6068:
3743:
6063:{\displaystyle {\begin{aligned}\sum _{k=0}^{\infty }&(k+r)(k+r-1)A_{k}z^{k+r-2}-{\frac {1}{z}}\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r-1}+\left({\frac {1}{z^{2}}}-{\frac {1}{z}}\right)\sum _{k=0}^{\infty }A_{k}z^{k+r}\\&=\sum _{k=0}^{\infty }(k+r)(k+r-1)A_{k}z^{k+r-2}-{\frac {1}{z}}\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r-1}+{\frac {1}{z^{2}}}\sum _{k=0}^{\infty }A_{k}z^{k+r}-{\frac {1}{z}}\sum _{k=0}^{\infty }A_{k}z^{k+r}\\&=\sum _{k=0}^{\infty }(k+r)(k+r-1)A_{k}z^{k+r-2}-\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r-2}+\sum _{k=0}^{\infty }A_{k}z^{k+r-2}-\sum _{k=0}^{\infty }A_{k}z^{k+r-1}\\&=\sum _{k=0}^{\infty }(k+r)(k+r-1)A_{k}z^{k+r-2}-\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r-2}+\sum _{k=0}^{\infty }A_{k}z^{k+r-2}-\sum _{k-1=0}^{\infty }A_{k-1}z^{k-1+r-1}\\&=\sum _{k=0}^{\infty }(k+r)(k+r-1)A_{k}z^{k+r-2}-\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r-2}+\sum _{k=0}^{\infty }A_{k}z^{k+r-2}-\sum _{k=1}^{\infty }A_{k-1}z^{k+r-2}\\&=\left\{\sum _{k=0}^{\infty }\left((k+r)(k+r-1)-(k+r)+1\right)A_{k}z^{k+r-2}\right\}-\sum _{k=1}^{\infty }A_{k-1}z^{k+r-2}\\&=\left\{\left(r(r-1)-r+1\right)A_{0}z^{r-2}+\sum _{k=1}^{\infty }\left((k+r)(k+r-1)-(k+r)+1\right)A_{k}z^{k+r-2}\right\}-\sum _{k=1}^{\infty }A_{k-1}z^{k+r-2}\\&=(r-1)^{2}A_{0}z^{r-2}+\left\{\sum _{k=1}^{\infty }(k+r-1)^{2}A_{k}z^{k+r-2}-\sum _{k=1}^{\infty }A_{k-1}z^{k+r-2}\right\}\\&=(r-1)^{2}A_{0}z^{r-2}+\sum _{k=1}^{\infty }\left((k+r-1)^{2}A_{k}-A_{k-1}\right)z^{k+r-2}\end{aligned}}}
20:
2004:
912:
1999:{\displaystyle {\begin{aligned}&z^{2}\sum _{k=0}^{\infty }(k+r-1)(k+r)A_{k}z^{k+r-2}+zp(z)\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r-1}+q(z)\sum _{k=0}^{\infty }A_{k}z^{k+r}\\={}&\sum _{k=0}^{\infty }(k+r-1)(k+r)A_{k}z^{k+r}+p(z)\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r}+q(z)\sum _{k=0}^{\infty }A_{k}z^{k+r}\\={}&\sum _{k=0}^{\infty }\\={}&\sum _{k=0}^{\infty }\leftA_{k}z^{k+r}\\={}&\leftA_{0}z^{r}+\sum _{k=1}^{\infty }\leftA_{k}z^{k+r}=0\end{aligned}}}
2928:
2345:
3736:
2923:{\displaystyle {\begin{aligned}I(k+r)A_{k}+\sum _{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \over (k-j)!}A_{j}&=0\\\sum _{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \over (k-j)!}A_{j}&=-I(k+r)A_{k}\\{1 \over -I(k+r)}\sum _{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \over (k-j)!}A_{j}&=A_{k}\end{aligned}}}
6337:
The previous example involved an indicial polynomial with a repeated root, which gives only one solution to the given differential equation. In general, the
Frobenius method gives two independent solutions provided that the indicial equation's roots are not separated by an integer (including zero).
3452:
3464:
2127: β 1 or, something else depending on the given differential equation. This detail is important to keep in mind. In the process of synchronizing all the series of the differential equation to start at the same index value (which in the above expression is
2336:
3179:
905:
455:
673:
6448:
2097:
6961:
In cases in which roots of the indicial polynomial differ by an integer (including zero), the coefficients of all series involved in second linearly independent solutions can be calculated straightforwardly from
780:
2350:
535:
A first contribution by
Frobenius to the theory was to show that - as regards a first, linearly independent solution, which then has the form of an analytical power series multiplied by an arbitrary power
6206:
3027:
3296:
296:
186:
3285:
3748:
3731:{\displaystyle {\begin{aligned}f&=\sum _{k=0}^{\infty }A_{k}z^{k+r}\\f'&=\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r-1}\\f''&=\sum _{k=0}^{\infty }(k+r)(k+r-1)A_{k}z^{k+r-2}\end{aligned}}}
3469:
917:
558:
A large part of
Frobenius' 1873 publication was devoted to proofs of convergence of all the series involved in the solutions, as well as establishing the radii of convergence of these series.
6771:
6266:
234:
2147:
2131: = 1), one can end up with complicated expressions. However, in solving for the indicial roots attention is focused only on the coefficient of the lowest power of
54:
6647:
6319:
3203:, we gain a solution to the differential equation. If the difference between the roots is not an integer, we get another, linearly independent solution in the other root.
6884:
6837:
6484:
6944:
6914:
6677:
6600:
6569:
6542:
6511:
569:
358:
6799:
328:
77:
6344:
2011:
784:
6946:
which can be set arbitrarily. If it is set to zero then with this differential equation all the other coefficients will be zero and we obtain the solution
3032:
551:
of the second linearly independent solution (see below) can be obtained by a procedure which is based on differentiation with respect to the parameter
363:
680:
6089:
6966:. These tandem relations can be constructed by further developing Frobenius' original invention of differentiating with respect to the parameter
458:
2946:
547:
A second contribution by
Frobenius was to show that, in cases in which the roots of the indicial equation differ by an integer, the general
6211:
7202:
106:
7043:
3214:
7158:"Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points"
6710:
6270:
Given some initial conditions, we can either solve the recurrence entirely or obtain a solution in power series form.
7132:
Fuchs, Lazarus
Immanuel (1866). "Zur Theorie der linearen Differentialgleichungen mit veranderlichen Coefficienten".
7117:
Fuchs, Lazarus
Immanuel (1865). "Zur Theorie der linearen Differentialgleichungen mit veranderlichen Coefficienten".
7101:
6326:
500:. The Frobenius method enables one to create a power series solution to such a differential equation, provided that
239:
3447:{\displaystyle f''-{1 \over z}f'+{1-z \over z^{2}}f=f''-{1 \over z}f'+\left({1 \over z^{2}}-{1 \over z}\right)f=0}
100:
19:
7074:
Frobenius, Ferdinand Georg (1968) . "Uber die
Integration der linearen Differentialgleichungen durch Reihen".
7035:
516:) are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist (and are finite).
191:
528:
of the series solutions involved (see below). These forms had all been established earlier, by Fuchs. The
6341:
If the root is repeated or the roots differ by an integer, then the second solution can be found using:
26:
6605:
92:
7031:
6276:
540:
of the independent variable (see below) - the coefficients of the generalized power series obey a
6842:
2340:
These coefficients must be zero, since they should be solutions of the differential equation, so
2331:{\displaystyle I(k+r)A_{k}+\sum _{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \over (k-j)!}A_{j},}
6984:
6956:
302:
6804:
6453:
23:
Some solutions of a differential equation having a regular singular point with indicial roots
6919:
6889:
6652:
6690:
6578:
6547:
6520:
6489:
336:
8:
6776:
307:
561:
59:
6970:, and using this approach to actually calculate the series coefficients in all cases.
