
249:
222:
220:
219:
214:
194:
179:is a field, the
178:
171:
159:
155:
146:
140:
132:rational numbers
87:
80:
76:
73:
67:
62:this article by
53:inline citations
40:
39:
32:
4456:
4455:
4451:
4450:
4449:
4447:
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4445:
4416:
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4407:
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4309:
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4288:
4278:
4276:
4274:
4250:
4246:
4242:
4236:
4217:
4200:
4196:
4191:
4158:
4148:(filters) with
4076:
4068:projective line
4040:coordinate ring
4027:
4018:
4003:
3994:
3988:indeterminates
3978:
3972:
3910:
3907:
3906:
3902:
3893:
3887:
3881:
3875:
3845:
3831:
3829:
3812:
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3808:
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3666:
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3655:
3609:
3605:
3604:
3599:
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3594:
3593:
3591:
3582:
3554:
3550:
3541:
3537:
3533:
3528:
3526:
3523:
3522:
3520:
3505:
3363:polynomial ring
3323:
3298:
3263:
3247:
3243:
3228:
3224:
3211:
3202:
3198:
3196:
3193:
3192:
3168:
3159:
3155:
3153:
3150:
3149:
3111:
3102:
3098:
3096:
3093:
3092:
3054:
3050:
3041:
3037:
3019:
3015:
3000:
2996:
2987:
2976:
2957:
2953:
2944:
2940:
2925:
2921:
2910:
2907:
2906:
2880:
2876:
2870:
2866:
2860:
2849:
2833:
2829:
2817:
2813:
2807:
2796:
2783:
2779:
2767:
2763:
2757:
2746:
2734:
2731:
2730:
2700:
2696:
2690:
2686:
2680:
2669:
2647:
2643:
2637:
2633:
2627:
2616:
2597:
2593:
2587:
2583:
2577:
2566:
2554:
2551:
2550:
2530:
2526:
2520:
2516:
2510:
2499:
2474:
2470:
2459:
2456:
2455:
2429:
2425:
2419:
2415:
2409:
2398:
2370:
2366:
2365:
2360:
2358:
2355:
2354:
2333:
2289:
2272:
2269:
2268:
2204:
2201:
2200:
2169:
2166:
2165:
2125:
2122:
2121:
2111:
2100:
2085:
2051:
2032:
2018:
2016:
1998:
1994:
1993:
1980:
1976:
1975:
1973:
1956:
1953:
1952:
1926:
1923:
1922:
1915:complex numbers
1881:
1877:
1876:
1863:
1859:
1858:
1856:
1839:
1836:
1835:
1805:
1802:
1801:
1778:
1776:
1773:
1772:
1748:
1724:
1720:
1718:
1715:
1714:
1679:
1675:
1668:
1652:
1648:
1647:
1645:
1628:
1625:
1624:
1618:
1617:
1616:
1615:
1614:
1589:
1585:
1584:
1562:
1558:
1557:
1555:
1547:
1544:
1543:
1536:
1527:
1526:
1525:
1497:
1493:
1486:
1470:
1466:
1465:
1463:
1455:
1452:
1451:
1444:
1435:
1434:
1428:
1417:
1416:
1371:In the case of
1358:critical values
1351:
1345:
1339:
1304:
1301:
1300:
1291:
1285:
1275:
1269:
1255:
1232:
1230:
1227:
1226:
1223:proper fraction
1193:
1190:
1189:
1164:
1161:
1160:
1117:
1113:
1112:
1096:
1092:
1091:
1089:
1086:
1083:
1082:
1054:
1040:
1038:
1035:
1032:
1031:
1030:. In this case
967:
964:
963:
935:
921:
919:
916:
913:
912:
884:
870:
868:
865:
862:
861:
837:
833:
831:
828:
827:
811:
808:
807:
790:
786:
784:
781:
780:
764:
761:
760:
741:
738:
737:
721:
718:
717:
701:
