This is a preprint of: “On the existence of Symmetric Bicircular Central Configurations of the 3n-Body Problem”, Montserrat Corbera, Clàudia Valls, J. Nonlinear Sci., vol. 31(6), 88:1–52, 2021. DOI: [10.1007/s00332-021-09743-z] PL h>1 1sAah1L*1 P6 auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa P6 h>1 3nĜ"P.u S_P"G1J JPLha1__h *P_"1_ L. *Gl.A oGGa #bi`+iX AM i?Bb TT2` r2 +QMbB/2` +2Mi`H +QM};m`iBQMb Q7 i?2 THM` 3nĜ#Q/v T`Q#H2K +QMbBbiBM; Q7 n Kbb2b i i?2 p2`iB+2b Q7  `2;mH` nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb r M/ 2n Kbb2b i i?2 p2`iB+2b Q7  b2+QM/ UMQi M2+2bb`BHv `2;mH`V +QM+2Mi`B+ 2nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb ar r?B+? `2 bvKK2i`B+ BM i?2 b2Mb2 i?i i?2 b2i Q7 TQbBiBQMb Q7 i?2 3n Kbb2b M/ i?2 b2i Q7 i?2 +Q``2bTQM/BM; Kbb2b `2 BMp`BMi mM/2` i?2 +iBQM Q7  }MBi2 bm#;`QmT Q7 O(2)X h?2`2 `2 irQ /Bz2`2Mi ivT2b Q7 bm+? +QM};m`iBQMbX AM i?2 }`bi ivT2- +HH2/ `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K- i?2 b2+QM/ 2nĜ;QM Bb `2;mH`- n Q7 i?2 p2`iB+2b Q7 i?2 b2+QM/ nĜ;QM `2 HB;M2/ rBi? i?2 p2`iB+2b Q7 i?2 }`bi `2;mH` nĜ;QM M/ i?2 Kbb2b i i?2 p2`iB+2b Q7 i?Bb 2nĜ;QM Hi2`Mi2 pHm2bX AM i?2 b2+QM/ ivT2- +HH2/ b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K- i?2 b2+QM/ 2nĜ;QM Bb b2KB`2;mH` M/ i?2 Kbb2b i Bib p2`iB+2b `2 HH Q7 i?2K 2[mHX  b2KB`2;mH` 2nĜ;QM ?b n TB` Q7 p2`iB+2b bvKK2i`B+ #v  `2~2+iBQM Q7 M M;H2 β rBi? 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 rv i?i i?2 `2bmHiBM; +QM};m`iBQM Bb +2Mi`H Ub22 7Q` KQ`2 /2iBHb (N)VX aBM+2 i?2M b2p2`H mi?Q`b ?p2 bim/B2/ +2Mi`H +QM};m`iBQMb ?pBM; i?2 Kbb2b i i?2 p2`iB+2b Q7 +QM+2Mi`B+ `2;mH` nĜ;QMbX bbmKBM; i?i i?2 Kbb2b QM 2+? nĜ;QM `2 2[mH M/ Kbb2b QM /Bz2`2Mi nĜ;QMb +QmH/ #2 /Bz2`2Mi- EH2KTH2`2` (3) BM RNed- ;p2 i?2 `2HiBQM #2ir22M i?2 `iBQ Q7 i?2 Kbb2b M/ i?2 `iBQ Q7 i?2 `/BB 7Q` irQ irBbi2/ `2;mH` nĜ;QMb +2Mi`H +QM};m`iBQMb r?2M n = 2, 3, 4X JQ`2 `2+2MiHv JQ2+F2H M/ aBKƦ BM (RR) T`Qp2/ i?i 7Q` Mv n M/ 7Q` 2p2`v `iBQ Q7 i?2 Kbb2b i?2`2 `2 2t+iHv irQ M2bi2/ `2;mH` nĜ;QMb +2Mi`H +QM};m`iBQMb QM2 rBi? i?2 `iBQb Q7 i?2 bBx2b Q7 i?2 nĜ;QMb H2bb i?M QM2 M/ i?2 Qi?2` rBi? i?2 `iBQ Q7 i?2 bBx2b ;`2i2` i?M QM2X h?2`2 `2 HbQ b2p2`H rQ`Fb bim/vBM; +2Mi`H +QM};m`iBQMb rBi? kn Kbb2b i i?2 p2`iB+2b Q7 k +QM+2Mi`B+ nĜ;QMb bm+? i?i HH i?2 Kbb2b QM i?2 bK2 nĜ;QM `2 2[mH M/ Kbb2b QM /Bz2`2Mi nĜ;QMb +QmH/ #2 /Bz2`2MiX h?2b2 +2Mi`H +QM};m`iBQMb `2 +HH2/  +`QrM Q7 k `BM;b Q7 n #Q/B2b Q` bBKTHv  (k, n)@+`QrM BM (k)- #mi bBM+2 QMHv 72r mi?Q`b mb2 i?Bb MQK2M+Him`2 r2 rBHH MQi mb2 Bi ?2`2X GHB#`2 M/ J2HQ BM (j) T`Qp2/ i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb Q7 i?