7179:
7097:
7051:
7039:
7025:
7006:
7003:
6979:
6322:
490:
7169:
6332:
6886:
has a power series starting with the power zero. In a power series starting with
96:
6989:
6773:
The roots of the indicial equation are β1 and 0. Two independent solutions are
6486:
is the first solution (based on the larger root in the case of unequal roots),
6916:
the recurrence relation places no restriction on the coefficient for the term
668:{\displaystyle u(z)=z^{r}\sum _{k=0}^{\infty }A_{k}z^{k},\qquad (A_{0}\neq 0)}
7196:
7183:
7021:
6957:
Tandem recurrence relations for series coefficients in the exceptional cases
7174:
7157:
6839:
so we see that the logarithm does not appear in any solution. The solution
2119:
th coefficient but, it is possible for the lowest possible exponent to be
84:
7094:
Linear
Differential Equations and Group Theory from Riemann to Poincare
6443:{\displaystyle y_{2}=Cy_{1}\ln x+\sum _{k=0}^{\infty }B_{k}x^{k+r_{2}}}
2115:
in the infinite series. In this case it happens to be that this is the
2092:{\displaystyle r\left(r-1\right)+p\left(0\right)r+q\left(0\right)=I(r)}
566:
The method of
Frobenius is to seek a power series solution of the form
6080:
we get a double root of 1. Using this root, we set the coefficient of
900:{\displaystyle u''(z)=\sum _{k=0}^{\infty }(k+r-1)(k+r)A_{k}z^{k+r-2}}
7011:
6649:, which can be set arbitrarily. This then determines the rest of the
3174:{\displaystyle z^{2}U_{r}(z)''+p(z)zU_{r}(z)'+q(z)U_{r}(z)=I(r)z^{r}}
562:
Explanation of
Frobenius Method: first, linearly independent solution
450:{\displaystyle u''+{\frac {p(z)}{z}}u'+{\frac {q(z)}{z^{2}}}u=0}
909:
Substituting the above differentiation into our original ODE:
775:{\displaystyle u'(z)=\sum _{k=0}^{\infty }(k+r)A_{k}z^{k+r-1}}
532:(see below) and its role had also been established by Fuchs.
6201:{\displaystyle (k+1-1)^{2}A_{k}-A_{k-1}=k^{2}A_{k}-A_{k-1}=0}
3183:
If we choose one of the roots to the indicial polynomial for
544:, so that they can always be straightforwardly calculated.
524:
Frobenius' contribution was not so much in all the possible
519:
3022:{\displaystyle U_{r}(z)=\sum _{k=0}^{\infty }A_{k}z^{k+r}}
7001:
2138:
Using this, the general expression of the coefficient of
7054:). Chapter 4 contains the full method including proofs.
7078:(in German). Berlin: Springer-Verlag. pp. 84β105.
6333:"The exceptional cases": roots separated by an integer
6086:
to be zero (for it to be a solution), which gives us:
242:
194:
7027:
Ordinary Differential Equations and Dynamical Systems
6922:
6892:
6845:
6807:
6779:
6713:
6655:
6608:
6581:
6550:
6523:
6492:
6456:
6347:
6279:
6214:
6092:
3746:
3467:
3299:
3217:
3035:
2949:
2348:
2150:
2014:
915:
787:
683:
572:
366:
339:
310:
109:
62:
29:
7052:https://www.mat.univie.ac.at/~gerald/ftp/book-ode/
6938:
6908:
6878:
6831:
6793:
6765:
6671:
6641:
6594:
6563:
6536:
6505:
6478:
6442:
6313:
6260:
6200:
6062:
3730:
3446:
3279:
3173:
3021:
2922:
2330:
2091:
1998:
899:
774:
667:
449:
352:
322:
290:
228:
180:
71:
48:
16:Method for solving ordinary differential equations
7194:
6689:: consider the following differential equation (
6571:is chosen (for example by setting it to 1) then
7134:Journal fΓΌr die reine und angewandte Mathematik
291:{\textstyle u''\equiv {\frac {d^{2}u}{dz^{2}}}}
6261:{\displaystyle A_{k}={\frac {A_{k-1}}{k^{2}}}}
360:to obtain a differential equation of the form
7155:
7121:(in German). University Of Michigan Library.
3454:which has the requisite singularity at
7119:Gesammelte Mathematische Werke von L. Fuchs
7156:van der Toorn, Ramses (27 December 2022).
2111:is the coefficient of the lowest power of
7173:
7073:
520:History: Frobenius' actual contributions
457:which will not be solvable with regular
181:{\displaystyle z^{2}u''+p(z)zu'+q(z)u=0}
18:
6602:are determined up to but not including
6325:, the power series can be written as a
6208:hence we have the recurrence relation:
7195:
7020:
6513:is the smaller root, and the constant
229:{\textstyle u'\equiv {\frac {du}{dz}}}
7131:
7116:
7002:
3280:{\displaystyle z^{2}f''-zf'+(1-z)f=0}
7151:
7149:
7147:
7091:
7087:
7085:
7069:
7067:
7050:(Draft version available online at
99:solution for a linear second-order
13:
6402:
5962:
5837:
5753:
5628:
5497:
5354:
5223:
5149:
5093:
5022:
4930:
4855:
4793:
4722:
4630:
4567:
4511:
4440:
4348:
4291:
4231:
4143:
4041:
3984:
3873:
3767:
3651:
3564:
3498:
2988:
1864:
1631:
1432:
1372:
1295:
1197:
1137:
1054:
947:
824:
720:
614:
14:
7214:
7144:
7082:
7064:
6995:
6327:generalized hypergeometric series
6766:{\displaystyle zu''+(2-z)u'-u=0}
6273:Since the ratio of coefficients
2107:. The general definition of the
49:{\displaystyle r={\frac {1}{2}}}
7203:Ordinary differential equations
6642:{\displaystyle B_{r_{1}-r_{2}}}
642:
7125:
7110:
6865:
6846:
6740:
6728:
6473:
6467:
6112:
6093:
5991:
5972:
5908:
5895:
5777:
5758:
5694:
5681:
5558:
5546:
5540:
5522:
5519:
5507:
5432:
5420:
5284:
5272:
5266:
5248:
5245:
5233:
5039:
5027:
4968:
4950:
4947:
4935:
4739:
4727:
4668:
4650:
4647:
4635:
4457:
4445:
4386:
4368:
4365:
4353:
4160:
4148:
4079:
4061:
4058:
4046:
3890:
3878:
3809:
3791:
3788:
3776:
3689:
3671:
3668:
3656:
3581:
3569:
3265:
3253:
3158:
3152:
3143:
3137:
3124:
3118:
3105:
3098:
3082:
3076:
3063:
3056:
2966:
2960:
2880:
2868:
2863:
2857:
2852:
2840:
2829:
2823:
2818:
2806:
2798:
2786:
2750:
2738:
2709:
2697:
2665:
2653:
2648:
2642:
2637:
2625:
2614:
2608:
2603:
2591:
2583:
2571:
2508:
2496:
2491:
2485:
2480:
2468:
2457:
2451:
2446:
2434:
2426:
2414:
2368:
2356:
2306:
2294:
2289:
2283:
2278:
2266:
2255:
2249:
2244:
2232:
2224:
2212:
2166:
2154:
2086:
2080:
1952:
1946:
1937:
1925:
1922:
1916:
1907:
1895:
1892:
1874:
1817:
1811:
1799:
1793:
1784:
1772:
1719:
1713:
1704:
1692:
1689:
1683:
1674:
1662:
1659:
1641:
1599:
1570:
1564:
1529:
1517:
1514:
1508:
1473:
1461:
1458:
1440:
1437:
1353:
1347:
1312:
1300:
1276:
1270:
1235:
1223:
1220:
1202:
1118:
1112:
1071:
1059:
1035:
1029:
985:
973:
970:
952:
862:
850:
847:
829:
802:
796:
737:
725:
698:
692:
662:
643:
582:
576:
422:
416:
390:
384:
166:
160:
140:
134:
101:ordinary differential equation
1:
7058:
7036:American Mathematical Society
6314:{\displaystyle A_{k}/A_{k-1}}
2103:, which is quadratic in
7:
6973:
6964:tandem recurrence relations
6879:{\displaystyle (e^{z}-1)/z}
6679:In some cases the constant
6544:are to be determined. Once
10:
7219:
3206:
2932:The series solution with
93:Ferdinand Georg Frobenius
6832:{\displaystyle e^{z}/z,}
6479:{\displaystyle y_{1}(x)}
3461:Use the series solution
7076:Gesammelte Abhandlungen
301:in the vicinity of the
7175:10.3390/axioms12010032
7096:. Boston: Birkhauser.