698:
697:
662:
658:
657:
641:
637:
636:
634:
616:
612:
610:
607:
606:
582:
578:
569:
566:
565:
544:
540:
531:
528:
527:
526:, then setting
509:
506:
505:
484:
481:
480:
462:
459:
458:
430:
427:
426:
410:
407:
406:
390:
387:
386:
362:
359:
358:
342:
339:
338:
318:
315:
314:
298:
295:
294:
264:
250:
248:
231:
228:
227:
208:
205:
204:
201:
192:
176:
169:
157:
153:
144:
138:
88:
77:
71:
68:
58:Please help to
57:
41:
37:
28:
17:
12:
11:
5:
4454:
4444:
4443:
4438:
4433:
4428:
4414:
4413:
4406:
4405:External links
4403:
4402:
4401:
4396:
4379:
4358:
4357:
4339:
4332:
4314:
4307:
4286:
4272:
4255:Linear Systems
4244:
4241:
4240:
4234:
4221:
4215:
4201:
4193:
4192:
4190:
4187:
4186:
4185:
4179:
4174:
4169:
4164:
4157:
4154:
4123:wave functions
4094:introduced by
4075:
4072:
4023:
4016:
3999:
3992:
3974:Main article:
3971:
3968:
3960:Riemann sphere
3942:Riemann sphere
3929:
3926:
3923:
3920:
3917:
3914:
3901:The domain of
3872:
3871:
3857:
3854:
3851:
3848:
3843:
3840:
3837:
3834:
3828:
3825:
3822:
3819:
3816:
3798:
3797:
3782:
3779:
3776:
3773:
3768:
3764:
3757:
3753:
3741:
3734:
3732:
3717:
3714:
3709:
3705:
3699:
3696:
3693:
3690:
3687:
3682:
3678:
3665:
3658:
3656:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3612:
3608:
3603:
3592:
3585:
3583:
3568:
3565:
3562:
3557:
3553:
3549:
3544:
3540:
3536:
3532:
3521:
3514:
3512:
3504:
3501:
3322:
3319:
3291:
3290:
3279:
3276:
3273:
3261:
3256:
3253:
3250:
3246:
3242:
3237:
3234:
3231:
3227:
3223:
3218:
3215:
3210:
3205:
3201:
3189:
3188:
3175:
3172:
3167:
3162:
3158:
3135:
3134:
3123:
3118:
3115:
3110:
3105:
3101:
3074:
3073:
3062:
3057:
3053:
3049:
3044:
3040:
3036:
3033:
3028:
3025:
3022:
3018:
3014:
3009:
3006:
3003:
2999:
2995:
2990:
2985:
2982:
2979:
2975:
2971:
2968:
2965:
2960:
2956:
2952:
2947:
2943:
2939:
2936:
2933:
2928:
2924:
2920:
2917:
2914:
2900:
2899:
2888:
2883:
2879:
2873:
2869:
2863:
2858:
2855:
2852:
2848:
2844:
2841:
2836:
2832:
2826:
2823:
2820:
2816:
2810:
2805:
2802:
2799:
2795:
2791:
2786:
2782:
2776:
2773:
2770:
2766:
2760:
2755:
2752:
2749:
2745:
2741:
2738:
2720:
2719:
2708:
2703:
2699:
2693:
2689:
2683:
2678:
2675:
2672:
2668:
2664:
2661:
2656:
2653:
2650:
2646:
2640:
2636:
2630:
2625:
2622:
2619:
2615:
2611:
2606:
2603:
2600:
2596:
2590:
2586:
2580:
2575:
2572:
2569:
2565:
2561:
2558:
2547:
2546:
2533:
2529:
2523:
2519:
2513:
2508:
2505:
2502:
2498:
2494:
2491:
2488:
2485:
2482:
2477:
2473:
2469:
2466:
2463:
2449:
2448:
2437:
2432:
2428:
2422:
2418:
2412:
2407:
2404:
2401:
2397:
2393:
2387:
2384:
2381:
2378:
2373:
2369:
2364:
2332:
2329:
2297:
2294:
2288:
2285:
2282:
2279:
2276:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2188:
2185:
2182:
2179:
2176:
2173:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2075:
2074:
2063:
2058:
2055:
2050:
2044:
2041:
2038:
2035:
2030:
2027:
2024:
2021:
2015:
2009:
2006:
2001:
1997:
1991:
1988:
1983:
1979:
1972:
1969:
1966:
1963:
1960:
1946:imaginary unit
1933:
1930:
1907:
1906:
1892:
1889:
1884:
1880:
1874:
1871:
1866:
1862:
1855:
1852:
1849:
1846:
1843:
1818:
1815:
1812:
1809:
1786:
1783:
1769:
1768:
1757:
1752:
1747:
1744:
1741:
1738:
1735:
1732:
1727:
1723:
1708:
1707:
1693:
1690:
1687:
1682:
1678:
1674:
1671:
1666:
1663:
1660:
1655:
1651:
1644:
1641:
1638:
1635:
1632:
1600:
1597:
1592:
1588:
1582:
1579:
1576:
1573:
1570:
1565:
1561:
1554:
1551:
1537:
1530:
1529:
1528:
1511:
1508:
1505:
1500:
1496:
1492:
1489:
1484:
1481:
1478:
1473:
1469:
1462:
1459:
1445:
1438:
1437:
1436:
1432:
1431:
1430:
1429:
1427:
1424:
1336:
1335:
1323:
1320:
1317:
1314:
1311:
1308:
1254:
1251:
1239:
1235:
1206:
1203:
1200:
1197:
1177:
1174:
1171:
1168:
1137:
1131:
1128:
1125:
1120:
1116:
1110:
1107:
1104:
1099:
1095:
1066:
1063:
1060:
1057:
1052:
1049:
1046:
1043:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
947:
944:
941:
938:
933:
930:
927:
924:
896:
893:
890:
887:
882:
879:
876:
873:
845:
840:
836:
815:
793:
789:
768:
748:
745:
725:
705:
694:
693:
682:
676:
673:
670:
665:
661:
655:
652:
649:
644:
640:
633:
630:
627:
624:
619:
615:
590:
585:
581:
577:
574:
552:
547:
543:
539:
536:
514:
489:
467:
443:
440:
437:
434:
414:
394:
366:
346:
322:
302:
291:
290:
276:
273:
270:
267:
262:
259:
256:
253:
247:
244:
241:
238:
235:
212:
200:
197:
110:, which is an
90:
89:
72:September 2015
44:
42:
35:
15:
9:
6:
4:
3:
2:
4453:
4442:
4439:
4437:
4434:
4432:
4429:
4427:
4424:
4423:
4421:
4412:
4409:
4408:
4399:
4393:
4389:
4385:
4380:
4376:
4372:
4371:
4366:
4362:
4361:
4350:
4343:
4335:
4329:
4325:
4318:
4310:
4308:3-540-19375-8
4304:
4300:
4296:
4290:
4275:
4269:
4265:
4261:
4257:
4256:
4248:
4237:
4231:
4227:
4222:
4218:
4212:
4208:
4203:
4202:
4198:
4194:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4159:
4153:
4151:
4147:
4143:
4139:
4135:
4131:
4126:
4124:
4120:
4116:
4112:
4107:
4105:
4101:
4097:
4093:
4089:
4088:approximation
4085:
4084:interpolation
4081:
4071:
4069:
4065:
4061:
4057:
4053:
4049:
4045:
4041:
4037:
4034:
4029:
4026:
4022:
4015:
4011:
4007:
4002:
3998:
3991:
3987:
3983:
3977:
3967:
3965:
3961:
3956:
3954:
3949:
3947:
3943:
3927:
3924:
3918:
3912:
3899:
3896:
3890:
3884:
3878:
3852:
3846:
3838:
3832:
3826:
3820:
3814:
3807:
3806:
3805:
3803:
3780:
3777:
3774:
3771:
3766:
3762:
3755:
3751:
3738:
3733:
3715:
3712:
3707:
3703:
3697:
3694:
3691:
3688:
3685:
3680:
3676:
3662:
3657:
3636:
3633:
3630:
3627:
3624:
3618:
3615:
3610:
3606:
3601:
3589:
3584:
3566:
3563:
3560:
3555:
3551:
3547:
3542:
3538:
3534:
3530:
3518:
3513:
3510:
3507:
3506:
3500:
3498:
3494:
3490:
3486:
3482:
3478:
3474:
3470:
3466:
3461:
3459:
3455:
3452:chosen to be
3451:
3447:
3443:
3439:
3435:
3431:
3428:, there is a
3427:
3423:
3419:
3415:
3411:
3407:
3403:
3399:
3395:
3391:
3387:
3383:
3379:
3375:
3371:
3367:
3364:
3360:
3356:
3352:
3348:
3344:
3340:
3336:
3332:
3328:
3318:
3316:
3312:
3306:
3302:
3296:
3277:
3274:
3271:
3254:
3251:
3248:
3244:
3240:
3235:
3232:
3229:
3225:
3216:
3213:
3208:
3203:
3199:
3191:
3190:
3173:
3170:
3165:
3160:
3156:
3148:
3147:
3146:
3144:
3140:
3121:
3116:
3113:
3108:
3103:
3099:
3091:
3090:
3089:
3087:
3086:constant term
3083:
3079:
3060:
3055:
3051:
3042:
3038:
3034:
3031:
3026:
3023:
3020:
3016:
3012:
3007:
3004:
3001:
2997:
2983:
2980:
2977:
2973:
2969:
2966:
2958:
2954:
2950:
2945:
2941:
2937:
2931:
2926:
2922:
2918:
2915:
2912:
2905:
2904:
2903:
2886:
2881:
2877:
2871:
2867:
2856:
2853:
2850:
2846:
2842:
2839:
2834:
2830:
2824:
2821:
2818:
2814:
2803:
2800:
2797:
2793:
2789:
2784:
2780:
2774:
2771:
2768:
2764:
2753:
2750:
2747:
2743:
2739:
2736:
2729:
2728:
2727:
2725:
2706:
2701:
2697:
2691:
2687:
2676:
2673:
2670:
2666:
2662:
2659:
2654:
2651:
2648:
2644:
2638:
2634:
2623:
2620:
2617:
2613:
2609:
2604:
2601:
2598:
2594:
2588:
2584:
2573:
2570:
2567:
2563:
2559:
2556:
2549:
2548:
2531:
2527:
2521:
2517:
2506:
2503:
2500:
2496:
2489:
2486:
2483:
2480:
2475:
2471:
2464:
2461:
2454:
2453:
2452:
2435:
2430:
2426:
2420:
2416:
2405:
2402:
2399:
2395:
2391:
2385:
2382:
2379:
2376:
2371:
2367:
2362:
2353:
2352:
2351:
2350:For example,
2348:
2346:
2342:
2338:
2337:Taylor series
2331:Taylor series
2328:
2326:
2322:
2318:
2314:
2295:
2292:
2286:
2280:
2274:
2265:
2263:
2259:
2254:
2252:
2236:
2230:
2224:
2221:
2218:
2212:
2206:
2186:
2183:
2177:
2171:
2148:
2142:
2139:
2133:
2127:
2120:
2115:
2107:
2103:
2098:
2092:
2088:
2083:
2078:
2061:
2056:
2053:
2048:
2042:
2039:
2036:
2033:
2028:
2025:
2022:
2019:
2013:
2007:
2004:
1999:
1995:
1989:
1986:
1981:
1977:
1970:
1964:
1958:
1951:
1950:
1949:
1947:
1931:
1928:
1920:
1916:
1912:
1890:
1887:
1882:
1878:
1872:
1869:
1864:
1860:
1853:
1847:
1841:
1834:
1833:
1832:
1829:
1816:
1807:
1784:
1781:
1755:
1750:
1745:
1742:
1739:
1733:
1730:
1725:
1721:
1713:
1712:
1711:
1688:
1685:
1680:
1676:
1669:
1664:
1661:
1658:
1653:
1649:
1642:
1636:
1630:
1623:
1622:
1621:
1598:
1595:
1590:
1586:
1580:
1577:
1574:
1571:
1568:
1563:
1559:
1552:
1549:
1541:
1534:
1506:
1503:
1498:
1494:
1487:
1482:
1479:
1476:
1471:
1467:
1460:
1457:
1449:
1442:
1423:
1421:
1413:
1409:
1404:
1402:
1398:
1393:
1391:
1387:
1382:
1380:
1379:
1374:
1369:
1367:
1363:
1359:
1354:
1348:
1342:
1321:
1318:
1312:
1306:
1299:
1298:
1297:
1294:
1288:
1283:
1278:
1272:
1267:
1263:
1258:
1250:
1237:
1224:
1220:
1217:and both are
1201:
1195:
1172:
1166:
1158:
1154:
1149:
1135:
1126:
1118:
1114:
1105:
1097:
1093:
1061:
1055:
1047:
1041:
1014:
1008:
1002:
996:
993:
987:
981:
975:
969:
942:
936:
928:
922:
891:
885:
877:
871:
859:
843:
838:
834:
813:
791:
787:
766:
746:
743:
723:
703:
680:
671:
663:
659:
650:
642:
638:
631:
625:
617:
613:
605:
604:
603:
588:
583:
579:
575:
572:
550:
545:
541:
537:
534:
512:
504:
487:
465:
455:
454:is not zero.
