2079:
3265:
1739:
2565:
4330:
934:
4134:
2982:
1515:
2772:
246:
678:
472:
1231:
1851:
3029:
1084:
2363:
1541:
751:
The three versions of the
Wirtinger inequality can all be proved by various means. This is illustrated in the following by a different kind of proof for each of the three Wirtinger inequalities given above. In each case, by a linear
3853:
4145:
786:
3946:
2150:
4797:
2622:
2844:
108:
3961:
545:
339:
4420:
1329:
2074:{\displaystyle |y(x)|=\left|\int _{0}^{x}y'(x)\,\mathrm {d} x\right|\leq \int _{0}^{x}|y'(x)|\,\mathrm {d} x\leq {\sqrt {x}}\left(\int _{0}^{x}y'(x)^{2}\,\mathrm {d} x\right)^{1/2},}
3260:{\displaystyle y(x)-y_{n}(x)=\int _{0}^{x}{\big (}y_{n}'(z)-g_{n}(z){\big )}\,\mathrm {d} z-\int _{0}^{1}\int _{0}^{w}(y_{n}'(z)-g_{n}(z){\big )}\,\mathrm {d} z\,\mathrm {d} w}
1095:
3642:
The second and third versions of the
Wirtinger inequality can be extended to statements about first Dirichlet and Neumann eigenvalues of the Laplace−Beltrami operator on
963:
3552:
These can also be extended to statements about higher-dimensional spaces. For example, the
Riemannian circle may be viewed as the one-dimensional version of either a
2560:{\displaystyle \int _{0}^{1}Tf(x)^{2}\,\mathrm {d} x\leq \pi ^{-2}\int _{0}^{1}f(x)Tf(x)\,\mathrm {d} x={\frac {1}{\pi ^{2}}}\int _{0}^{1}(Tf)'(x)^{2}\,\mathrm {d} x}
1744:
This proves the
Wirtinger inequality, since the second integral is clearly nonnegative. Furthermore, equality in the Wirtinger inequality is seen to be equivalent to
4425:
which is the isoperimetric inequality. Furthermore, equality in the isoperimetric inequality implies both equality in the
Wirtinger inequality and also the equality
1734:{\displaystyle \int _{0}^{\pi }y(x)^{2}\,\mathrm {d} x+\int _{0}^{\pi }{\big (}y'(x)-y(x)\cot x{\big )}^{2}\,\mathrm {d} x=\int _{0}^{\pi }y'(x)^{2}\,\mathrm {d} x.}
54:
for curves in the plane. A variety of closely related results are today known as
Wirtinger's inequality, all of which can be viewed as certain forms of the
22:
3759:
4325:{\displaystyle \int _{0}^{2\pi }{\big (}x'(t)+y(t){\big )}^{2}\,\mathrm {d} t+\int _{0}^{2\pi }{\big (}y'(t)^{2}-y(t)^{2}{\big )}\,\mathrm {d} t.}
929:{\displaystyle y(x)={\frac {1}{2}}a_{0}+\sum _{n\geq 1}\left(a_{n}{\frac {\sin nx}{\sqrt {\pi }}}+b_{n}{\frac {\cos nx}{\sqrt {\pi }}}\right),}
4675:
1236:
and since the summands are all nonnegative, the
Wirtinger inequality is proved. Furthermore it is seen that equality holds if and only if
4371:
have average value zero. Then the
Wirtinger inequality can be applied to see that the second integral is also nonnegative, and therefore
4722:
4659:
3872:
1293:
2767:{\displaystyle y_{n}(x)=\int _{0}^{x}g_{n}(z)\,\mathrm {d} z-\int _{0}^{1}\int _{0}^{w}g_{n}(z)\,\mathrm {d} z\,\mathrm {d} w.}
1521:
2090:
727:
Despite their differences, these are closely related to one another, as can be seen from the account given below in terms of
3379:
2977:{\displaystyle \int _{0}^{1}y_{n}(x)^{2}\,\mathrm {d} x\leq {\frac {1}{\pi ^{2}}}\int _{0}^{1}y_{n}'(x)^{2}\,\mathrm {d} x.}
4873:
241:{\displaystyle \int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{4\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x,}
4129:{\displaystyle {\frac {L^{2}}{2\pi }}-2A=\int _{0}^{2\pi }{\big (}x'(t)^{2}+y'(t)^{2}+2y(t)x'(t){\big )}\,\mathrm {d} t}
3326:
2326:
673:{\displaystyle \int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x.}
467:{\displaystyle \int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x,}
3293:
1510:{\displaystyle y(x)^{2}+{\big (}y'(x)-y(x)\cot x{\big )}^{2}=y'(x)^{2}-{\frac {d}{dx}}{\big (}y(x)^{2}\cot x{\big )}.}
753:
4830:
4764:
4377:
1795:
There is a subtlety in the above application of the fundamental theorem of calculus, since it is not the case that
756:
in the integrals involved, there is no loss of generality in only proving the theorem for one particular choice of
4863:
777:
3710:
51:
3564:(of arbitrary dimension). The Wirtinger inequality, in the first version given here, can then be seen as the
4752:
1767:
3285:
954:
732:
55:
1226:{\displaystyle \int _{0}^{2\pi }y'(x)^{2}\,\mathrm {d} x=\sum _{n=1}^{\infty }n^{2}(a_{n}^{2}+b_{n}^{2})}
4868:
3609:, and the corresponding eigenfunctions are the restrictions of the homogeneous quadratic polynomials on
3557:
2354:
740:
35:
2575:
1835:
3370:, the three versions of the Wirtinger inequality above can be rephrased as theorems about the first
3284:
is a function for which equality in the
Wirtinger inequality holds, then a standard argument in the
3420:, and the corresponding eigenfunctions are the linear combinations of the two coordinate functions.