`22 irBbi2/ `2;mH` nĜ;QMb rBi? 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+QM+2Mi`B+ TQHv;QM Bb MQi M2+2bb`BHv 2[mHX a22 #2HQr 7Q`  KQ`2 T`2+Bb2 2tTHMiBQM Q7 JQMiH/BǶb rQ`FX h?2`2 `2 bQK2 Qi?2` rQ`Fb bim/vBM; +2Mi`H +QM};m`iBQMb rBi? i?2 Kbb2b i i?2 p2`iB+2b Q7 +QM+2Mi`B+ TQHv;QMb ?pBM; /Bz2`2Mi MmK#2` Q7 p2`iB+2bX um M/ w?M; BM (R3) T`Qp2/ i?i B7  +QM};m`iBQM rBi? n@2[mH Kbb2b i i?2 p2`iB+2b Q7  `2;mH` nĜ;QM M/ ℓ 2[mH Kbb2b i i?2 p2`iB+2b Q7  `2;mH` ℓĜ;QM rBi?  +QKKQM +2Mi2` Bb +2Mi`H- i?2M n = ℓX aBHmbxvF BM (Rj- R9) bim/B2/ +2Mi`H +QM};m`iBQMb +QMbBbiBM; Q7 n 2[mH Kbb2b i i?2 p2`iB+2b Q7  `2;mH` nĜ;QM M/ 2n Kbb2b i i?2 p2`iB+2b Q7  b2+QM/ +QM+2Mi`B+ 2nĜ;QM rBi?  +QKKQM +2Mi2`X AM T`iB+mH` BM (Rj) i?2 mi?Q` 7QmM/ i?2 2tT`2bbBQMb Q7 i?2 Kbb2b BM 7mM+iBQM Q7 i?2 bBx2b Q7 i?2 nĜ;QMb M/ BM (R9)- #v mbBM; +QKTmi2` bbBbi2/ K2i?Q/b- b?2 bim/B2/ i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb Q7 i?Bb ivT2 7Q` n = 2X J`+?2bBM BM (Ry) T`Qp2/ i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb +QMbBbiBM; Q7 4 2[mH Kbb2b i i?2 p2`iB+2b Q7 M b[m`2- 4 2[mH Kbb2b i i?2 p2`iB+2b Q7  b2+QM/ +QM+2Mi`B+ b[m`2 irBbi2/ #v M M;H2 Q7 π/4 M/ 8 2[mH Kbb2b QM  i?B`/ +QM+2Mi`B+ TQHv;QM ?pBM; i?2 p2`iB+2b i i?2 #Bb2+iQ`b Q7 i?2 M;H2b 7Q`K2/ #v 2+? TB` Q7 TQbBiBQM p2+iQ`b Q7 irQ +QMb2+miBp2 Kbb2b Q7 i?2 T`2pBQmb irQ b[m`2bX 6`QK MQr QM  +2Mi`H +QM};m`iBQM Bb bvKK2i`B+ B7 i?2 b2i Q7 TQbBiBQMb M/ i?2 b2i Q7 Kbb2b `2 BMp`BMi mM/2` i?2 +iBQM Q7  }MBi2 bm#;`QmT Q7 O(2)X AMbTB`2/ #v i?2 rQ`Fb Q7 aBHmbxvF- JQMiH/B M/ J`+?2bBM- BM i?Bb rQ`F r2 +QMbB/2` bvKK2i`B+ +2Mi`H +QM};m`iBQMb +QMbBbiBM; Q7 n 2[mH Kbb2b PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa j i i?2 p2`iB+2b Q7  `2;mH` nĜ;QM M/ 2n Kbb2b UMQi HH i?2K M2+2bb`BHv 2[mHV i i?2 p2`iB+2b Q7  b2+QM/ UMQi M2+2bb`BHv `2;mH`V +QM+2Mi`B+ 2nĜ;QMX q2 +HH i?Bb FBM/ Q7 +QM};m`iBQMb bvKK2i`B+ #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2KX LQiB+2 i?i #v (R3) B7 i?2 2nĜ;QM Bb `2;mH`- i?2M i?2 Kbb2b i i?Bb 2nĜ;QM +MMQi #2 HH Q7 i?2K 2[mHX AM 7+i- r?2M i?2 2nĜ;QM Bb `2;mH` i?2 bvKK2i`B+ #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K +M #2 i?Qm;?i b i?2 HBKBi +b2 r?2M a2 → a3 Q7 i?2 +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K +QMbBbiBM; Q7 n Kbb2b 2[mH iQ m1 i i?2 p2`iB+2b Q7  `2;mH` nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb a1 - n Kbb2b 2[mH iQ m2 i i?2 p2`iB+2b Q7  M2bi2/ `2;mH` nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb a2 M/ n //BiBQMH Kbb2b 2[mH iQ m3 UrBi? m2 6= m3 #v (R3)V- i i?2 p2`iB+2b Q7  `2;mH` nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb a3 irBbi2/ #v M M;H2 Q7 π/n rBi? `2bT2+i i?2 T`2pBQmb irQX h?mb r?2M i?2 2nĜ;QM Bb `2;mH` i?2 Kbb2b i i?2 p2`iB+2b Q7 i?Bb 2nĜ;QM Hi2`Mi2 i?2 pHm2b m2 M/ m3 X lbBM; i?2 MQK2M+Him`2 BM (k)- i?2 bvKK2i`B+ #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K `2  (3, n)Ĝ+`QrMX kX aii2K2Mi Q7 i?2 T`Q#H2K M/ Q7 i?2 KBM `2bmHib ++Q`/BM; iQ JQMiH/B (Rk) r2 ?p2 irQ /Bz2`2Mi ivT2b Q7 bvKK2i`B+ #B+B`+mH` +2Mi`H +QM};m`@ iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2KX AM/22/- BM JQMiH/B (Rk) Bi Bb T`Qp2/ i?i  ;2M2`B+ bvKK2i`B+ THM` +2Mi`H +QM};m`iBQM +QMbBbib Q7 kO UrBi? kO = 0 Q` kO = 1V Kbb2b i i?2 Q`B;BM- n · kN Kbb2b i i?2 p2`iB+2b Q7 kN `2;mH` M2bi2/ nĜ;QMb- n · kT Kbb2b i i?2 p2`iB+2b Q7 kT `2;mH` nĜ;QMb irBbi2/ #v M M;H2 Q7 π/n- M/ 2n · kS Kbb2b i i?2 p2`iB+2b Q7 kS M2bi2/ b2KB`2;mH` 2nĜ;QMb- HH +2Mi2`2/ i i?2 Q`B;BMX JQ`2Qp2` i?2 Kbb2b QM 2+? TQHv;QM Kmbi #2 2[mH #mi Kbb2b QM /Bz2`2Mi TQHv;QMb +M #2 /Bz2`2MiX  b2KB`2;mH` 2nĜ;QM Bb  bvKK2i`B+ TQHv;QM Q7 2n p2`iB+2b r?B+? Bb BMp`BMi #v `2~2+iBQMb rBi? `2bT2+i iQ HH bvKK2i`v tBb Q7 i?2 `2;mH` nĜ;QMX h?2 bvKK2i`B+ #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K /Q2b MQi +QMiBM Mv Kbb i i?2 Q`B;BM- bQ kO = 0X JQ`2Qp2` bBM+2 BM i?2b2 +QM};m`iBQMb r2 ?p2 3nĜ#Q/B2b- kN +kT +2 kS = 3X h?mb r2 ?p2 QMHv i?`22 TQbbB#BHBiB2b, 2Bi?2` kN = 1- kT = 2 M/ kS = 0- Q` kN = 2- kT = 1 M/ kS = 0- Q` kN = 1- kT = 0 M/ kS = 1X aBM+2 +2Mi`H +QM};m`iBQMb `2 BMp`BMi mM/2` `QiiBQMb i?2 +b2 kN = 2- kT = 1 M/ kS = 0 M/ i?2 +b2 kN = 1- kT = 2 M/ kS = 0 T`QpB/2 i?2 bK2 +QM};m`iBQM mT iQ  `QiiBQMX aQ r2 ?p2 QMHv irQ /Bz2`2Mi ivT2b Q7 bvKK2i`B+ #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K- i?2 +QM};m`iBQMb rBi? kN = 1- kT = 2 M/ kS = 0 r?B+? `2 +HH2/ `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K M/ i?2 +QM};m`iBQMb rBi? kN = 1- kT = 0 M/ kS = 1 r?B+? `2 +HH2/ b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2KX AM b?Q`i-  `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQM Q7 i?2 3nĜ#Q/v T`Q#H2K +QMbBbib Q7 n #Q/B2b rBi? Kbb2b 2[mH iQ m1 i i?2 p2`iB+2b Q7  `2;mH` nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb a1 = r M/ 2n #Q/B2b rBi? Hi2`MiBM; Kbb2b m2 M/ m3 rBi? m2 6= m3 i i?2 p2`iB+2b Q7  b2+QM/ `2;mH` 2nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb a2 = ar rBi?  +QKKQM +2Mi2` ?pBM; n p2`iB+2b HB;M2/ rBi? i?2 p2`iB+2b Q7 i?2 }`bi `2;mH` nĜ;QM- b22 6B;X RUVX JQ`2Qp2`-  b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQM Q7 i?2 3nĜ#Q/v T`Q#H2K +QMbBbib Q7 n #Q/B2b rBi? Kbb2b 2[mH iQ m1 i i?2 p2`iB+2b Q7  `2;mH` nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb a1 = r M/ 2n #Q/B2b rBi? Kbb2b 2[mH iQ m2 i i?2 p2`iB+2b Q7  b2+QM/ +QM+2Mi`B+ b2KB`2;mH` 2nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb a2 = ar ?pBM; n TB` Q7 p2`iB+2b bvKK2i`B+ #v  `2~2+iBQM Q7 M;H2 β rBi? `2bT2+i iQ i?2 tBb Q7 bvKK2i`v Q7 i?2 `2;mH` nĜ;QM- b22 6B;X RU#VX qBi?Qmi HQbb Q7 ;2M2`HBiv r2 +?QQb2 i?2 mMBi Q7 Kbb M/ i?2 mMBi Q7 H2M;i? bQ i?i m1 = 1 M/ r = 1X LQiB+2 i?i i?2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K `2 i?2 QM2b bim/B2/ BM (Rj- R9)- #mi i?2 mi?Q` BM i?2b2 irQ rQ`Fb /Q2b MQi MHvx2 r?2i?2` i?2 2tT`2bbBQMb Q7 i?2 Kb22b `2 TQbBiBp2 Q` M2;iBp2 M/ BM +b2 Q7 #2BM; TQbBiBp2 7Q` r?B+? pHm2b Q7 i?2 `iBQb Q7 i?2 `/Bmb i?2v `2 TQbBiBp2 M/ +QMb2[m2MiHv i?2v T`QpB/2 +2Mi`H +QM};m`iBQMbX AM i?2 }`bi T`i Q7 i?2 TT2` Ua2+iBQM jV r2 ;Q  bi2T 7m`i?2` BM i?2 bim/v Q7 i?2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2KX AM T`iB+mH` r2 T`Qp2 MHviB+HHv 7Q` HH n ≥ 3 i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb 7Q` 2Bi?2` bm{+B2MiHv bKHH pHm2b- Q` bm{+B2MiHv H`;2 pHm2b Q7 i?2 `iBQ Q7 i?2 `/Bmb Q7 i?2 +B`+H2b a M/ 7Q` n = 2 r2 T`Qp2 i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb 7Q` bm{+B2MiHv H`;2 pHm2b Q7 aX h?2M mbBM; +QKTmi2` bbBbi2/ K2i?Q/b r2 }M/ i?2 +QKTH2i2 b2i Q7 pHm2b Q7 a 7Q` r?B+? i?2`2 2tBbi `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K 7Q` 9 JX *P_"1_ L. *X oGGa 6B;XR  _2;mH` #B+B`+mH` 6B;XR # a2KB`2;mH` #B+B`+mH` +QM};m`iBQMb +QM};m`iBQMb 6B;m`2 RX 1tKTH2b Q7 #B+B`+mH` +QM};m`iBQMb n = 2, 3, 4, 5 M/ r2 MHvx2 i?2b2 7KBHB2b Q7 +2Mi`H +QM};m`iBQMbX 6BMHHv r2 KF2  MmK2`B+H 2tTHQ`iBQM Q7 i?2 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb Q7 i?2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K 7Q` n = 6, ..., 500X q2 `2K`F i?i i?2 `2bmHi Q#iBM2/ ?2`2 7Q` n = 2 /Q2b MQi +QBM+B/2 rBi? i?2 QM2 ;Bp2M BM (R9)X AM i?2 b2+QM/ T`i Q7 i?2 