438:
432:
412:
392:
384:
380:
379:zero function
364:
344:
336:
320:
300:
271:
265:
257:
251:
245:
239:
233:
226:
225:
224:
210:
196:
190:
186:
182:
173:
167:
163:
151:
147:
137:
133:
129:
125:
121:
117:
113:
109:
105:
101:
97:
86:
83:
75:
65:
61:
55:
54:
48:
43:
34:
33:
30:
26:
22:
4387:
4368:
4342:
4326:. Springer.
4323:
4317:
4298:
4295:Bourbaki, N.
4289:
4277:. Retrieved
4254:
4247:
4225:
4206:
4197:
4127:
4119:spectroscopy
4117:in physics,
4108:
4077:
4074:Applications
4063:
4059:
4055:
4043:
4035:
4030:
4024:
4020:
4013:
4009:
4005:
4000:
3996:
3989:
3985:
3979:
3957:
3950:
3900:
3894:
3888:
3882:
3876:
3873:
3799:
3496:
3492:
3488:
3484:
3480:
3472:
3468:
3464:
3462:
3449:
3445:
3441:
3437:
3433:
3421:
3417:
3413:
3409:
3405:
3401:
3397:
3393:
3389:
3385:
3381:
3377:
3373:
3369:
3365:
3354:
3346:
3342:
3338:
3334:
3324:
3304:
3300:
3292:
3143:coefficients
3138:
3136:
3077:
3075:
2901:
2723:
2721:
2450:
2349:
2334:
2324:
2320:
2312:
2266:
2255:
2250:
2116:
2105:
2101:
2090:
2086:
2079:
2076:
1918:
1911:real numbers
1908:
1830:
1770:
1709:
1619:
1415:
1405:
1400:
1394:
1383:
1376:
1370:
1357:
1352:
1346:
1340:
1337:
1292:
1286:
1282:lowest terms
1276:
1270:
1261:
1259:
1256:
1152:
1150:
695:
457:However, if
456:
292:
202:
174:
142:
128:coefficients
107:
99:
93:
78:
69:
50:
29:
4138:z-transform
3982:polynomials
1917:, since if
1362:at infinity
826:to that of
377:is not the
203:A function
199:Definitions
160:. Then the
156:containing
124:polynomials
120:denominator
96:mathematics
64:introducing
4420:Categories
4333:9048194431
4299:Algebra II
4279:5 November
4235:0133323048
4216:0203911377
4189:References
4096:Henri Padé
3509:Julia sets
3483:, because
2345:like terms
1944:(i.e. the
47:references
4375:EMS Press
3925:≠
3772:−
3686:−
3631:−
3625:−
3275:≥
3252:−
3241:−
3233:−
3024:−
3013:−
3005:−
2989:∞
2974:∑
2951:−
2862:∞
2847:∑
2822:−
2809:∞
2794:∑
2790:−
2772:−
2759:∞
2744:∑
2726:, we get
2682:∞
2667:∑
2629:∞
2614:∑
2610:−
2579:∞
2564:∑
2512:∞
2497:∑
2481:−
2411:∞
2396:∑
2377:−
2225:
2034:−
2020:−
1929:−
1814:∞
1811:→
1746:±
1737:⇔
1686:−
1659:−
1596:−
1578:−
1569:−
1542: 3:
1504:−
1477:−
1450: 3:
1356:, called
150:variables
116:numerator
4297:(1990).