3383:
1079:{\displaystyle \int _{0}^{2\pi }y(x)^{2}\,\mathrm {d} x=\sum _{n=1}^{\infty }(a_{n}^{2}+b_{n}^{2})}
4718:
4671:
3391:
3367:
728:
43:
4335:
The first integral is clearly nonnegative. Without changing the area or length of the curve,
3354:
To make this argument fully formal and precise, it is necessary to be more careful about the
4842:
4806:
4776:
4736:
4710:
4689:
4663:
4838:
4772:
4732:
4685:
4846:
4834:
4810:
4780:
4768:
4740:
4728:
4693:
4681:
3686:
3661:
3647:
2217:
4756:
4706:
3863:
3355:
4714:
3729:
be a differentiable embedding of the circle in the plane. Parametrizing the circle by
4857:
4788:
3714:
3375:
47:
4792:
3654:
the first
Dirichlet eigenvalue of the Laplace−Beltrami operator on the unit ball in
3598:
3679:
the first Neumann eigenvalue of the Laplace−Beltrami operator on the unit ball in
3709:
In the first form given above, the Wirtinger inequality can be used to prove the
3325:, and the regularity theory of such equations, followed by the usual analysis of
4822:
4818:
4748:
4651:
3643:
3423:
the first Dirichlet eigenvalue of the Laplace–Beltrami operator on the interval
31:
4667:
3487:
the first Neumann eigenvalue of the Laplace–Beltrami operator on the interval
3371:
2599:
be a sequence of compactly supported continuously differentiable functions on
4541:
4539:
4537:
3848:{\displaystyle \int _{0}^{2\pi }{\sqrt {x'(t)^{2}+y'(t)^{2}}}\,\mathrm {d} t}
3002:, and thereby prove the Wirtinger inequality, as soon as it is verified that
3634:, and the corresponding eigenfunctions are arbitrary linear combinations of
3685:
is the square of the smallest positive zero of the first derivative of the
42:
is an important inequality for functions of a single variable, named after
3583:, and the corresponding eigenfunctions are the linear combinations of the
3575:
the first eigenvalue of the Laplace–Beltrami operator on the unit-radius
731:. They can also all be regarded as special cases of various forms of the
739:
identified explicitly. The middle version is also a special case of the
4656:
Functional analysis, Sobolev spaces and partial differential equations
3553:
66:
There are several inequivalent versions of the Wirtinger inequality:
4556:
4554:
3597:-dimensional real projective space (with normalization given by the
4619:
4617:
3561:
2822:
has average value zero. So application of the above inequality to
2238:
converges to zero. In exactly the same way, it can be proved that
4829:. Princeton Mathematical Series. Vol. 32. Princeton, NJ:
3618:
the first eigenvalue of the Laplace–Beltrami operator on the
3593:
the first eigenvalue of the Laplace–Beltrami operator on the
3390:
the first eigenvalue of the Laplace–Beltrami operator on the
4705:. Pure and Applied Mathematics. Vol. 115. Orlando, FL:
526:
be a continuous and differentiable function on the interval
305:
be a continuous and differentiable function on the interval
74:
be a continuous and differentiable function on the interval
3941:{\displaystyle -\int _{0}^{2\pi }y(t)x'(t)\,\mathrm {d} t.}
3615:
to the unit sphere (and then to the real projective space).
1304:
Consider the second Wirtinger inequality given above. Take
1292:. This is equivalent to the stated condition by use of the
3327:
ordinary differential equations with constant coefficients
2327:
ordinary differential equations with constant coefficients
2268:
Consider the third Wirtinger inequality given above. Take
768:
Consider the first Wirtinger inequality given above. Take
2578:. Finally, for any continuously differentiable function
2145:{\displaystyle \int _{0}^{\pi }y'(x)^{2}\,\mathrm {d} x}
3273:
This proves the Wirtinger inequality. In the case that
4798:
Comptes Rendus des Séances de l'Académie des Sciences
4380:
4148:
3964:
3875:
3762:
3032:
2847:
2625:
2366:
2093:
1854:
1544:
1332:
1098:
966:
789:
548:
342:
111:
4827:
Introduction to Fourier analysis on Euclidean spaces
4635:
4545:
3638:-fold products of the eigenfunctions on the circles.
2574:
of average value zero, where the equality is due to
3660:is the square of the smallest positive zero of the
1834:. This is resolved as follows. It follows from the
4414:
4324:
4128:
3940:
3847:
3517:and the corresponding eigenfunctions are given by
3453:and the corresponding eigenfunctions are given by
3259:
2976:
2766:
2559:
2144:
2073:
1733:
1509:
1225:
1078:
928:
672:
466:
240:
21:For other inequalities named after Wirtinger, see
3023:. This is verified in a standard way, by writing
3270:and applying the Hölder or Jensen inequalities.
1766:, the general solution of which (as computed by
743:, again with the optimal constant identified.
4415:{\displaystyle {\frac {L^{2}}{4\pi }}\geq A,}
4305:
4252:
4209:
4169:
4112:
4019:
3951:Since the integrand of the integral defining
3234:
3138:
3087:
1657:
1608:
1499:
1464:
1406:
1357:
8:
4763:(Second edition of 1934 original ed.).
3705:Application to the isoperimetric inequality
4584:
4512:. These equations mean that the image of
4387:
4381:
4379:
4311:
4310:
4304:
4303:
4297:
4275:
4251:
4250:
4241:
4236:
4221:
4220:
4214:
4208:
4207:
4168:
4167:
4158:
4153:
4147:
4118:
4117:
4111:
4110:
4069:
4042:
4018:
4017:
4008:
4003:
3971:
3965:
3963:
3927:
3926:
3888:
3883:
3874:
3837:
3836:
3828:
3801:
3781:
3772:
3767:
3761:
3249:
3248:
3240:
3239:
3233:
3232:
3217:
3192:
3179:
3174:
3164:
3159:
3144:
3143:
3137:
3136:
3121:
3096:
3086:
3085:
3079:
3074:
3052:
3031:
2963:
2962:
2956:
2937:
2927:
2922:
2910:
2901:
2890:
2889:
2883:
2867:
2857:
2852:
2846:
2753:
2752:
2744:
2743:
2728:
2718:
2713:
2703:
2698:
2683:
2682:
2667:
2657:
2652:
2630:
2624:
2549:
2548:
2542:
2509:
2504:
2492:
2483:
2472:
2471:
2438:
2433:
2420:
2405:
2404:
2398:
2376:
2371:
2365:
2303:which is of average value zero, and with
2134:
2133:
2127:
2103:
2098:
2092:
2058:
2054:
2041:
2040:
2034:
2010:
2005:
1988:
1977:
1976:
1971:
1949:
1943:
1938:
1918:
1917:
1894:
1889:
1872:
1855:
1853:
1720:
1719:
1713:
1689:
1684:
1669:
1668:
1662:
1656:
1655:
1607:
1606:
1600:
1595:
1580:
1579:
1573:
1554:
1549:
1543:
1498:
1497:
1482:
1463:
1462:
1447:
1438:
1411:
1405:
1404:
1356:
1355:
1346:
1331:
1214:
1209:
1196:
1191:
1178:
1168:
1157:
1142:
1141:
1135:
1108:
1103:
1097:
1067:
1062:
1049:
1044:
1031:
1020:
1005:
1004:
998:
976:
971:
965:
895:
889:
859:
853:
832:
819:
805:
788:
659:
658:
652:
628:
623:
611:
601:
595:
584:
583:
577:
558:
553:
547:
453:
452:
446:
422:
417:
405:
395:
389:
378:
377:
371:
352:
347:
341:
227:
226:
220:
196:
191:
178:
164:
158:
147:
146:
140:
121:
116:
110:
4639:
4533:
3862:enclosed by the curve is given (due to
939:and the fact that the average value of
4623:
4608:
4596:
4572:
4560:
3626:-fold product of the circle of length
3713:for curves in the plane, as found by
7:
4793:"Sur le problème des isopérimètres"
50:in 1901 to give a new proof of the
4703:Eigenvalues in Riemannian geometry
4636:Hardy, Littlewood & Pólya 1952
4546:Hardy, Littlewood & Pólya 1952
4312:
4222:
4119:
3928:
3838:
3250:
3241:
3145:
2964:
2891:
2754:
2745:
2684:
2550:
2473:
2406:
2135:
2042:
1978:
1919:
1721:
1670:
1581:
1169:
1143:
1032:
1006:
684:and equality holds if and only if
660:
585:
478:and equality holds if and only if
454:
379:
252:and equality holds if and only if
228:
148:
14:
3687:Bessel function of the first kind
3662:Bessel function of the first kind
3622:-dimensional torus (given as the
2187:converges to zero is zero. Since
84:with average value zero and with
4524:is a round circle in the plane.