TT2` Ua2+iBQM 9V r2 bim/v i?2 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m@ `iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2KX AM T`iB+mH`- r2 b?Qr i?i i?2 QMHv pHm2b Q7 β i?i +M T`QpB/2 +2Mi`H +QM};m`iBQMb `2 β ∈ (π/2n, π/n) M/ r2 T`Qp2 MHviB+HHv i?2 2tBbi2M+2 Q7 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K 7Q` HH pHm2b Q7 β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 π/2n M/ Q7 π/n 7Q` HH n ≥ 2X h?2M r2 KF2  MmK2`B+H 2tTHQ`iBQM Q7 i?2 7KBHB2b Q7 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K 7Q` n = 2, . . . , 100X L2ti r2 ;Bp2  +QKTH2i2 bmKK`v Q7 i?2 Q#iBM2/ `2bmHibX kXRX amKK`v Q7 i?2 `2bmHib +QM+2`MBM; `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2KX h?2 MHviB+H `2bmHib i?i r2 ?p2 Q#iBM2/ 7Q` i?2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K `2 bmKK`Bx2/ BM i?2 7QHHQrBM; i?2Q`2KX h?2Q`2K RX 6Q` 2+? n > 2 i?2`2 2tBbi  MQM2KTivb2i D M/ 7mM+iBQMb m2 = m2 (a) M/ m3 = m3 (a) i?i T`QpB/2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K 7Q` HH a ∈ DX AM T`iB+mH`- UV A7 n = 2- i?2M r2 +M bbm`2 i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb i H2bi 7Q` HH a ∈ (a∗ , ∞)- r?2`2 a∗ ∈ (1, ∞) Bb i?2 H`;2bi x2`Q Q7 m3 X AM i?Bb +QM};m`iBQMb m3 → 0 M/ m2 → m2 (a∗ ) r?2M a → a∗ + M/ m2 , m3 → ∞ r?2M a → ∞X U#V A7 n > 3- i?2M r2 +M bbm`2 i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb i H2bi 7Q` HH a ∈ (0, a∗1 ) ∪ (a∗2 , ∞)- r?2`2 a∗1 Bb i?2 KBMBKmK #2ir22M i?2 }`bi TQbBiBp2 x2`Q M/ i?2 TQH2 Q7 m2 M/ a∗2 Bb i?2 H`;2bi x2`Q Q7 m3 X AM i?2b2 +QM};m`iBQMb m2 , m3 → 0 r?2M a → 0+ - m3 → 0 M/ m2 → m2 (a∗2 ) r?2M a → a∗2 + - M/ m2 , m3 → ∞ r?2M a → ∞X h?2 +QKTmi2/ bbBbi2/ `2bmHib i?i r2 ?p2 Q#iBM2/ 7Q` i?2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`@ iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K rBi? n = 2, 3, 4, 5 `2 bmKK`Bx2/ BM i?2 7QHHQrBM; `2bmHibX h?2Q`2K kX 6Q` n = 2 r2 +M }M/ 7mM+iBQMb m2 = m2 (a) M/ m3 = m3 (a) i?i T`QpB/2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 6Ĝ#Q/v T`Q#H2K 7Q` HH a ∈ D = (a1 , a2 ) ∪ (a∗ , ∞) r?2`2 a1 Bb i?2 TQH2 Q7 m2 - a2 Bb i?2 }`bi x2`Q Q7 m2 M/ a∗ Bb i?2 H`;2bi x2`Q Q7 m3 X h?2 TT`QtBKi2 pHm2b Q7 a1 - a2 M/ a∗ `2 ;Bp2M BM h#H2 RX JQ`2Qp2` i?2 7mM+iBQMb m2 M/ m3 biBb7v i?2 7QHHQrBM; T`QT2`iB2b Ub22 6B;X kV, UV m2 , m3 → ∞ rBi? m2 /m3 → 1− r?2M a → a+ 1c U#V BM i?2 BMi2`pH (a1 , a2 ) i?2 7mM+iBQM m2 Bb /2+`2bBM; M/ m3 ?b  mMB[m2 +`BiB+H TQBMi i a = a0 = 