4156:See also
4104:software
3962:creates
3458:fraction
2084:such as
1426:Examples
166:codomain
118:and the
104:function
4377:, 2001
4048:Zariski
3412:, when
3361:of the
3349:or, in
3080:in the
2262:subring
1388:of the
1373:complex
1266:degrees
381:. The
187:of the
183:of the
102:is any
60:improve
4394:
4330:
4305:
4270:
4232:
4213:
4132:, the
4115:forces
4111:fields
4019:,...,
3995:,...,
3475:by (a
3408:, and
3269:
2256:Every
2117:Every
1540:degree
1448:degree
1401:degree
1399:, the
1386:degree
1262:degree
1253:Degree
1157:degree
383:domain
293:where
162:domain
126:. The
49:, but
4352:(PDF)
4052:dense
3980:Like
3781:0.025
3716:0.917
3454:monic
3440:with
3424:is a
3376:with
3331:field
3299:1 / (
2097:value
2093:) = π
1390:graph
191:over
143:over
136:field
4392:ISBN
4328:ISBN
4303:ISBN
4281:2022
4268:ISBN
4230:ISBN
4211:ISBN
4113:and
4086:and
4082:for
3886:and
3444:and
3353:, a
3341:, a
1410:and
1384:The
1338:has
1274:and
911:and
779:and
564:and
479:and
357:and
333:are
313:and
185:ring
122:are
98:, a
4260:doi
4128:In
4042:of
4028:).
3948:).
3800:In
3695:0.7
3689:0.2
3325:In
3265:for
2251:not
2222:sin
2099:of
1800:as
1406:In
1368:).
1290:is
1225:in
1159:of
385:of
337:of
168:is
94:In
4422::
4386:,
4373:,
4367:,
4266:.
4070:.
3966:.
3955:.
3499:.
3479:)
3418:QR
3416:=
3414:PS
3404:,
3400:,
3317:.
3303:+
3301:ax
3278:2.
2187:1.
2114:.
2080:A
1422:.
1381:.
1151:A
195:.
172:.
4354:.
4336:.
4311:.
4283:.
4262::
4238:.
4219:.
4064:U
4060:f
4056:V
4050:-
4044:V
4036:V
4025:n
4021:X
4017:1
4014:X
4012:(
4010:F
4006:F
4001:n
3997:X
3993:1
3990:X
3986:n
3944:(
3928:0
3922:)
3919:z
3916:(
3913:Q
3903:f
3895:f
3889:Q
3883:P
3877:Q
3856:)
3853:z
3850:(
3847:Q
3842:)
3839:z
3836:(
3833:P
3827:=
3824:)
3821:z
3818:(
3815:f
3778:+
3775:z
3767:9
3763:z
3756:2
3752:z
3713:+
3708:2
3704:z
3698:i
3692:+
3681:2
3677:z
3640:)
3637:i
3634:3
3628:3
3622:(
3619:z
3616:+
3611:3
3607:z
3602:1
3567:z
3564:b
3561:+
3556:3
3552:z
3548:+
3543:5
3539:z
3535:a
3531:1
3497:X
3493:F
3489:X
3487:(
3485:F
3481:X
3473:F
3469:X
3467:(
3465:F
3450:Q
3446:Q
3442:P
3438:Q
3436:/
3434:P
3422:F
3410:S
3406:R
3402:Q
3398:P
3394:S
3392:/
3390:R
3386:Q
3384:/
3382:P
3378:Q
3374:Q
3372:/
3370:P
3366:F
3339:X
3335:F
3307:)
3305:b
3272:k
3260:)
3255:2
3249:k
3245:a
3236:1
3230:k
3226:a
3222:(
3217:2
3214:1
3209:=
3204:k
3200:a
3174:4
3171:1
3166:=
3161:1
3157:a
3139:x
3122:.
3117:2
3114:1
3109:=
3104:0
3100:a
3078:x
3061:.
3056:k
3052:x
3048:)
3043:k
3039:a
3035:2
3032:+
3027:1
3021:k
3017:a
3008:2
3002:k
2998:a
2994:(
2984:2
2981:=
2978:k
2970:+
2967:x
2964:)
2959:0
2955:a
2946:1
2942:a
2938:2
2935:(
2932:+
2927:0
2923:a
2919:2
2916:=
2913:1
2887:.
2882:k
2878:x
2872:k
2868:a
2857:0
2854:=
2851:k
2843:2
2840:+
2835:k
2831:x
2825:1
2819:k
2815:a
2804:1
2801:=
2798:k
2785:k
2781:x
2775:2
2769:k
2765:a
2754:2
2751:=
2748:k
2740:=
2737:1
2724:x
2707:.