3601:from the unit-radius sphere) is
3747:has constant speed, the length
3292:must be a weak solution of the
2345:, the largest of which is then
1522:fundamental theorem of calculus
1294:trigonometric addition formulas
4294:
4287:
4272:
4265:
4203:
4197:
4188:
4182:
4107:
4101:
4090:
4084:
4066:
4059:
4039:
4032:
3955:is assumed constant, there is
3923:
3917:
3906:
3900:
3825:
3818:
3798:
3791:
3571:case of any of the following:
3544:for arbitrary nonzero numbers
3480:for arbitrary nonzero numbers
3229:
3223:
3207:
3201:
3185:
3133:
3127:
3111:
3105:
3064:
3058:
3042:
3036:
2953:
2946:
2880:
2873:
2740:
2734:
2679:
2673:
2642:
2636:
2539:
2532:
2525:
2515:
2468:
2462:
2453:
2447:
2395:
2388:
2276:. Given a continuous function
2124:
2117:
2031:
2024:
1972:
1968:
1962:
1950:
1914:
1908:
1873:
1869:
1863:
1856:
1710:
1703:
1642:
1636:
1627:
1621:
1570:
1563:
1479:
1472:
1435:
1428:
1391:
1385:
1376:
1370:
1343:
1336:
1312:. Any differentiable function
1220:
1184:
1132:
1125:
1073:
1037:
995:
988:
799:
793:
649:
642:
574:
567:
443:
436:
368:
361:
217:
210:
137:
130:
1:
4715:10.1016/s0079-8169(08)x6051-9
2838:is legitimate and shows that
2809:, which in turn implies that
2212:for small positive values of
536:with average value zero. Then
4563:, pp. 511–513, 576–578.
2788:has average value zero with
2084:which shows that as long as
1524:and the boundary conditions
3382:on various one-dimensional
2588:of average value zero, let
2286:of average value zero, let
4890:
4831:Princeton University Press
4765:Cambridge University Press
4658:. Universitext. New York:
4139:which can be rewritten as
2987:It is possible to replace
4668:10.1007/978-0-387-70914-7
3753:of the curve is given by
3380:Laplace–Beltrami operator
2325:. From basic analysis of
3711:isoperimetric inequality
2155:is finite, the limit of
1809:extends continuously to
1788:for an arbitrary number
52:isoperimetric inequality
4575:, Sections I.3 and I.5.
3579:-dimensional sphere is
3294:Euler–Lagrange equation
1768:separation of variables
1323:satisfies the identity
1260:, which is to say that
4701:Chavel, Isaac (1984).
4585:Stein & Weiss 1971
4494:for arbitrary numbers
4416:
4326:
4130:
3942:
3849:
3286:calculus of variations
3261:
2978:
2768:
2561:
2216:, it follows from the
2146:
2075:
1735:
1520:Integration using the
1511:
1227:
1173:
1080:
1036:
930:
780:are met, we can write
778:Dirichlet's conditions
674:
468:
242:
23:Wirtinger's inequality
4417:
4327:
4131:
3943:
3850:
3590:coordinate functions.
3558:real projective space
3262:
2979:
2769:
2562:
2355:self-adjoint operator
2341:for nonzero integers
2329:, the eigenvalues of
2252:converges to zero as
2234:converges to zero as
2147:
2076:
1736:
1512:
1228:
1153:
1081:
1016:
931:
741:Friedrichs inequality
675:
469:
243:
4874:Theorems in analysis
4378:
4146:
3962:
3873:
3760:
3732:[0, 2π]
3384:Riemannian manifolds
3030:
2845:
2623:
2576:integration by parts
2364:
2293:denote the function
2091:
1852:
1542:
1330:
1300:Integration by parts
1096:
964:
787:
546:
340:
109:
40:Wirtinger inequality
4444:, which amounts to
4347:can be replaced by
4249:
4166:
4016:
3896:
3780:
3366:In the language of
3200:
3184:
3169:
3104:
3084:
2945:
2932:
2862:
2723:
2708:
2662:
2514:
2443:
2381:
2264:Functional analysis
2108:
2015:
1948:
1899:
1823:for every function
1694:
1605:
1559:
1219:
1201:
1116:
1072:
1054:
984:
955:Parseval's identity
943:is zero means that
754:change of variables
735:, with the optimal
733:Poincaré inequality
633:
563:
427:
357:
201:
126:
56:Poincaré inequality
16:Theorem in analysis
4412:
4322:
4232:
4149:
4126:
3999:
3938:
3879:
3845:
3763:
3374:and corresponding
3257:
3188:
3170:
3155:
3092:
3070:
2974:
2933:
2918:
2848:
2764:
2709:
2694:
2648:
2605:which converge in
2557:
2500:
2429:
2367:
2357:, it follows that
2142:
2094:
2071:
2001:
1934:
1885:
1731:
1680:
1591:
1545:
1507:
1223:
1205:
1187:
1099:
1076:
1058:
1040:
967:
926:
843:
670:
619:
549:
464:
413:
343:
238:
187:
112:
4753:Littlewood, J. E.
4677:978-0-387-70913-0
4626:, Theorem II.5.4.