0.5670013389.. r?B+? Bb  KBMBKmK rBi? m3 (a0 ) = 4.7014182338..X JQ`2Qp2` m2 < m 3 c PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa 8 n k j 9 8 D (a1 , a2 ) ∪ (a∗ , ∞) (0, a∗1 ) ∪ (a1 , a2 ) ∪ (a∗2 , ∞) (0, a∗1 ) ∪ (a1 , a2 ) ∪ (a∗2 , ∞) (0, a∗1 ) ∪ (a∗2 , ∞) a∗1 = 0.13753297.. a∗1 = 0.35631015.. a∗1 = 0.63131956.. a1 = 0.53178607.. a1 = 0.58231561.. a1 = 0.61035220.. a2 = 0.61614478.. a2 = 0.63460382.. a2 = 0.61188342... a∗ = 3.52823222.. a∗2 = 2.63616425.. a∗2 = 2.26854434.. a∗2 = 2.06282906.. h#H2 RX h?2 b2i D T`QpB/BM; `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ #Q/v T`Q#H2KX h?2 pHm2b Q7 a BM i?2 +2HHb +QHQ`2/ BM HB;?i ;`v +Q``2bTQM/ iQ x2`Qb Q7 i?2 7mM+iBQM m2 - i?2 QM2b BM i?2 +2HHb +QHQ`2/ BM /`F ;`v +Q``2bTQM/ iQ TQH2b Q7 i?2 7mM+iBQM m2 M/ i?2 QM2b BM i?2 MQM +QHQ`2/ +2HHb +Q``2bTQM/ iQ i?2 H`;2bi x2`Q Q7 i?2 7mM+iBQM m3 X U+V m2 → 0 M/ m3 → 5.950134407.. r?2M a → a− 2c U/V m2 → 1.6921282709.. M/ m3 → 0 r?2M a → a∗ + c U2V BM i?2 BMi2`pH (a∗ , ∞) #Qi? 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM; M/ m2 > m3 c U7V m2 , m3 → ∞ rBi? m2 /m3 → 1+ r?2M a → ∞X h?2Q`2K jX 6Q` n = 3 r2 +M }M/ 7mM+iBQMb m2 = m2 (a) M/ m3 = m3 (a) i?i T`QpB/2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 9Ĝ#Q/v T`Q#H2K 7Q` HH a ∈ D = (0, a∗1 ) ∪ (a1 , a2 ) ∪ (a∗2 , ∞) r?2`2 a∗1 Bb i?2 }`bi x2`Q Q7 m2 - a1 Bb i?2 TQH2 Q7 m2 - a2 Bb i?2 b2+QM/ x2`Q Q7 m2 (a) M/ a∗2 Bb i?2 H`;2bi x2`Q Q7 m3 X h?2 TT`QtBKi2 pHm2b Q7 a∗1 - a1 - a2 M/ a∗2 `2 ;Bp2M BM h#H2 RX JQ`2Qp2` i?2 7mM+iBQMb m2 M/ m3 biBb7v i?2 7QHHQrBM; T`QT2`iB2b Ub22 6B;X jV, UV m2 , m3 → 0 rBi? m2 /m3 → 1− r?2M a → 0+ c U#V BM i?2 BMi2`pH (0, a∗1 ) i?2 7mM+iBQM m2 ?b  mMB[m2 +`BiB+H TQBMi i a = a0 = 0.1030896914.. r?B+? Bb  KtBKmK rBi? m2 (a0 ) = 0.0003095830..X JQ`2Qp2` m3 Bb BM+`2bBM; M/ m2 < m3 c U+V m2 → 0 M/ m3 → 0.0060680996.. r?2M a → a∗1 − c U/V m2 , m3 → ∞ r?2M a → a+ 1 rBi? lima→a+ m2 /m3 = 1− c 1 U2V BM i?2 BMi2`pH (a1 , a2 ) i?2 7mM+iBQM m2 Bb /2+`2bBM; M/ m3 ?b  mMB[m2 +`BiB+H TQBMi i a = a3 = a3 = 0.6096781095.. r?B+? Bb  KBMBKmK rBi? m3 (a3 ) = 4.7014182338..X JQ`2Qp2` m2 < m3 c U7V m2 → 0 M/ m3 → 4.4805332525.. r?2M a → a− 2c U;V m2 → 1.3553872894.. M/ m3 → 0 r?2M a → a∗2 c U?V BM i?2 BMi2`pH (a∗2 , ∞) #Qi? 