6985:Regular singular point
6940:
6939:{\displaystyle z^{0},}
6910:
6909:{\displaystyle z^{-1}}
6880:
6833:
6795:
6767:
6673:
6672:{\displaystyle B_{k}.}
6643:
6596:
6565:
6538:
6507:
6480:
6444:
6406:
6315:
6262:
6202:
6064:
5966:
5841:
5757:
5632:
5501:
5358:
5227:
5153:
5097:
5026:
4934:
4859:
4797:
4726:
4634:
4571:
4515:
4444:
4352:
4295:
4235:
4147:
4045:
3988:
3877:
3771:
3732:
3655:
3568:
3502:
3448:
3281:
3175:
3023:
2992:
2924:
2782:
2567:
2410:
2332:
2208:
2093:
2000:
1868:
1635:
1436:
1376:
1299:
1201:
1141:
1058:
951:
901:
828:
776:
724:
669:
618:
451:
354:
324:
303:regular singular point
292:
230:
182:
95:, is a way to find an
80:
73:
50:
7092:Gray, Jeremy (1986).
6941:
6911:
6881:
6834:
6796:
6768:
6674:
6644:
6597:
6595:{\displaystyle B_{k}}
6566:
6564:{\displaystyle B_{0}}
6539:
6537:{\displaystyle B_{k}}
6517:and the coefficients
6508:
6506:{\displaystyle r_{2}}
6481:
6445:
6386:
6316:
6263:
6203:
6065:
5946:
5821:
5737:
5612:
5481:
5338:
5207:
5133:
5077:
5006:
4914:
4833:
4777:
4706:
4614:
4551:
4495:
4424:
4332:
4275:
4215:
4127:
4025:
3968:
3857:
3751:
3733:
3635:
3548:
3482:
3449:
3289:Divide throughout by
3282:
3176:
3024:
2972:
2925:
2756:
2541:
2384:
2333:
2182:
2094:
2001:
1848:
1615:
1416:
1356:
1279:
1181:
1121:
1038:
931:
902:
808:
777:
704:
670:
598:
452:
355:
353:{\displaystyle z^{2}}
325:
293:
231:
183:
74:
51:
22:
6920:
6890:
6843:
6805:
6777:
6711:
6653:
6606:
6579:
6548:
6521:
6490:
6454:
6345:
6277:
6212:
6090:
3744:
3465:
3297:
3215:
3033:
2947:
2346:
2148:
2012:
913:
785:
681:
570:
459:power series methods
364:
337:
308:
240:
192:
107:
60:
27:
6794:{\displaystyle 1/z}
2109:indicial polynomial
2101:indicial polynomial
555:, mentioned above.
542:recurrence relation
530:indicial polynomial
323:{\displaystyle z=0}
89:method of Frobenius
7007:"Frobenius Method"
7004:Weisstein, Eric W.
6936:
6906:
6876:
6829:
6791:
6763:
6669:
6639:
6592:
6561:
6534:
6503:
6476:
6440:
6311:
6258:
6198:
6060:
6058:
3740:Now, substituting
3728:
3726:
3444:
3277:
3171:
3019:
2920:
2918:
2328:
2089:
1996:
1994:
897:
772:
665:
447:
350:
333:One can divide by
320:
288:
226:
178:
81:
72:{\displaystyle -1}
69:
46:
7045:978-0-8218-8328-0
6691:Kummer's equation
6323:rational function
6256:
4273:
4213:
4125:
3961:
3948:
3855:
3428:
3415:
3382:
3355:
3319:
2887:
2754:
2672:
2515:
2313:
677:Differentiating:
436:
397:
286:
224:
44:
7210:
7188:
7187:
7177:
7153:
7142:
7141:
7129:
7123:
7122:
7114:
7108:
7107:
7089:
7080:
7079:
7071:
7049:
7017:
7016:
6952:
6945:
6943:
6942:
6937:
6932:
6931:
6915:
6913:
6912:
6907:
6905:
6904:
6885:
6883:
6882:
6877:
6872:
6858:
6857:
6838:
6836:
6835:
6830:
6822:
6817:
6816:
6800:
6798:
6797:
6792:
6787:
6772:
6770:
6769:
6764:
6750:
6724:
6706:
6699:
6682:
6678:
6676:
6675:
6670:
6665:
6664:
6648:
6646:
6645:
6640:
6638:
6637:
6636:
6635:
6623:
6622:
6601:
6599:
6598:
6593:
6591:
6590:
6574:
6570:
6568:
6567:
6562:
6560:
6559:
6543:
6541:
6540:
6535:
6533:
6532:
6516:
6512:
6510:
6509:
6504:
6502:
6501:
6485:
6483:
6482:
6477:
6466:
6465:
6449:
6447:
6446:
6441:
6439:
6438:
6437:
6436:
6416:
6415:
6405:
6400:
6373:
6372:
6357:
6356:
6320:
6318:
6317:
6312:
6310:
6309:
6294:
6289:
6288:
6267:
6265:
6264:
6259:
6257:
6255:
6254:
6245:
6244:
6229:
6224:
6223:
6207:
6205:
6204:
6199:
6191:
6190:
6172:
6171:
6162:
6161:
6149:
6148:
6130:
6129:
6120:
6119:
6085:
6079:
6069:
6067:
6066:
6061:
6059:
6055:
6054:
6033:
6029:
6028:
6027:
6009:
6008:
5999:
5998:
5965:
5960:
5942:
5941:
5926:
5925:
5916:
5915:
5888:
5884:
5880:
5879:
5878:
5857:
5856:
5840:
5835:
5817:
5816:
5795:
5794:
5785:
5784:
5756:
5751:
5728:
5727:
5712:
5711:
5702:
5701:
5674:
5670:
5669:
5648:
5647:
5631:
5626:
5608:
5604:
5603:
5602:
5581:
5580:
5571:
5567:
5500:
5495:
5477:
5476:
5461:
5460:
5451:
5447:
5400:
5396:
5395:
5374:
5373:
5357:
5352:
5334:
5330:
5329:
5328:
5307:
5306:
5297:
5293:
5226:
5221:
5195:
5191:
5190:
5169:
5168:
5152:
5147:
5129:
5128:
5107:
5106:
5096:
5091:
5073:
5072:
5051:
5050:
5025:
5020:
5002:
5001:
4980:
4979:
4933:
4928:
4907:
4903:
4902:
4875:
4874:
4858:
4853:
4829:
4828:
4807:
4806:
4796:
4791:
4773:
4772:
4751:
4750:
4725:
4720:
4702:
4701:
4680:
4679:
4633:
4628:
4607:
4603:
4602:
4581:
4580:
4570:
4565:
4547:
4546:
4525:
4524:
4514:
4509:
4491:
4490:
4469:
4468:
4443:
4438:
4420:
4419:
4398:
4397:
4351:
4346:
4325:
4321:
4320:
4305:
4304:
4294:
4289:
4274:
4266:
4261:
4260:
4245:
4244:
4234:
4229:
4214:
4212:
4211:
4199:
4194:
4193:
4172:
4171:
4146:
4141:
4126:
4118:
4113:
4112:
4091:
4090:
4044:
4039:
4018:
4014:
4013:
3998:
3997:
3987:
3982:
3967:
3963:
3962:
3954:
3949:
3947:
3946:
3934:
3924:
3923:
3902:
3901:
3876:
3871:
3856:
3848:
3843:
3842:
3821:
3820:
3770:
3765:
3737:
3735:
3734:
3729:
3727:
3723:
3722:
3701:
3700:
3654:
3649:
3627:
3615:
3614:
3593:
3592:
3567:
3562:
3540:
3528:
3527:
3512:
3511:
3501:
3496:
3458: = 0.