4401:
3985:
3834:
3392:Riemannian circle
3368:spectral geometry
3362:Spectral geometry
2916:
2498:
2353:is a bounded and
1993:
1836:Hölder inequality
1460:
916:
915:
880:
879:
828:
813:
737:Poincaré constant
729:spectral geometry
617:
411:
288:for some numbers
185:
46:. It was used by
44:Wilhelm Wirtinger
4881:
4864:Fourier analysis
4850:
4814:
4784:
4744:
4697:
4643:
4633:
4627:
4621:
4612:
4606:
4600:
4594:
4588:
4582:
4576:
4570:
4564:
4558:
4549:
4543:
4523:
4511:
4502:
4493:
4465:
4443:
4421:
4419:
4418:
4413:
4402:
4400:
4392:
4391:
4382:
4370:
4367:, so as to make
4366:
4363:for some number
4362:
4346:
4331:
4329:
4328:
4323:
4315:
4309:
4308:
4302:
4301:
4280:
4279:
4264:
4256:
4255:
4248:
4240:
4225:
4219:
4218:
4213:
4212:
4181:
4173:
4172:
4165:
4157:
4135:
4133:
4132:
4127:
4122:
4116:
4115:
4100:
4074:
4073:
4058:
4047:
4046:
4031:
4023:
4022:
4015:
4007:
3986:
3984:
3976:
3975:
3966:
3954:
3947:
3945:
3944:
3939:
3931:
3916:
3895:
3887:
3861:
3854:
3852:
3851:
3846:
3841:
3835:
3833:
3832:
3817:
3806:
3805:
3790:
3782:
3779:
3771:
3752:
3746:
3734:
3733:
3728:
3699:
3684:
3675:
3659:
3637:
3633:
3630:with itself) is
3629:
3625:
3621:
3614:
3608:
3596:
3589:
3582:
3578:
3570:
3547:
3543:
3542:
3540:
3539:
3534:
3531:
3516:
3515:
3513:
3512:
3507:
3504:
3496:
3495:
3483:
3479:
3478:
3476:
3475:
3470:
3467:
3452:
3451:
3449:
3448:
3443:
3440:
3432:
3431:
3419:
3418:
3416:
3415:
3410:
3407:
3399:
3350:
3347:for some number
3346:
3324:
3313:
3291:
3283:
3266:
3264:
3263:
3258:
3253:
3244:
3238:
3237:
3222:
3221:
3196:
3183:
3178:
3168:
3163:
3148:
3142:
3141:
3126:
3125:
3100:
3091:
3090:
3083:
3078:
3057:
3056:
3022:
3016:
3012:
3001:
2997:
2983:
2981:
2980:
2975:
2967:
2961:
2960:
2941:
2931:
2926:
2917:
2915:
2914:
2902:
2894:
2888:
2887:
2872:
2871:
2861:
2856:
2837:
2821:
2808:
2787:
2773:
2771:
2770:
2765:
2757:
2748:
2733:
2732:
2722:
2717:
2707:
2702:
2687:
2672:
2671:
2661:
2656:
2635:
2634:
2615:
2608:
2604:
2603:
2598:
2587:
2586:
2581:
2573:
2566:
2564:
2563:
2558:
2553:
2547:
2546:
2531:
2513:
2508:
2499:
2497:
2496:
2484:
2476:
2442:
2437:
2428:
2427:
2409:
2403:
2402:
2380:
2375:
2352:
2348:
2344:
2340:
2332:
2324:
2313:
2302:
2301:
2296:
2292:
2285:
2284:
2279:
2275:
2271:
2259:
2255:
2251:
2237:
2233:
2215:
2211:
2210:
2208:
2207:
2202:
2199:
2186:
2182:
2173:
2171:
2170:
2165:
2162:
2151:
2149:
2148:
2143:
2138:
2132:
2131:
2116:
2107:
2102:
2080:
2078:
2077:
2072:
2067:
2066:
2062:
2053:
2049:
2045:
2039:
2038:
2023:
2014:
2009:
1994:
1989:
1981:
1975:
1961:
1953:
1947:
1942:
1930:
1926:
1922:
1907:
1898:
1893:
1876:
1859:
1844:
1833:
1822:
1815:
1808:
1791:
1787:
1765:
1740:
1738:
1737:
1732:
1724:
1718:
1717:
1702:
1693:
1688:
1673:
1667:
1666:
1661:
1660:
1620:
1612:
1611:
1604:
1599:
1584:
1578:
1577:
1558:
1553:
1534:
1516:
1514:
1513:
1508:
1503:
1502:
1487:
1486:
1468:
1467:
1461:
1459:
1448:
1443:
1442:
1427:
1416:
1415:
1410:
1409:
1369:
1361:
1360:
1351:
1350:
1322:
1311:
1307:
1291:
1259:
1252:
1232:
1230:
1229:
1224:
1218:
1213:
1200:
1195:
1183:
1182:
1172:
1167:
1146:
1140:
1139:
1124:
1115:
1107:
1085:
1083:
1082:
1077:
1071:
1066:
1053:
1048:
1035:
1030:
1009:
1003:
1002:
983:
975:
952:
942:
935:
933:
932:
927:
922:
918:
917:
911:
910:
896:
894:
893:
881:
875:
874:
860:
858:
857:
842:
824:
823:
814:
806:
775:
771:
759:
722:
719:for some number
718:
717:
715:
714:
709:
706:
679:
677:
676:
671:
663:
657:
656:
641:
632:
627:
618:
616:
615:
606:
605:
596:
588:
582:
581:
562:
557:
535:
534:
525:
516:
513:for some number
512:
511:
509:
508:
503:
500:
473:
471:
470:
465:
457:
451:
450:
435:
426:
421:
412:
410:
409:
400:
399:
390:
382:
376:
375:
356:
351:
329:
314:
313:
304:
295:
291:
287:
286:
284:
283:
278:
275:
247:
245:
244:
239:
231:
225:
224:
209:
200:
195:
186:
184:
183:
182:
169:
168:
159:
151:
145:
144:
125:
120:
98:
83:
82:
73:
4889:
4888:
4884:
4883:
4882:
4880:
4879:
4878:
4854:
4853:
4819:Stein, Elias M.
4817:
4787:
4747:
4725:
4700:
4678:
4650:
4647:
4646:
4638:, Section 7.7;
4634:
4630:
4622:
4615:
4611:, Section II.2.
4607:
4603:
4595:
4591:
4587:, Chapter IV.2.