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM; M/ m2 > m3 c UBV m2 , m3 → ∞ rBi? m2 /m3 → 1+ r?2M a → ∞X h?2Q`2K 9X 6Q` n = 4 r2 +M }M/ 7mM+iBQMb m2 = m2 (a) M/ m3 = m3 (a) i?i T`QpB/2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 12Ĝ#Q/v T`Q#H2K 7Q` HH a ∈ D = (0, a∗1 ) ∪ (a1 , a2 ) ∪ (a∗2 , ∞) r?2`2 a∗1 Bb i?2 }`bi x2`Q Q7 m2 - a1 Bb i?2 b2+QM/ x2`Q Q7 m2 - a2 Bb i?2 TQH2 Q7 m2 M/ a∗2 Bb i?2 H`;2bi x2`Q 7Q m3 X h?2 TT`QtBKi2 pHm2b Q7 a∗1 - a1 - a2 M/ a∗2 `2 ;Bp2M BM h#H2 RX JQ`2Qp2` i?2 7mM+iBQMb m2 M/ m3 biBb7v i?2 7QHHQrBM; T`QT2`iB2b Ub22 6B;X 9V, UV m2 , m3 → 0 rBi? m2 /m3 → 1− r?2M a → 0+ c U#V BM i?2 BMi2`pH (0, a∗1 ) i?2 7mM+iBQM m2 ?b  mMB[m2 +`BiB+H TQBMi i a = a0 = 0.2746698699.. r?B+? Bb  KtBKmK rBi? m2 (a0 ) = 0.0085881109.. M/ i?2 7mM+iBQM m3 Bb BM+`2bBM;X JQ`2@ Qp2` m2 < m3 c U+V m2 → 0 M/ m3 → 0.1238514421.. r?2M a → a∗1 − c U/V m2 → 0 M/ m3 → 2.3831374646.. r?2M a → a+ 1c U2V BM i?2 BMi2`pH (a1 , a2 ) #Qi? 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM; M/ m2 < m3 c U7V m2 , m3 → ∞ rBi? m2 /m3 → 1− r?2M a → a− 2c U;V m2 → 1.0670996767.. M/ m3 → 0 r?2M a → a∗2 + c U?V BM i?2 BMi2`pH (a∗2 , ∞) #Qi? 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM; M/ m2 > m3 c UBV m2 , m3 → ∞ rBi? m2 /m3 → 1+ r?2M a → ∞X e JX *P_"1_ L. *X oGGa h?2Q`2K 8X 6Q` n = 5 r2 +M }M/ 7mM+iBQMb m2 = m2 (a) M/ m3 = m3 (a) i?i T`QpB/2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 15Ĝ#Q/v T`Q#H2K 7Q` HH a ∈ D = (0, a∗1 ) ∪ (a∗2 , ∞) r?2`2 a∗1 Bb i?2 TQH2 Q7 m2 M/ a∗2 Bb i?2 H`;2bi x2`Q Q7 m3 X h?2 TT`QtBKi2 pHm2b Q7 a∗1 - a2 M/ a∗2 `2 ;Bp2M BM h#H2 RX JQ`2Qp2` i?2 7mM+iBQMb m2 M/ m3 biBb7v i?2 7QHHQrBM; T`QT2`iB2b Ub22 6B;X 8V, UV m2 , m3 → 0 rBi? m2 /m3 → 1− r?2M a → 0+ c U#V BM i?2 BMi2`pH (0, a∗1 ) #Qi? 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM; M/ m2 < m3 c U+V m2 , m3 → ∞ rBi? m2 /m3 → 1− r?2M a → a∗1 − c U/V m2 → 0.8426164718.. M/ m3 → 0 r?2M a → a∗2 + c U2V BM i?2 BMi2`pH (a∗2 , ∞) #Qi? 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM; M/ m2 > m3 c U7V m2 , m3 → ∞ rBi? m2 /m3 → 1+ r?2M a → ∞X h?2 7mM+iBQMb m2 = m2 (a) M/ m3 = m3 (a) BM h?2Q`2Kb R@8 `2 2tT`2bb2/ #v i?2 7Q`KmHb BM U9VX 7i2`  MmK2`B+H 2tTHQ`iBQM Q7 i?2 +b2b n = 6, . . . , 500 r2 b22 i?i i?2 Tii2`M Q#b2`p2/ 7Q` n = 6, ..., 500 Bb i?2 bK2 b i?2 QM2 Q#b2`p2/ 7Q` n = 5X aQ r2 KF2 i?2 7QHHQrBM; +QMD2+im`2X *QMD2+im`2 eX 6Q` HH n ≥ 5 i?2`2 2tBbi 7mM+iBQMb m2 = m2 (a) M/ m3 = m3 (a) i?i T`QpB/2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3n@#Q/v T`Q#H2K 7Q` HH a BM i?2 b2i D ⊂ (0, a∗1 ) ∪ (a∗2 , ∞) r?2`2 a∗1 Bb i?2 TQH2 Q7 m2 M/ a∗2 Bb i?2 