3453:
3451:
3450:
3445:
3434:
3430:
3429:
3421:
3416:
3414:
3413:
3401:
3391:
3383:
3375:
3370:
3356:
3354:
3353:
3344:
3333:
3328:
3320:
3312:
3307:
3286:
3284:
3283:
3278:
3249:
3235:
3227:
3226:
3202:
3180:
3178:
3177:
3172:
3170:
3169:
3136:
3135:
3111:
3097:
3096:
3069:
3055:
3054:
3045:
3044:
3028:
3026:
3025:
3020:
3018:
3017:
3002:
3001:
2991:
2986:
2959:
2958:
2942:
2929:
2927:
2926:
2921:
2919:
2915:
2914:
2898:
2897:
2888:
2886:
2866:
2856:
2855:
2822:
2821:
2784:
2781:
2770:
2755:
2753:
2727:
2721:
2720:
2683:
2682:
2673:
2671:
2651:
2641:
2640:
2607:
2606:
2569:
2566:
2555:
2526:
2525:
2516:
2514:
2494:
2484:
2483:
2450:
2449:
2412:
2409:
2398:
2380:
2379:
2337:
2335:
2334:
2329:
2324:
2323:
2314:
2312:
2292:
2282:
2281:
2248:
2247:
2210:
2207:
2196:
2178:
2177:
2143:
2123: β 2,
2099:is known as the
2098:
2096:
2095:
2090:
2073:
2053:
2036:
2032:
2005:
2003:
2002:
1997:
1995:
1985:
1984:
1969:
1968:
1959:
1955:
1867:
1862:
1844:
1843:
1834:
1833:
1824:
1820:
1761:
1752:
1751:
1736:
1735:
1726:
1722:
1634:
1629:
1610:
1598:
1597:
1582:
1581:
1557:
1556:
1541:
1540:
1501:
1500:
1485:
1484:
1435:
1430:
1411:
1402:
1401:
1386:
1385:
1375:
1370:
1340:
1339:
1324:
1323:
1298:
1293:
1263:
1262:
1247:
1246:
1200:
1195:
1176:
1167:
1166:
1151:
1150:
1140:
1135:
1105:
1104:
1083:
1082:
1057:
1052:
1019:
1018:
997:
996:
950:
945:
930:
929:
919:
906:
904:
903:
898:
896:
895:
874:
873:
827:
822:
795:
781:
779:
778:
773:
771:
770:
749:
748:
723:
718:
691:
674:
672:
671:
666:
655:
654:
638:
637:
628:
627:
617:
612:
597:
596:
499:
488:
474:
456:
454:
453:
448:
437:
435:
434:
425:
411:
406:
398:
393:
379:
374:
359:
357:
356:
351:
349:
348:
329:
327:
326:
321:
297:
295:
294:
289:
287:
285:
284:
283:
270:
266:
265:
255:
250:
235:
233:
232:
227:
225:
223:
215:
207:
202:
187:
185:
184:
179:
153:
127:
119:
118:
78:
76:
75:
70:
55:
53:
52:
47:
45:
37:
7218:
7217:
7213:
7212:
7211:
7209:
7208:
7207:
7193:
7192:
7191:
7154:
7145:
7130:
7126:
7115:
7111:
7104:
7090:
7083:
7072:
7065:
7061:
7046:
6998:
6976:
6959:
6947:
6927:
6923:
6921:
6918:
6917:
6897:
6893:
6891:
6888:
6887:
6868:
6853:
6849:
6844:
6841:
6840:
6818:
6812:
6808:
6806:
6803:
6802:
6783:
6778:
6775:
6774:
6743:
6717:
6712:
6709:
6708:
6701:
6694:
6683:must be zero.
6680:
6660:
6656:
6654:
6651:
6650:
6631:
6627:
6618:
6614:
6613:
6609:
6607:
6604:
6603:
6586:
6582:
6580:
6577:
6576:
6572:
6555:
6551:
6549:
6546:
6545:
6528:
6524:
6522:
6519:
6518:
6514:
6497:
6493:
6491:
6488:
6487:
6461:
6457:
6455:
6452:
6451:
6432:
6428:
6421:
6417:
6411:
6407:
6401:
6390:
6368:
6364:
6352:
6348:
6346:
6343:
6342:
6335:
6299:
6295:
6290:
6284:
6280:
6278:
6275:
6274:
6250:
6246:
6234:
6230:
6228:
6219:
6215:
6213:
6210:
6209:
6180:
6176:
6167:
6163:
6157:
6153:
6138:
6134:
6125:
6121:
6115:
6111:
6091:
6088:
6087:
6081:
6073:
6057:
6056:
6038:
6034:
6017:
6013:
6004:
6000:
5994:
5990:
5971:
5967:
5961:
5950:
5931:
5927:
5921:
5917:
5911:
5907:
5886:
5885:
5862:
5858:
5846:
5842:
5836:
5825:
5800:
5796:
5790:
5786:
5780:
5776:
5752:
5741:
5736:
5732:
5717:
5713:
5707:
5703:
5697:
5693:
5672:
5671:
5653:
5649:
5637:
5633:
5627:
5616:
5586:
5582:
5576:
5572:
5506:
5502:
5496:
5485:
5466:
5462:
5456:
5452:
5416:
5412:
5411:
5407:
5398:
5397:
5379:
5375:
5363:
5359:
5353:
5342:
5312:
5308:
5302:
5298:
5232:
5228:
5222:
5211:
5206:
5202:
5193:
5192:
5174:
5170:
5158:
5154:
5148:
5137:
5112:
5108:
5102:
5098:
5092:
5081:
5056:
5052:
5046:
5042:
5021:
5010:
4985:
4981:
4975:
4971:
4929:
4918:
4905:
4904:
4880:
4876:
4864:
4860:
4854:
4837:
4812:
4808:
4802:
4798:
4792:
4781:
4756:
4752:
4746:
4742:
4721:
4710:
4685:
4681:
4675:
4671:
4629:
4618:
4605:
4604:
4586:
4582:
4576:
4572:
4566:
4555:
4530:
4526:
4520:
4516:
4510:
4499:
4474:
4470:
4464:
4460:
4439:
4428:
4403:
4399:
4393:
4389:
4347:
4336:
4323:
4322:
4310:
4306:
4300:
4296:
4290:
4279:
4265:
4250:
4246:
4240:
4236:
4230:
4219:
4207:
4203:
4198:
4177:
4173:
4167:
4163:
4142:
4131:
4117:
4096:
4092:
4086:
4082:
4040:
4029:
4016:
4015:
4003:
3999:
3993:
3989:
3983:
3972:
3953:
3942:
3938:
3933:
3932:
3928:
3907:
3903:
3897:
3893:
3872:
3861:
3847:
3826:
3822:
3816:
3812:
3772:
3766:
3755:
3747:
3745:
3742:
3741:
3725:
3724:
3706:
3702:
3696:
3692:
3650:
3639:
3628:
3620:
3617:
3616:
3598:
3594:
3588:
3584:
3563:
3552:
3541:
3533:
3530:
3529:
3517:
3513:
3507:
3503:
3497:
3486:
3475:
3468:
3466:
3463:
3462:
3420:
3409:
3405:
3400:
3399:
3395:
3384:
3374:
3363:
3349:
3345:
3334:
3332:
3321:
3311:
3300:
3298:
3295:
3294:
3242:
3228:
3222:
3218:
3216:
3213:
3212:
3209:
3196:
3188:
3165:
3161:
3131:
3127:
3104:
3092:
3088:
3062:
3050:
3046:
3040:
3036:
3034:
3031:
3030:
3007:
3003:
2997:
2993:
2987:
2976:
2954:
2950:
2948:
2945:
2944:
2941:
2933:
2917:
2916:
2910:
2906:
2899:
2893:
2889:
2867:
2839:
2835:
2805:
2801:
2785:
2783:
2771:
2760:
2731:
2726:
2723:
2722:
2716:
2712:
2684:
2678:
2674:
2652:
2624:
2620:
2590:
2586:
2570:
2568:
2556:
2545:
2538:
2537:
2527:
2521:
2517:
2495:
2467:
2463:
2433:
2429:
2413:
2411:
2399:
2388:
2375:
2371:
2349:
2347:
2344:
2343:
2319:
2315:
2293:
2265:
2261:
2231:
2227:
2211:
2209:
2197:
2186:
2173:
2169:
2149:
2146:
2145:
2139:
2063:
2043:
2022:
2018:
2013:
2010:
2009:
2008:The expression
1993:
1992:
1974:
1970:
1964:
1960:
1873:
1869:
1863:
1852:
1839:
1835:
1829:
1825:
1768:
1764:
1762:
1760:
1754:
1753:
1741:
1737:
1731:
1727:
1640:
1636:
1630:
1619:
1611:
1609:
1603:
1602:
1587:
1583:
1577:
1573:
1546:
1542:
1536:
1532:
1490:
1486:
1480:
1476:
1431:
1420:
1412:
1410:
1404:
1403:
1391:
1387:
1381:
1377:
1371:
1360:
1329:
1325:
1319:
1315:
1294:
1283:
1252:
1248:
1242:
1238:
1196:
1185:
1177:
1175:
1169:
1168:
1156:
1152:
1146:
1142:
1136:
1125:
1088:
1084:
1078:
1074:
1053:
1042:
1002:
998:
992:
988:
946:
935:
925:
921:
916:
914:
911:
910:
879:
875:
869:
865:
823:
812:
788:
786:
783:
782:
754:
750:
744:
740:
719:
708:
684:
682:
679:
678:
650:
646:
633:
629:
623:
619:
613:
602:
592:
588:
571:
568:
567:
564:
522:
494:
476:
462:
430:
426:
412:
410:
399:
380:
378:
367:
365:
362:
361:
344:
340:
338:
335:
334:
309:
306:
305:
279:
275:
271:
261:
257:
256:
254:
243:
241:
238:
237:
216:
208:
206:
195:
193:
190:
189:
146:
120:
114:
110:
108:
105:
104:
97:infinite series
61:
58:
57:
36:
28:
25:
24:
17:
12:
11:
5:
7216:
7206:
7205:
7190:
7189:
7143:
7124:
7109:
7102:
7081:
7062:
7060:
7057:
7056:
7055:
7044:
7022:Teschl, Gerald
7018:
6997:
6996:External links
6994:
6993:
6992:
6990:Laurent series
6987:
6982:
6980:Fuchs' theorem
6975:
6972:
6958:
6955:
6935:
6930:
6926:
6903:
6900:
6896:
6875:
6871:
6867:
6864:
6861:
6856:
6852:
6848:
6828:
6825:
6821:
6815:
6811:
6790:
6786:
6782:
6762:
6759:
6756:
6753:
6749:
6746:
6742:
6739:
6736:
6733:
6730:
6727:
6723:
6720:
6716:
6668:
6663:
6659:
6634:
6630:
6626:
6621:
6617:
6612:
6589:
6585:
6558:
6554:
6531:
6527:
6500:
6496:
6475:
6472:
6469:
6464:
6460:
6435:
6431:
6427:
6424:
6420:
6414:
6410:
6404:
6399:
6396:
6393:
6389:
6385:
6382:
6379:
6376:
6371:
6367:
6363:
6360:
6355:
6351:
6334:
6331:
6308:
6305:
6302:
6298:
6293:
6287:
6283:
6253:
6249:
6243:
6240:
6237:
6233:
6227:
6222:
6218:
6197:
6194:
6189:
6186:
6183:
6179:
6175:
6170:
6166:
6160:
6156:
6152:
6147:
6144:
6141:
6137:
6133:
6128:
6124:
6118:
6114:
6110:
6107:
6104:
6101:
6098:
6095:
6053:
6050:
6047:
6044:
6041:
6037:
6032:
6026:
6023:
6020:
6016:
6012:
6007:
6003:
5997:
5993:
5989:
5986:
5983:
5980:
5977:
5974:
5970:
5964:
5959:
5956:
5953:
5949:
5945:
5940:
5937:
5934:
5930:
5924:
5920:
5914:
5910:
5906:
5903:
5900:
5897:
5894:
5891:
5889:
5887:
5883:
5877:
5874:
5871:
5868:
5865:
5861:
5855:
5852:
5849:
5845:
5839:
5834:
5831:
5828:
5824:
5820:
5815:
5812:
5809:
5806:
5803:
5799:
5793:
5789:
5783:
5779:
5775:
5772:
5769:
5766:
5763:
5760:
5755:
5750:
5747:
5744:
5740:
5735:
5731:
5726:
5723:
5720:
5716:
5710:
5706:
5700:
5696:
5692:
5689:
5686:
5683:
5680:
5677:
5675:
5673:
5668:
5665:
5662:
5659:
5656:
5652:
5646:
5643:
5640:
5636:
5630:
5625:
5622:
5619:
5615:
5611:
5607:
5601:
5598:
5595:
5592:
5589:
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5579:
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5542:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5509:
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5499:
5494:
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5434:
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5428:
5425:
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5318:
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5280:
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2240:
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2088:
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1980:
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1973:
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1958:
1954:
1951:
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1942:
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429:
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392:
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373:
370:
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319:
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274:
269:
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260:
253:
249:
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222:
219:
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205:
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177:
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171:
168:
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15:
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5492:
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5218:
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4953:
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4925:
4922:
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4097:
4093:
4087:
4083:
4076:
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4070:
4067:
4064:
4055:
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4049:
4036:
4033:
4030:
4026:
4022:
4020:
4010:
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4000:
3994:
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3979:
3976:
3973:
3969:
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3925:
3920:
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3643:
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3406:
3402:
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3301:
3292:
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3037:
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2998:
2994:
2983:
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2969:
2963:
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2110:
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2057:
2054:
2050:
2047:
2044:
2040:
2037:
2033:
2029:
2026:
2023:
2019:
2015:
2006:
1989:
1986:
1981:
1978:
1975:
1971:
1965:
1961:
1956:
1949:
1943:
1940:
1934:
1931:
1928:
1919:
1913:
1910:
1904:
1901:
1898:
1889:
1886:
1883:
1880:
1877:
1870:
1859:
1856:
1853:
1849:
1845:
1840:
1836:
1830:
1826:
1821:
1814:
1808:
1805:
1802:
1796:
1790:
1787:
1781:
1778:
1775:
1769:
1765:
1757:
1748:
1745:
1742:
1738:
1732:
1728:
1723:
1716:
1710:
1707:
1701:
1698:
1695:
1686:
1680:
1677:
1671:
1668:
1665:
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1653:
1650:
1647:
1644:
1637:
1626:
1623:
1620:
1616:
1613:
1606:
1594:
1591:
1588:
1584:
1578:
1574:
1567:
1561:
1558:
1553:
1550:
1547:
1543:
1537:
1533:
1526:
1523:
1520:
1511:
1505:
1502:
1497:
1494:
1491:
1487:
1481:
1477:
1470:
1467:
1464:
1455:
1452:
1449:
1446:
1443:
1427:
1424:
1421:
1417:
1414:
1407:
1398:
1395:
1392:
1388:
1382:
1378:
1367:
1364:
1361:
1357:
1350:
1344:
1341:
1336:
1333:
1330:
1326:
1320:
1316:
1309:
1306:
1303:
1290:
1287:
1284:
1280:
1273:
1267:
1264:
1259:
1256:
1253:
1249:
1243:
1239:
1232:
1229:
1226:
1217:
1214:
1211:
1208:
1205:
1192:
1189:
1186:
1182:
1179:
1172:
1163:
1160:
1157:
1153:
1147:
1143:
1132:
1129:
1126:
1122:
1115:
1109:
1106:
1101:
1098:
1095:
1092:
1089:
1085:
1079:
1075:
1068:
1065:
1062:
1049:
1046:
1043:
1039:
1032:
1026:
1023:
1020:
1015:
1012:
1009:
1006:
1003:
999:
993:
989:
982:
979:
976:
967:
964:
961:
958:
955:
942:
939:
936:
932:
926:
922:
907:
892:
889:
886:
883:
880:
876:
870:
866:
859:
856:
853:
844:
841:
838:
835:
832:
819:
816:
813:
809:
805:
799:
792:
789:
767:
764:
761:
758:
755:
751:
745:
741:
734:
731:
728:
715:
712:
709:
705:
701:
695:
688:
685:
675:
659:
656:
651:
647:
639:
634:
630:
624:
620:
609:
606:
603:
599:
593:
589:
585:
579:
573:
559:
556:
554:
550:
545:
543:
539:
533:
531:
527:
517:
515:
511:
507:
503:
497:
492:
487:
483:
479:
473:
469:
465:
460:
444:
441:
438:
431:
427:
419:
413:
407:
403:
400:
394:
387:
381:
375:
371:
368:
345:
341:
331:
317:
314:
311:
304:
299:
280:
276:
272:
267:
262:
258:
251:
247:
244:
220:
217:
212:
209:
203:
199:
196:
175:
172:
169:
163:
157:
154:
150:
147:
143:
137:
131:
128:
124:
121:
115:
111:
102:
98:
94:
90:
86:
66:
63:
41:
38:
33:
30:
21:
7165:
7161:
7137:
7133:
7127:
7118:
7112:
7093:
7075:
7026:
7010:
6967:
6963:
6960:
6949:
6702:
6695:
6686:
6685:
6340:
6336:
6272:
6269:
6082:
6075:
6071:
3739:
3460:
3455:
3290:
3288:
3210:
3198:
3193:
3189:
3184:
3182:
2938:
2934:
2931:
2342:
2339:
2140:
2137:
2132:
2128:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2007:
908:
676:
565:
557:
552:
548:
546:
541:
537:
534:
529:
525:
523:
513:
509:
505:
501:
495:
485:
481:
477:
471:
467:
463:
332:
300:
103:of the form
88:
82:
85:mathematics
7140:: 159β204.
7059:References
7032:Providence
3029:satisfies
461:if either
7184:2075-1680
7168:(1): 32.
7012:MathWorld
6899:−
6860:−
6752:−
6735:−
6625:−
6403:∞
6388:∑
6378:
6304:−
6239:−
6185:−
6174:−
6143:−
6132:−
6106:−
6049:−
6022:−
6011:−
5985:−
5963:∞
5948:∑
5936:−
5902:−
5873:−
5851:−
5838:∞
5823:∑
5819:−
5811:−
5771:−
5754:∞
5739:∑
5722:−
5688:−
5664:−
5642:−
5629:∞
5614:∑
5610:−
5597:−
5544:−
5535:−
5498:∞
5483:∑
5471:−
5436:−
5427:−
5390:−
5368:−
5355:∞
5340:∑
5336:−
5323:−
5270:−
5261:−
5224:∞
5209:∑
5185:−
5163:−
5150:∞
5135:∑
5131:−
5123:−
5094:∞
5079:∑
5067:−
5023:∞
5008:∑
5004:−
4996:−
4963:−
4931:∞
4916:∑
4897:−
4885:−
4869:−
4856:∞
4842:−
4835:∑
4831:−
4823:−
4794:∞
4779:∑
4767:−
4723:∞
4708:∑
4704:−
4696:−
4663:−
4631:∞
4616:∑
4597:−
4568:∞
4553:∑
4549:−
4541:−
4512:∞
4497:∑
4485:−
4441:∞
4426:∑
4422:−
4414:−
4381:−
4349:∞
4334:∑
4292:∞
4277:∑
4263:−
4232:∞
4217:∑
4188:−
4144:∞
4129:∑
4115:−
4107:−
4074:−
4042:∞
4027:∑
3985:∞
3970:∑
3951:−
3918:−
3874:∞
3859:∑
3845:−
3837:−
3804:−
3768:∞
3753:∑
3717:−
3684:−
3652:∞
3637:∑
3609:−
3565:∞
3550:∑
3499:∞
3484:∑
3418:−
3372:−
3339:−
3309:−
3260:−
3237:−
2989:∞
2974:∑
2875:−
2847:−
2813:−
2776:−
2758:∑
2733:−
2692:−
2660:−
2632:−
2598:−
2561:−
2543:∑
2503:−
2475:−
2441:−
2404:−
2386:∑
2301:−
2273:−
2239:−
2202:−
2184:∑
2027:−
1887:−
1865:∞
1850:∑
1779:−
1654:−
1632:∞
1617:∑
1453:−
1433:∞
1418:∑
1373:∞
1358:∑
1296:∞
1281:∑
1215:−
1198:∞
1183:∑
1138:∞
1123:∑
1099:−
1055:∞
1040:∑
1013:−
965:−
948:∞
933:∑
890:−
842:−
825:∞
810:∑
765:−
721:∞
706:∑
657:≠
615:∞
600:∑
252:≡
204:≡
64:−
7197:Category
7024:(2012).
6974:See also
6748:′
6722:″
6575:and the
6078:β 1) = 0
3625:″
3538:′
3389:′
3368:″
3326:′
3305:″
3293:to give
3247:′
3233:″
3109:′
3067:″
793:″
689:′
493:at
491:analytic
404:′
372:″
248:″
200:′
151:′
125:″
6687:Example
3207:Example
2943:above,
489:is not
7182:
7162:Axioms
7100:
7042:
6450:where
508:) and
87:, the
6693:with
6321:is a
6072:From
526:forms
188:with
7180:ISSN
7098:ISBN
7040:ISBN
6801:and
6700:and
549:form
236:and
56:and
7170:doi
6707:):
6705:= 2
6698:= 1
3187:in
2144:is
498:= 0
475:or
330:.
83:In
7199::
7178:.
7166:12
7164:.
7160:.
7146:^
7138:66
7136:.
7084:^
7066:^
7038:.
7034::
7030:.
7009:.
6953:.
6948:1/
6375:ln
6329:.
2135:.
484:)/
470:)/
298:.
7186:.
7172::
7106:.
7048:.
7015:.
6968:r
6950:z
6934:,
6929:0
6925:z
6902:1
6895:z
6874:z
6870:/
6866:)
6863:1
6855:z
6851:e
6847:(
6827:,
6824:z
6820:/
6814:z
6810:e
6789:z
6785:/
6781:1
6761:0
6758:=
6755:u
6745:u
6741:)
6738:z
6732:2
6729:(
6726:+
6719:u
6715:z
6703:b
6696:a
6681:C
6667:.