4583:
4579:
4571:
4567:
4559:
4552:
4544:
4535:
4530:
4513:
4510:
4504:
4501:
4495:
4492:
4481:
4467:
4459:
4445:
4426:
4393:
4383:
4376:
4375:
4368:
4364:
4348:
4336:
4293:
4271:
4257:
4206:
4174:
4144:
4143:
4093:
4065:
4051:
4038:
4024:
3977:
3967:
3960:
3959:
3952:
3909:
3871:
3870:
3859:
3824:
3810:
3797:
3783:
3758:
3757:
3748:
3736:
3731:
3730:
3718:
3707:
3698:
3689:
3680:
3674:
3664:
3655:
3648:Euclidean space
3635:
3631:
3627:
3623:
3619:
3610:
3602:
3594:
3584:
3580:
3576:
3565:
3545:
3535:
3532:
3526:
3525:
3523:
3518:
3508:
3505:
3502:
3501:
3499:
3498:
3489:
3488:
3481:
3471:
3468:
3462:
3461:
3459:
3454:
3444:
3441:
3438:
3437:
3435:
3434:
3425:
3424:
3411:
3408:
3405:
3404:
3402:
3401:
3395:
3364:
3356:function spaces
3348:
3330:
3315:
3296:
3289:
3274:
3213:
3117:
3048:
3028:
3027:
3018:
3014:
3011:
3003:
2999:
2996:
2988:
2952:
2906:
2879:
2863:
2843:
2842:
2835:
2823:
2819:
2810:
2806:
2797:
2789:
2786:
2778:
2724:
2663:
2626:
2621:
2620:
2610:
2606:
2601:
2600:
2597:
2589:
2584:
2583:
2579:
2571:
2538:
2524:
2488:
2416:
2394:
2362:
2361:
2350:
2346:
2342:
2334:
2330:
2315:
2304:
2299:
2298:
2294:
2287:
2282:
2281:
2277:
2273:
2269:
2266:
2257:
2253:
2239:
2235:
2221:
2218:squeeze theorem
2213:
2203:
2200:
2197:
2196:
2194:
2188:
2184:
2166:
2163:
2160:
2159:
2157:
2156:
2123:
2109:
2089:
2088:
2030:
2016:
2000:
1996:
1995:
1954:
1900:
1884:
1880:
1850:
1849:
1839:
1824:
1817:
1810:
1796:
1789:
1771:
1745:
1709:
1695:
1654:
1613:
1569:
1540:
1539:
1525:
1478:
1452:
1434:
1420:
1403:
1362:
1342:
1328:
1327:
1313:
1309:
1305:
1302:
1286:
1275:
1261:
1254:
1249:
1242:
1237:
1174:
1131:
1117:
1094:
1093:
994:
962:
961:
950:
944:
940:
897:
885:
861:
849:
848:
844:
815:
785:
784:
773:
769:
766:
757:
749:
720:
710:
707:
701:
700:
698:
685:
648:
634:
607:
597:
573:
544:
543:
528:
527:
523:
514:
504:
501:
495:
494:
492:
479:
442:
428:
401:
391:
367:
338:
337:
316:
307:
306:
302:
293:
289:
279:
276:
269:
268:
266:
253:
216:
202:
174:
170:
160:
136:
107:
106:
85:
76:
75:
71:
64:
17:
12:
11:
5:
4887:
4885:
4877:
4876:
4871:
4866:
4856:
4855:
4852:
4851:
4815:
4785:
4745:
4723:
4707:Academic Press
4698:
4676:
4645:
4644:
4628:
4613:
4601:
4589:
4577:
4565:
4550:
4548:, Section 7.7.
4532:
4531:
4529:
4526:
4508:
4499:
4490:
4479:
4457:
4423:
4422:
4411:
4408:
4405:
4399:
4396:
4390:
4386:
4333:
4332:
4321:
4318:
4314:
4307:
4300:
4296:
4292:
4289:
4286:
4283:
4278:
4274:
4270:
4267:
4263:
4260:
4254:
4247:
4244:
4239:
4235:
4231:
4228:
4224:
4217:
4211:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4180:
4177:
4171:
4164:
4161:
4156:
4152:
4137:
4136:
4125:
4121:
4114:
4109:
4106:
4103:
4099:
4096:
4092:
4089:
4086:
4083:
4080:
4077:
4072:
4068:
4064:
4061:
4057:
4054:
4050:
4045:
4041:
4037:
4034:
4030:
4027:
4021:
4014:
4011:
4006:
4002:
3998:
3995:
3992:
3989:
3983:
3980:
3974:
3970:
3949:
3948:
3937:
3934:
3930:
3925:
3922:
3919:
3915:
3912:
3908:
3905:
3902:
3899:
3894:
3891:
3886:
3882:
3878:
3864:Stokes theorem
3856:
3855:
3844:
3840:
3831:
3827:
3823:
3820:
3816:
3813:
3809:
3804:
3800:
3796:
3793:
3789:
3786:
3778:
3775:
3770:
3766:
3706:
3703:
3702:
3701:
3693:
3677:
3668:
3640:
3639:
3616:
3591:
3550:
3549:
3485:
3421:
3376:eigenfunctions
3363:
3360:
3268:
3267:
3256:
3252:
3247:
3243:
3236:
3231:
3228:
3225:
3220:
3216:
3212:
3209:
3206:
3203:
3199:
3195:
3191:
3187:
3182:
3177:
3173:
3167:
3162:
3158:
3154:
3151:
3147:
3140:
3135:
3132:
3129:
3124:
3120:
3116:
3113:
3110:
3107:
3103:
3099:
3095:
3089:
3082:
3077:
3073:
3069:
3066:
3063:
3060:
3055:
3051:
3047:
3044:
3041:
3038:
3035:
3007:
2992:
2985:
2984:
2973:
2970:
2966:
2959:
2955:
2951:
2948:
2944:
2940:
2936:
2930:
2925:
2921:
2913:
2909:
2905:
2900:
2897:
2893:
2886:
2882:
2878:
2875:
2870:
2866:
2860:
2855:
2851:
2831:
2815:
2802:
2793:
2782:
2775:
2774:
2763:
2760:
2756:
2751:
2747:
2742:
2739:
2736:
2731:
2727:
2721:
2716:
2712:
2706:
2701:
2697:
2693:
2690:
2686:
2681:
2678:
2675:
2670:
2666:
2660:
2655:
2651:
2647:
2644:
2641:
2638:
2633:
2629:
2616:. Then define
2593:
2585:[0, 1]
2568:
2567:
2556:
2552:
2545:
2541:
2537:
2534:
2530:
2527:
2523:
2520:
2517:
2512:
2507:
2503:
2495:
2491:
2487:
2482:
2479:
2475:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2441:
2436:
2432:
2426:
2423:
2419:
2415:
2412:
2408:
2401:
2397:
2393:
2390:
2387:
2384:
2379:
2374:
2370:
2300:[0, 1]
2283:[0, 1]
2265:
2262:
2153:
2152:
2141:
2137:
2130:
2126:
2122:
2119:
2115:
2112:
2106:
2101:
2097:
2082:
2081:
2070:
2065:
2061:
2057:
2052:
2048:
2044:
2037:
2033:
2029:
2026:
2022:
2019:
2013:
2008:
2004:
1999:
1992:
1987:
1984:
1980:
1974:
1970:
1967:
1964:
1960:
1957:
1952:
1946:
1941:
1937:
1933:
1929:
1925:
1921:
1916:
1913:
1910:
1906:
1903:
1897:
1892:
1888:
1883:
1879:
1875:
1871:
1868:
1865:
1862:
1858:
1742:
1741:
1730:
1727:
1723:
1716:
1712:
1708:
1705:
1701:
1698:
1692:
1687:
1683:
1679:
1676:
1672:
1665:
1659:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1619:
1616:
1610:
1603:
1598:
1594:
1590:
1587:
1583:
1576:
1572:
1568:
1565:
1562:
1557:
1552:
1548:
1518:
1517:
1506:
1501:
1496:
1493:
1490:
1485:
1481:
1477:
1474:
1471:
1466:
1458:
1455:
1451:
1446:
1441:
1437:
1433:
1430:
1426:
1423:
1419:
1414:
1408:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1368:
1365:
1359:
1354:
1349:
1345:
1341:
1338:
1335:
1301:
1298:
1284:
1273:
1247:
1240:
1234:
1233:
1222:
1217:
1212:
1208:
1204:
1199:
1194:
1190:
1186:
1181:
1177:
1171:
1166:
1163:
1160:
1156:
1152:
1149:
1145:
1138:
1134:
1130:
1127:
1123:
1120:
1114:
1111:
1106:
1102:
1087:
1086:
1075:
1070:
1065:
1061:
1057:
1052:
1047:
1043:
1039:
1034:
1029:
1026:
1023:
1019:
1015:
1012:
1008:
1001:
997:
993:
990:
987:
982:
979:
974:
970:
948:
937:
936:
925:
921:
914:
909:
906:
903:
900:
892:
888:
884:
878:
873:
870:
867:
864:
856:
852:
847:
841:
838:
835:
831:
827:
822:
818:
812:
809:
804:
801:
798:
795:
792:
765:
764:Fourier series
762:
748:
745:
725:
724:
682:
681:
680:
669:
666:
662:
655:
651:
647:
644:
640:
637:
631:
626:
622:
614:
610:
604:
600:
594:
591:
587:
580:
576:
572:
569:
566:
561:
556:
552:
538:
537:
519:
518:
476:
475:
474:
463:
460:
456:
449:
445:
441:
438:
434:
431:
425:
420:
416:
408:
404:
398:
394:
388:
385:
381:
374:
370:
366:
363:
360:
355:
350:
346:
332:
331:
298:
297:
250:
249:
248:
237:
234:
230:
223:
219:
215:
212:
208:
205:
199:
194:
190:
181:
177:
173:
167:
163:
157:
154:
150:
143:
139:
135:
132:
129:
124:
119:
115:
101:
100:
63:
60:
28:
27:
15:
13:
10:
9:
6:
4:
3:
2:
4886:
4875:
4872:
4870:
4867:
4865:
4862:
4861:
4859:
4848:
4844:
4840:
4836:
4832:
4828:
4824:
4820:
4816:
4812:
4808:
4804:
4800:
4799:
4794:
4790:
4786:
4782:
4778:
4774:
4770:
4766:
4762:
4758:
4754:
4750:
4746:
4742:
4738:
4734:
4730:
4726:
4724:0-12-170640-0
4720:
4716:
4712:
4708:
4704:
4699:
4695:
4691:
4687:
4683:
4679:
4673:
4669:
4665:
4661:
4657:
4653:
4649:
4648:
4641:
4637:
4632:
4629:
4625:
4620:
4618:
4614:
4610:
4605:
4602:
4599:, p. 36.
4598:
4593:
4590:
4586:
4581:
4578:
4574:
4569:
4566:
4562:
4557:
4555:
4551:
4547:
4542:
4540:
4538:
4534:
4527:
4525:
4521:
4517:
4507:
4498:
4489:
4485:
4478:
4474:
4470:
4463:
4456:
4452:
4448:
4441:
4437:
4433:
4429:
4409:
4406:
4403:
4397:
4394:
4388:
4384:
4374:
4373:
4372:
4360:
4356:
4352:
4344:
4340:
4319:
4316:
4298:
4290:
4284:
4281:
4276:
4268:
4261:
4258:
4245:
4242:
4237:
4233:
4229:
4226:
4215:
4200:
4194:
4191:
4185:
4178:
4175:
4162:
4159:
4154:
4150:
4142:
4141:
4140:
4123:
4104:
4097:
4094:
4087:
4081:
4078:
4075:
4070:
4062:
4055:
4052:
4048:
4043:
4035:
4028:
4025:
4012:
4009:
4004:
4000:
3996:
3993:
3990:
3987:
3981:
3978:
3972:
3968:
3958:
3957:
3956:
3935:
3932:
3920:
3913:
3910:
3903:
3897:
3892:
3889:
3884:
3880:
3876:
3869:
3868:
3867:
3865:
3858:and the area
3842:
3829:
3821:
3814:
3811:
3807:
3802:
3794:
3787:
3784:
3776:
3773:
3768:
3764:
3756:
3755:
3754:
3751:
3744:
3740:
3726:
3722:
3717:in 1901. Let
3716:
3715:Adolf Hurwitz
3712:
3704:
3696:
3692:
3688:
3683:
3678:
3672:
3667:
3663:
3658:
3653:
3652:
3651:
3649:
3645:
3617:
3613:
3606:
3600:
3592:
3587:
3574:
3573:
3572:
3568:
3563:
3559:
3555:
3538:
3530:
3521:
3511:
3493:
3486:
3474:
3466:
3457:
3447:
3429:
3422:
3414:
3398:
3393:
3389:
3388:
3387:
3385:
3381:
3377:
3373:
3369:
3361:
3359:
3358:in question.