2702:k
2698:x
2692:k
2688:a
2677:0
2674:=
2671:k
2663:2
2660:+
2655:1
2652:+
2649:k
2645:x
2639:k
2635:a
2624:0
2621:=
2618:k
2605:2
2602:+
2599:k
2595:x
2589:k
2585:a
2574:0
2571:=
2568:k
2560:=
2557:1
2532:k
2528:x
2522:k
2518:a
2507:0
2504:=
2501:k
2493:)
2490:2
2487:+
2484:x
2476:2
2472:x
2468:(
2465:=
2462:1
2436:.
2431:k
2427:x
2421:k
2417:a
2406:0
2403:=
2400:k
2392:=
2386:2
2383:+
2380:x
2372:2
2368:x
2363:1
2325:x
2323:/
2321:x
2313:x
2296:x
2293:x
2287:=
2284:)
2281:x
2278:(
2275:f
2237:,
2234:)
2231:x
2228:(
2219:=
2216:)
2213:x
2210:(
2207:f
2184:=
2181:)
2178:x
2175:(
2172:Q
2152:)
2149:x
2146:(
2143:P
2140:=
2137:)
2134:x
2131:(
2128:f
2112:x
2108:)
2106:x
2104:(
2102:f
2091:x
2089:(
2087:f
2062:,
2057:0
2054:1
2049:=
2043:1
2040:+
2037:1
2029:2
2026:+
2023:1
2014:=
2008:1
2005:+
2000:2
1996:i
1990:2
1987:+
1982:2
1978:i
1971:=
1968:)
1965:i
1962:(
1959:f
1932:1
1919:x
1891:1
1888:+
1883:2
1879:x
1873:2
1870:+
1865:2
1861:x
1854:=
1851:)
1848:x
1845:(
1842:f
1817:.
1808:x
1785:2
1782:x
1756:.
1751:5
1743:=
1740:x
1734:5
1731:=
1726:2
1722:x
1692:)
1689:5
1681:2
1677:x
1673:(
1670:2
1665:x
1662:2
1654:3
1650:x
1643:=
1640:)
1637:x
1634:(
1631:f
1599:4
1591:2
1587:x
1581:2
1575:x
1572:3
1564:2
1560:x
1553:=
1550:y
1510:)
1507:5
1499:2
1495:x
1491:(
1488:2
1483:x
1480:2
1472:3
1468:x
1461:=
1458:y
1353:w
1347:z
1341:d
1322:w
1319:=
1316:)
1313:z
1310:(
1307:f
1293:d
1287:f
1277:Q
1271:P
1238:.
1234:Q
1205:)
1202:x
1199:(
1196:Q
1176:)
1173:x
1170:(
1167:P
1136:.
1130:)
1127:x
1124:(
1119:1
1115:Q
1109:)
1106:x
1103:(
1098:1
1094:P
1065:)
1062:x
1059:(
1056:Q
1051:)
1048:x
1045:(
1042:P
1018:)
1015:x
1012:(
1009:C
1006:)
1003:x
1000:(
997:B
994:=
991:)
988:x
985:(
982:D
979:)
976:x
973:(
970:A
946:)
943:x
940:(
937:D
932:)
929:x
926:(
923:C
895:)
892:x
889:(
886:B
881:)
878:x
875:(
872:A
844:.
839:1
835:f
814:f
792:1
788:f
767:f
747:.
744:f
724:f
704:f
681:,
675:)
672:x
669:(
664:1
660:Q
654:)
651:x
648:(
643:1
639:P
632:=
629:)
626:x
623:(
618:1
614:f
589:R
584:1
580:Q
576:=
573:Q
551:R
546:1
542:P
538:=
535:P
513:R
488:Q
466:P
442:)
439:x
436:(
433:Q
413:x
393:f
365:Q
345:x
321:Q
301:P
275:)
272:x
269:(
266:Q
261:)
258:x
255:(
252:P
246:=
243:)
240:x
237:(
234:f
211:f
193:K
177:K
170:L
158:K
154:L
145:K
139:K
85:)
79:(
74:)
70:(
56:.
27:.
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