6662:k
6658:B
6633:2
6629:r
6620:1
6616:r
6611:B
6588:k
6584:B
6573:C
6557:0
6553:B
6530:k
6526:B
6515:C
6499:2
6495:r
6474:)
6471:x
6468:(
6463:1
6459:y
6434:2
6430:r
6426:+
6423:k
6419:x
6413:k
6409:B
6398:0
6395:=
6392:k
6384:+
6381:x
6370:1
6366:y
6362:C
6359:=
6354:2
6350:y
6307:1
6301:k
6297:A
6292:/
6286:k
6282:A
6252:2
6248:k
6242:1
6236:k
6232:A
6226:=
6221:k
6217:A
6196:0
6193:=
6188:1
6182:k
6178:A
6169:k
6165:A
6159:2
6155:k
6151:=
6146:1
6140:k
6136:A
6127:k
6123:A
6117:2
6113:)
6109:1
6103:1
6100:+
6097:k
6094:(
6083:z
6076:r
6074:(
6052:2
6046:r
6043:+
6040:k
6036:z
6031:)
6025:1
6019:k
6015:A
6006:k
6002:A
5996:2
5992:)
5988:1
5982:r
5979:+
5976:k
5973:(
5969:(
5958:1
5955:=
5952:k
5944:+
5939:2
5933:r
5929:z
5923:0
5919:A
5913:2
5909:)
5905:1
5899:r
5896:(
5893:=
5882:}
5876:2
5870:r
5867:+
5864:k
5860:z
5854:1
5848:k
5844:A
5833:1
5830:=
5827:k
5814:2
5808:r
5805:+
5802:k
5798:z
5792:k
5788:A
5782:2
5778:)
5774:1
5768:r
5765:+
5762:k
5759:(
5749:1
5746:=
5743:k
5734:{
5730:+
5725:2
5719:r
5715:z
5709:0
5705:A
5699:2
5695:)
5691:1
5685:r
5682:(
5679:=
5667:2
5661:r
5658:+
5655:k
5651:z
5645:1
5639:k
5635:A
5624:1
5621:=
5618:k
5606:}
5600:2
5594:r
5591:+
5588:k
5584:z
5578:k
5574:A
5569:)
5565:1
5562:+
5559:)
5556:r
5553:+
5550:k
5547:(
5541:)
5538:1
5532:r
5529:+
5526:k
5523:(
5520:)
5517:r
5514:+
5511:k
5508:(
5504:(
5493:1
5490:=
5487:k
5479:+
5474:2
5468:r
5464:z
5458:0
5454:A
5449:)
5445:1
5442:+
5439:r
5433:)
5430:1
5424:r
5421:(
5418:r
5414:(
5409:{
5405:=
5393:2
5387:r
5384:+
5381:k
5377:z
5371:1
5365:k
5361:A
5350:1
5347:=
5344:k
5332:}
5326:2
5320:r
5317:+
5314:k
5310:z
5304:k
5300:A
5295:)
5291:1
5288:+
5285:)
5282:r
5279:+
5276:k
5273:(
5267:)
5264:1
5258:r
5255:+
5252:k
5249:(
5246:)
5243:r
5240:+
5237:k
5234:(
5230:(
5219:0
5216:=
5213:k
5204:{
5200:=
5188:2
5182:r
5179:+
5176:k
5172:z
5166:1
5160:k
5156:A
5145:1
5142:=
5139:k
5126:2
5120:r
5117:+
5114:k
5110:z
5104:k
5100:A
5089:0
5086:=
5083:k
5075:+
5070:2
5064:r
5061:+
5058:k
5054:z
5048:k
5044:A
5040:)
5037:r
5034:+
5031:k
5028:(
5018:0
5015:=
5012:k
4999:2
4993:r
4990:+
4987:k
4983:z
4977:k
4973:A
4969:)
4966:1
4960:r
4957:+
4954:k
4951:(
4948:)
4945:r
4942:+
4939:k
4936:(
4926:0
4923:=
4920:k
4912:=
4900:1
4894:r
4891:+
4888:1
4882:k
4878:z
4872:1
4866:k
4862:A
4851:0
4848:=
4845:1
4839:k
4826:2
4820:r
4817:+
4814:k
4810:z
4804:k
4800:A
4789:0
4786:=
4783:k
4775:+
4770:2
4764:r
4761:+
4758:k
4754:z
4748:k
4744:A
4740:)
4737:r
4734:+
4731:k
4728:(
4718:0
4715:=
4712:k
4699:2
4693:r
4690:+
4687:k
4683:z
4677:k
4673:A
4669:)
4666:1
4660:r
4657:+
4654:k
4651:(
4648:)
4645:r
4642:+
4639:k
4636:(
4626:0
4623:=
4620:k
4612:=
4600:1
4594:r
4591:+
4588:k
4584:z
4578:k
4574:A
4563:0
4560:=
4557:k
4544:2
4538:r
4535:+
4532:k
4528:z
4522:k
4518:A
4507:0
4504:=
4501:k
4493:+
4488:2
4482:r
4479:+
4476:k
4472:z
4466:k
4462:A
4458:)
4455:r
4452:+
4449:k
4446:(
4436:0
4433:=
4430:k
4417:2
4411:r
4408:+
4405:k
4401:z
4395:k
4391:A
4387:)
4384:1
4378:r
4375:+
4372:k
4369:(
4366:)
4363:r
4360:+
4357:k
4354:(
4344:0
4341:=
4338:k
4330:=
4318:r
4315:+
4312:k
4308:z
4302:k
4298:A
4287:0
4284:=
4281:k
4271:z
4268:1
4258:r
4255:+
4252:k
4248:z
4242:k
4238:A
4227:0
4224:=
4221:k
4209:2
4205:z
4201:1
4196:+
4191:1
4185:r
4182:+
4179:k
4175:z
4169:k
4165:A
4161:)
4158:r
4155:+
4152:k
4149:(
4139:0
4136:=
4133:k
4123:z
4120:1
4110:2
4104:r
4101:+
4098:k
4094:z
4088:k
4084:A
4080:)
4077:1
4071:r
4068:+
4065:k
4062:(
4059:)
4056:r
4053:+
4050:k
4047:(
4037:0
4034:=
4031:k
4023:=
4011:r
4008:+
4005:k
4001:z
3995:k
3991:A
3980:0
3977:=
3974:k
3965:)
3959:z
3956:1
3944:2
3940:z
3936:1
3930:(
3926:+
3921:1
3915:r
3912:+
3909:k
3905:z
3899:k
3895:A
3891:)
3888:r
3885:+
3882:k
3879:(
3869:0
3866:=
3863:k
3853:z
3850:1
3840:2
3834:r
3831:+
3828:k
3824:z
3818:k
3814:A
3810:)
3807:1
3801:r
3798:+
3795:k
3792:(
3789:)
3786:r
3783:+
3780:k
3777:(
3763:0
3760:=
3757:k
3720:2
3714:r
3711:+
3708:k
3704:z
3698:k
3694:A
3690:)
3687:1
3681:r
3678:+
3675:k
3672:(
3669:)
3666:r
3663:+
3660:k
3657:(
3647:0
3644:=
3641:k
3633:=
3622:f
3612:1
3606:r
3603:+
3600:k
3596:z
3590:k
3586:A
3582:)
3579:r
3576:+
3573:k
3570:(
3560:0
3557:=
3554:k
3546:=
3535:f
3525:r
3522:+
3519:k
3515:z
3509:k
3505:A
3494:0
3491:=
3488:k
3480:=
3473:f
3456:z
3442:0
3439:=
3436:f
3432:)
3426:z
3423:1
3411:2
3407:z
3403:1
3397:(
3393:+
3386:f
3380:z
3377:1
3365:f
3361:=
3358:f
3351:2
3347:z
3342:z
3336:1
3330:+
3323:f
3317:z
3314:1
3302:f
3291:z
3275:0
3272:=
3269:f
3266:)
3263:z
3257:1
3254:(
3251:+
3244:f
3240:z
3230:f
3224:2
3220:z
3201:)
3199:z
3197:(
3194:r
3190:U
3185:r
3167:r
3163:z
3159:)
3156:r
3153:(
3150:I
3147:=
3144:)
3141:z
3138:(
3133:r
3129:U
3125:)
3122:z
3119:(
3116:q
3113:+
3106:)
3102:z
3099:(
3094:r
3090:U
3086:z
3083:)
3080:z
3077:(
3074:p
3071:+
3064:)
3060:z
3057:(
3052:r
3048:U
3042:2
3038:z
3015:r
3012:+
3009:k
3005:z
2999:k
2995:A
2984:0
2981:=
2978:k
2970:=
2967:)
2964:z
2961:(
2956:r
2952:U
2939:k
2935:A
2912:k
2908:A
2904:=
2895:j
2891:A
2884:!