3357:
3352:
3345:
3341:
3337:
3333:
3329:, shows that
3328:
3322:
3318:
3311:
3307:
3303:
3299:
3295:
3287:
3281:
3277:
3271:
3254:
3245:
3226:
3218:
3214:
3210:
3204:
3197:
3193:
3189:
3180:
3175:
3171:
3165:
3160:
3156:
3152:
3149:
3130:
3122:
3118:
3114:
3108:
3101:
3097:
3093:
3080:
3075:
3071:
3067:
3061:
3053:
3049:
3045:
3039:
3033:
3026:
3025:
3024:
3021:
3013:converges in
3010:
3006:
2995:
2991:
2971:
2968:
2957:
2949:
2942:
2938:
2934:
2928:
2923:
2919:
2911:
2907:
2903:
2898:
2895:
2884:
2876:
2868:
2864:
2858:
2853:
2849:
2841:
2840:
2839:
2834:
2830:
2826:
2818:
2814:
2805:
2801:
2796:
2792:
2785:
2781:
2761:
2758:
2749:
2737:
2729:
2725:
2719:
2714:
2710:
2704:
2699:
2695:
2691:
2688:
2676:
2668:
2664:
2658:
2653:
2649:
2645:
2639:
2631:
2627:
2619:
2618:
2617:
2613:
2596:
2592:
2577:
2554:
2543:
2535:
2528:
2521:
2518:
2510:
2505:
2501:
2493:
2489:
2485:
2480:
2477:
2465:
2459:
2456:
2450:
2444:
2439:
2434:
2430:
2424:
2421:
2417:
2413:
2410:
2399:
2391:
2385:
2382:
2377:
2372:
2368:
2360:
2359:
2358:
2356:
2338:
2328:
2322:
2318:
2311:
2307:
2290:
2263:
2261:
2256:converges to
2250:
2246:
2242:
2232:
2228:
2224:
2219:
2206:
2192:
2180:
2176:
2169:
2139:
2128:
2120:
2113:
2110:
2104:
2099:
2095:
2087:
2086:
2085:
2068:
2063:
2059:
2055:
2050:
2046:
2035:
2027:
2020:
2017:
2011:
2006:
2002:
1997:
1990:
1985:
1982:
1965:
1958:
1955:
1944:
1939:
1935:
1931:
1927:
1923:
1911:
1904:
1901:
1895:
1890:
1886:
1881:
1877:
1866:
1860:
1848:
1847:
1846:
1842:
1837:
1831:
1827:
1820:
1813:
1807:
1803:
1799:
1793:
1786:
1782:
1778:
1774:
1769:
1764:
1760:
1756:
1752:
1748:
1728:
1725:
1714:
1706:
1699:
1696:
1690:
1685:
1681:
1677:
1674:
1663:
1651:
1648:
1645:
1639:
1633:
1630:
1624:
1617:
1614:
1601:
1596:
1592:
1588:
1585:
1574:
1566:
1560:
1555:
1550:
1546:
1538:
1537:
1536:
1532:
1528:
1523:
1504:
1494:
1491:
1488:
1483:
1475:
1469:
1456:
1453:
1449:
1444:
1439:
1431:
1424:
1421:
1417:
1412:
1400:
1397:
1394:
1388:
1382:
1379:
1373:
1366:
1363:
1352:
1347:
1339:
1333:
1326:
1325:
1324:
1320:
1316:
1299:
1297:
1295:
1290:
1283:
1279:
1272:
1268:
1264:
1257:
1250:
1243:
1215:
1210:
1206:
1202:
1197:
1192:
1188:
1179:
1175:
1164:
1161:
1158:
1154:
1150:
1147:
1136:
1128:
1121:
1118:
1112:
1109:
1104:
1100:
1092:
1091:
1090:
1068:
1063:
1059:
1055:
1050:
1045:
1041:
1027:
1024:
1021:
1017:
1013:
1010:
999:
991:
985:
980:
977:
972:
968:
960:
959:
958:
956:
947:
923:
919:
912:
907:
904:
901:
898:
890:
886:
882:
876:
871:
868:
865:
862:
854:
850:
845:
839:
836:
833:
829:
825:
820:
816:
810:
807:
802:
796:
790:
783:
782:
781:
779:
763:
761:
755:
746:
744:
742:
738:
734:
730:
713:
705:
696:
692:
688:
683:
667:
664:
653:
645:
638:
635:
629:
624:
620:
612:
608:
602:
598:
592:
589:
578:
570:
564:
559:
554:
550:
542:
541:
540:
539:
532:
521:
520:
507:
499:
490:
486:
482:
477:
461:
458:
447:
439:
432:
429:
423:
418:
414:
406:
402:
396:
392:
386:
383:
372:
364:
358:
353:
348:
344:
336:
335:
334:
333:
327:
323:
319:
311:
300:
299:
282:
273:
264:
260:
256:
251:
235:
232:
221:
213:
206:
203:
197:
192:
188:
179:
175:
171:
165:
161:
155:
152:
141:
133:
127:
122:
117:
113:
105:
104:
103:
102:
96:
92:
88:
80:
69:
68:
67:
61:
59:
57:
53:
49:
48:Adolf Hurwitz
45:
41:
37:
33:
26:
24:
19:
18:
4869:Inequalities
4826:
4823:Weiss, Guido
4802:
4796:
4761:Inequalities
4760:
4749:Hardy, G. H.
4702:
4655:
4652:Brezis, Haim
4640:Hurwitz 1901
4631:
4604:
4592:
4580:
4568:
4519:
4515:
4505:
4496:
4487:
4486:– α) +
4483:
4476:
4472:
4468:
4461:
4454:
4450:
4446:
4439:
4435:
4431:
4427:
4424:
4358:
4354:
4350:
4342:
4338:
4334:
4138:
3950:
3857:
3749:
3742:
3738:
3724:
3720:
3708:
3694:
3690:
3681:
3670:
3665:
3656:
3644:metric balls
3641:
3611:
3604:
3599:covering map
3585:
3566:
3551:
3536:
3528:
3519:
3509:
3491:
3472:
3464:
3455:
3445:
3427:
3412:
3396:
3365:
3353:
3343:
3339:
3335:
3331:
3320:
3316:
3309:
3305:
3301:
3297:
3279:
3275:
3272:
3269:
3019:
3008:
3004:
2993:
2989:
2986:
2832:
2828:
2824:
2816:
2812:
2803:
2799:
2794:
2790:
2783:
2779:
2776:
2611:
2594:
2590:
2569:
2336:
2320:
2316:
2309:
2305:
2288:
2267:
2248:
2244:
2240:
2230:
2226:
2222:
2204:
2190:
2178:
2174:
2167:
2154:
2083:
1840:
1829:
1825:
1818:
1811:
1805:
1801:
1797:
1794:
1784:
1780:
1776:
1772:
1762:
1758:
1754:
1750:
1746:
1743:
1533:(π) = 0
1530:
1526:
1519:
1318:
1314:
1303:
1288:
1281:
1277:
1270:
1266:
1262:
1255:
1245:
1238:
1235:
1088:
945:
938:
767:
750:
736:
726:
711:
703:
694:
690:
686:
530:
505:
497:
488:
484:
480:
325:
321:
317:
309:
280:
271:
262:
258:
254:
94:
90:
86:
78:
65:
39:
32:mathematical
29:
20:
4805:: 401–403.
4789:Hurwitz, A.
4624:Chavel 1984
4609:Chavel 1984
4597:Chavel 1984
4573:Chavel 1984
4561:Brezis 2011
1535:then shows
4858:Categories
4847:0232.42007
4811:32.0386.01
4781:0047.05302
4741:0551.53001
4694:1220.46002
4528:References
3394:of length
3372:eigenvalue
3342:cos π
3288:says that
2777:Then each
2349:. Because
4757:Pólya, G.
4466:and then
4464:– α)
4404:≥
4398:π
4282:−
4246:π
4234:∫
4163:π
4151:∫
4013:π
4001:∫
3988:−
3982:π
3893:π
3881:∫
3877:−
3777:π
3765:∫
3211:−
3172:∫
3157:∫
3153:−
3115:−
3072:∫
3046:−
2920:∫
2908:π
2899:≤
2850:∫
2711:∫
2696:∫
2692:−
2650:∫
2502:∫
2490:π
2431:∫
2422:−
2418:π
2414:≤
2369:∫
2105:π
2096:∫
2003:∫
1986:≤
1936:∫
1932:≤
1887:∫
1691:π
1682:∫
1649:
1631:−
1602:π
1593:∫
1556:π
1547:∫
1492:
1445:−
1398:
1380:−
1170:∞
1155:∑
1113:π
1101:∫
1033:∞
1018:∑
981:π
969:∫
913:π
902:
877:π
866:
837:≥
830:∑
621:∫
609:π
593:≤
551:∫
415:∫
403:π
387:≤
345:∫
274:− α)
189:∫
176:π
156:≤
114:∫
34:field of
4825:(1971).
4791:(1901).
4759:(1952).
4660:Springer
4654:(2011).
4262:′
4179:′
4098:′
4056:′
4029:′
3914:′
3815:′
3788:′
3735:so that
3490:[0,
3426:[0,
3323:′(1) = 0
3198:′
3102:′
2943:′
2807:′(1) = 0
2570:for all
2529:′
2323:′(1) = 0
2114:′
2021:′
1959:′
1905:′
1821:= π
1700:′
1618:′
1425:′
1367:′
1253:for all
1122:′
776:. Since
639:′
529:[0,
433:′
308:[0,
270:2π(
207:′
77:[0,
36:analysis
4839:0304972
4773:0046395
4733:0768584
4686:2759829
3628:2π
3541:
3524:
3514:
3500:
3477:
3460:
3450:
3436:
3417:
3406:4π
3403:
3378:of the
3319:′(0) =
2798:′(0) =
2339:π)
2319:′(0) =
2209:
2195:
2172:
2158:
1843:(0) = 0
774:2π
716:
699:
510:
493:
285:
267:
62:Theorem
30:In the
4845:
4837:
4809:
4779:
4771:
4739:
4731:
4721:
4692:
4684:
4674:
3673:− 2)/2
3554:sphere
3527:π
3503:π
3463:π
3439:π
2602:(0, 1)
2347:π
2272:to be
2258:π
2247:) cot
2229:) cot
1804:) cot
1761:) cot
1529:(0) =
1310:π
1308:to be
772:to be
747:Proofs
702:π
496:π
330:. Then
320:(0) =
294:α
99:. Then
89:(0) =
38:, the
4442:) = 0
3866:) by
3562:torus
3560:, or
3494:]
3430:]
3314:with
3312:) = 0
2308:′′ +
2220:that
2193:<
1845:that
1770:) is
953:. By
533:]
328:) = 0
315:with
312:]
81:]
4719:ISBN
4672:ISBN
4503:and
4482:cos(
4475:) =
4460:sin(
4453:) =
4434:) +
3522:cos
3458:sin
3338:) =
3304:) +
2333:are
2314:and
2189:cot
1838:and
1816:and
1783:sin
1779:) =
1753:) =
1287:cos
1276:sin
1269:) =
1089:and
697:cos
693:) =
522:Let
491:sin
487:) =
301:Let
292:and
265:sin
261:) =
70:Let
4843:Zbl
4807:JFM
4803:132
4777:Zbl
4737:Zbl
4711:doi
4690:Zbl
4664:doi
3646:in
3607:+ 2
3588:+ 1
3569:= 1
3497:is
3433:is
3400:is
3300:′′(
3017:to
2998:by
2827:= −
2609:to
2582:on
2312:= 0
2297:on
2280:on
2183:as
1814:= 0
1646:cot
1489:cot
1395:cot
1258:≥ 2
1251:= 0
951:= 0
899:cos
863:sin
4860::
4841:.
4835:MR
4833:.
4821:;
4801:.
4795:.
4775:.
4769:MR
4767:.
4755:;
4751:;
4735:.
4729:MR
4727:.
4717:.
4709:.
4688:.
4682:MR
4680:.
4670:.
4662:.
4616:^
4553:^
4536:^
4518:,
4430:′(
4357:+
4353:,
4341:,
3741:,
3723:,
3697:/2
3650::
3556:,
3386::
3351:.
2836:′′
2820:′′
2289:Tf
2260:.
1792:.
1749:′(
1296:.
1280:+
1244:=
957:,
760:.
58:.
4849:.
4813:.
4783:.
4743:.
4713::
4696:.
4666::
4642:.
4522:)
4520:y
4516:x
4514:(
4509:2
4506:c
4500:1
4497:c
4491:2
4488:c
4484:t
4480:1
4477:c
4473:t
4471:(
4469:x
4462:t
4458:1
4455:c
4451:t
4449:(
4447:y
4440:t
4438:(
4436:y
4432:t
4428:x
4410:,
4407:A
4395:4
4389:2
4385:L
4369:y
4365:z
4361:)
4359:z
4355:y
4351:x
4349:(
4345:)
4343:y
4339:x
4337:(
4320:.
4317:t
4313:d
4306:)
4299:2
4295:)
4291:t
4288:(
4285:y
4277:2
4273:)
4269:t
4266:(
4259:y
4253:(
4243:2
4238:0
4230:+
4227:t
4223:d
4216:2
4210:)
4204:)
4201:t
4198:(
4195:y
4192:+
4189:)
4186:t
4183:(
4176:x
4170:(
4160:2
4155:0
4124:t
4120:d
4113:)
4108:)
4105:t
4102:(
4095:x
4091:)
4088:t
4085:(
4082:y
4079:2
4076:+
4071:2
4067:)
4063:t
4060:(
4053:y
4049:+
4044:2
4040:)
4036:t
4033:(
4026:x
4020:(
4010:2
4005:0
3997:=
3994:A
3991:2
3979:2
3973:2
3969:L
3953:L
3936:.
3933:t
3929:d
3924:)
3921:t
3918:(
3911:x
3907:)
3904:t
3901:(
3898:y
3890:2
3885:0
3860:A
3843:t
3839:d
3830:2
3826:)
3822:t
3819:(
3812:y
3808:+
3803:2
3799:)
3795:t
3792:(
3785:x
3774:2
3769:0
3750:L
3745:)
3743:y
3739:x
3737:(
3727:)
3725:y
3721:x
3719:(
3700:.
3695:n
3691:J
3682:R
3676:.
3671:n
3669:(
3666:J
3657:R
3636:n
3632:1
3624:n
3620:n
3612:R
3605:n
3603:2
3595:n
3586:n
3581:n
3577:n
3567:n
3548:.
3546:c
3537:L
3533:/
3529:x
3520:c
3510:L
3506:/
3492:L
3484:.
3482:c
3473:L
3469:/
3465:x
3456:c
3446:L
3442:/
3428:L
3413:L
3409:/
3397:L
3349:c
3344:x
3340:c
3336:x
3334:(
3332:y
3321:y
3317:y
3310:x
3308:(
3306:y
3302:x
3298:y
3290:y
3282:)
3280:x
3278:(
3276:y
3255:w
3251:d
3246:z
3242:d
3235:)
3230:)
3227:z
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