2881:)
2878:j
2872:k
2869:(
2864:)
2861:0
2858:(
2853:)
2850:j
2844:k
2841:(
2837:q
2833:+
2830:)
2827:0
2824:(
2819:)
2816:j
2810:k
2807:(
2803:p
2799:)
2796:r
2793:+
2790:j
2787:(
2779:1
2773:k
2768:0
2765:=
2762:j
2751:)
2748:r
2745:+
2742:k
2739:(
2736:I
2729:1
2718:k
2714:A
2710:)
2707:r
2704:+
2701:k
2698:(
2695:I
2689:=
2680:j
2676:A
2669:!
2666:)
2663:j
2657:k
2654:(
2649:)
2646:0
2643:(
2638:)
2635:j
2629:k
2626:(
2622:q
2618:+
2615:)
2612:0
2609:(
2604:)
2601:j
2595:k
2592:(
2588:p
2584:)
2581:r
2578:+
2575:j
2572:(
2564:1
2558:k
2553:0
2550:=
2547:j
2535:0
2532:=
2523:j
2519:A
2512:!
2509:)
2506:j
2500:k
2497:(
2492:)
2489:0
2486:(
2481:)
2478:j
2472:k
2469:(
2465:q
2461:+
2458:)
2455:0
2452:(
2447:)
2444:j
2438:k
2435:(
2431:p
2427:)
2424:r
2421:+
2418:j
2415:(
2407:1
2401:k
2396:0
2393:=
2390:j
2382:+
2377:k
2373:A
2369:)
2366:r
2363:+
2360:k
2357:(
2354:I
2326:,
2321:j
2317:A
2310:!
2307:)
2304:j
2298:k
2295:(
2290:)
2287:0
2284:(
2279:)
2276:j
2270:k
2267:(
2263:q
2259:+
2256:)
2253:0
2250:(
2245:)
2242:j
2236:k
2233:(
2229:p
2225:)
2222:r
2219:+
2216:j
2213:(
2205:1
2199:k
2194:0
2191:=
2188:j
2180:+
2175:k
2171:A
2167:)
2164:r
2161:+
2158:k
2155:(
2152:I
2141:z
2133:z
2129:k
2125:r
2121:r
2117:r
2113:z
2105:r
2087:)
2084:r
2081:(
2078:I
2075:=
2071:)
2068:0
2065:(
2061:q
2058:+
2055:r
2051:)
2048:0
2045:(
2041:p
2038:+
2034:)
2030:1
2024:r
2020:(
2016:r
1990:0
1987:=
1982:r
1979:+
1976:k
1972:z
1966:k
1962:A
1957:]
1953:)
1950:z
1947:(
1944:q
1941:+
1938:)
1935:r
1932:+
1929:k
1926:(
1923:)
1920:z
1917:(
1914:p
1911:+
1908:)
1905:r
1902:+
1899:k
1896:(
1893:)
1890:1
1884:r
1881:+
1878:k
1875:(
1871:[
1860:1
1857:=
1854:k
1846:+
1841:r
1837:z
1831:0
1827:A
1822:]
1818:)
1815:z
1812:(
1809:q
1806:+
1803:r
1800:)
1797:z
1794:(
1791:p
1788:+
1785:)
1782:1
1776:r
1773:(
1770:r
1766:[
1758:=
1749:r
1746:+
1743:k
1739:z
1733:k
1729:A
1724:]
1720:)
1717:z
1714:(
1711:q
1708:+
1705:)
1702:r
1699:+
1696:k
1693:(
1690:)
1687:z
1684:(
1681:p
1678:+
1675:)
1672:r
1669:+
1666:k
1663:(
1660:)
1657:1
1651:r
1648:+
1645:k
1642:(
1638:[
1627:0
1624:=
1621:k
1607:=
1600:]
1595:r
1592:+
1589:k
1585:z
1579:k
1575:A
1571:)
1568:z
1565:(
1562:q
1559:+
1554:r
1551:+
1548:k
1544:z
1538:k
1534:A
1530:)
1527:r
1524:+
1521:k
1518:(
1515:)
1512:z
1509:(
1506:p
1503:+
1498:r
1495:+
1492:k
1488:z
1482:k
1478:A
1474:)
1471:r
1468:+
1465:k
1462:(
1459:)
1456:1
1450:r
1447:+
1444:k
1441:(
1438:[
1428:0
1425:=
1422:k
1408:=
1399:r
1396:+
1393:k
1389:z
1383:k
1379:A
1368:0
1365:=
1362:k
1354:)
1351:z
1348:(
1345:q
1342:+
1337:r
1334:+
1331:k
1327:z
1321:k
1317:A
1313:)
1310:r
1307:+
1304:k
1301:(
1291:0
1288:=
1285:k
1277:)
1274:z
1271:(
1268:p
1265:+
1260:r
1257:+
1254:k
1250:z
1244:k
1240:A
1236:)
1233:r
1230:+
1227:k
1224:(
1221:)
1218:1
1212:r
1209:+
1206:k
1203:(
1193:0
1190:=
1187:k
1173:=
1164:r
1161:+
1158:k
1154:z
1148:k
1144:A
1133:0
1130:=
1127:k
1119:)
1116:z
1113:(
1110:q
1107:+
1102:1
1096:r
1093:+
1090:k
1086:z
1080:k
1076:A
1072:)
1069:r
1066:+
1063:k
1060:(
1050:0
1047:=
1044:k
1036:)
1033:z
1030:(
1027:p
1024:z
1021:+
1016:2
1010:r
1007:+
1004:k
1000:z
994:k
990:A
986:)
983:r
980:+
977:k
974:(
971:)
968:1
962:r
959:+
956:k
953:(
943:0
940:=
937:k
927:2
923:z
893:2
887:r
884:+
881:k
877:z
871:k
867:A
863:)
860:r
857:+
854:k
851:(
848:)
845:1
839:r
836:+
833:k
830:(
820:0
817:=
814:k
806:=
803:)
800:z
797:(
790:u
768:1
762:r
759:+
756:k
752:z
746:k
742:A
738:)
735:r
732:+
729:k
726:(
716:0
713:=
710:k
702:=
699:)
696:z
693:(
686:u
663:)
660:0
652:0
648:A
644:(
640:,
635:k
631:z
625:k
621:A
610:0
607:=
604:k
594:r
590:z
586:=
583:)
580:z
577:(
574:u
553:r
538:r
514:z
512:(
510:q
506:z
504:(
502:p
496:z
486:z
482:z
480:(
478:q
472:z
468:z
466:(
464:p
445:0
442:=
439:u
432:2
428:z
423:)
420:z
417:(
414:q
408:+
401:u
395:z
391:)
388:z
385:(
382:p
376:+
369:u
346:2
342:z
318:0
315:=
312:z
281:2
277:z
273:d
268:u
263:2
259:d
245:u
221:z
218:d
213:u
210:d
197:u
176:0
173:=
170:u
167:)
164:z
161:(
158:q
155:+
148:u
144:z
141:)
138:z
135:(
132:p
129:+
122:u
116:2
112:z
79:.
67:1
42:2
39:1
34:=
31:r
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