H`;2bi x2`Q Q7 m3 X JQ`2Qp2` i?2 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM; BM D M/ i?2v biBb7v i?i m2 < m3 r?2M a ⊂ (0, a∗1 )- m2 > m3 r?2M a ⊂ (a∗2 , ∞)- m2 , m3 → 0 rBi? m2 /m3 → 1 r?2M a → 0+ c m2 , m3 → ∞ rBi? m2 /m3 → 1 r?2M a → a∗1 − c m3 → 0 r?2M a → a∗2 + c M/ m2 , m3 → ∞ rBi? m2 /m3 → 1 r?2M a → ∞X AM b?Q`i- r?2M n = 2 r2 ?p2 T`Qp2/ MHviB+HHv Ub22 h?2Q`2K RUVV i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb 7Q` HH a BM i?2 BMi2`pH (a∗ , ∞) r?2`2 a∗ > 1 Bb i?2 H`;2bi x2`Q Q7 m3 X lbBM; +QKTmi2` bbBbi2/ `2bmHib r2 T`Qp2 i?2 2tBbi2M+2 Q7 M //BiBQMH BMi2`pH (a1 , a2 ) T`QpB/BM; +2Mi`H +QM};m`iBQMb- r?2`2 a1 < 1 Bb i?2 TQH2 Q7 m2 M/ a2 < 1 Bb  x2`Q Q7 m2 Ub22 h?2Q`2K kVX q?2M n ≥ 3 r2 ?p2 T`Qp2/ MHviB+HHv Ub22 h?2Q`2K RU#VV i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb 7Q` a ∈ (0, a∗1 ) ∪ (a∗2 , ∞) r?2`2 0 < a∗1 < 1 Bb i?2 KBMBKmK #2ir22M i?2 }`bi TQbBiBp2 x2`Q M/ i?2 TQH2 Q7 m2 M/ a∗2 > 1 Bb i?2 H`;2bi x2`Q Q7 m3 X 6Q` n = 3, 4 mbBM; +QKTmi2` bbBbi2/ `2bmHib r2 ?p2 T`Qp2/ i?2 2tBbi2M+2 Q7 M //BiBQMH BMi2`pH (a1 , a2 ) rBi? a1 , a2 < 1 T`QpB/BM; +2Mi`H +QM};m`iBQMbX JQ`2Qp2` 7Q` n = 3 r2 ?p2 T`Qp2/ i?i a∗1 Bb i?2 }`bi TQbBiBp2 x2`Q Q7 m2 - a1 Bb i?2 TQH2 Q7 m2 M/ a2 Bb i?2 b2+QM/ x2`Q Q7 m2 Ub22 h?2Q`2K jVX q?2M n = 4 r2 ?p2 T`Qp2/ i?i a∗1 Bb i?2 }`bi x2`Q Q7 m2 - a1 Bb i?2 b2+QM/ x2`Q Q7 m2 M/ a2 Bb i?2 TQH2 Q7 m2 Ub22 h?2Q`2K 9VX 6Q` n = 5 r2 ?p2 T`Qp2/ mbBM; +QKTmi2` bbBbi2/ `2bmHib i?i i?2 QMHv b2i Q7 pHm2b Q7 a T`QpB/BM; +2Mi`H +QM};m`iBQMb Bb (0, a∗1 ) ∪ (a∗2 , ∞)c i?i Bb- i?2 b2i ;Bp2M BM h?2Q`2K RU#V Ub22 h?2Q`2K 8VX 6BMHHv 7i2`  MmK2`B+H 2tTHQ`iBQMb Q7 i?2 +b2b n = 6, . . . , 500 r2 +QM+Hm/2 i?i 7Q` HH n = 6, . . . , 500 i?i i?2 QMHv b2i Q7 pHm2b Q7 a T`QpB/BM; +2Mi`H +QM};m`iBQMb Bb i?2 QM2 ;Bp2M BM h?2Q`2K RU#VX q2 Kmbi `2K`F i?i }t2/  pHm2 Q7 n i?2`2 `2 MQ +2Mi`H +QM};m`iBQMb Q7 i?2 `2;mH` #B+B`+mH` 3nĜ#Q/v T`Q#H2K rBi? a → 1X q2 ?p2 Q#b2`p2/ i?i b n BM+`2b2b i?2 pHm2 a∗1 BM+`2b2b M/ i?2 pHm2 Q7 a∗2 /2+`2b2b- #mi r2 /Q MQi FMQr r?2i?2` a∗1 → 1− M/ a∗2 → 1+ r?2M n → ∞ Q` QM i?2 +QMi``v i?2v +QmH/ i2M/ iQ  pHm2 /Bz2`2Mi 7`QK QM2X h?Bb [m2biBQM `2KBMb b M QT2M T`Q#H2KX PM i?2 Qi?2` ?M/- b n BM+`2b2b i?2 pHm2b Q7 m2 M/ m3 `2 ;2iiBM; +HQb2`X q2 FMQr 7`QK `2bmHib BM (R3) i?i m2 6= m3 #2+mb2 r2 ?p2 irQ +QM+2Mi`B+ `2;mH` nĜ;QMb rBi? /Bz2`2Mi MmK#2` Q7 p2`iB+2b #mi Bi b22Kb i?i B7 n → ∞ i?2M m2 → m3 X kXkX amKK`v Q7 i?2 `2bmHib +QM+2`MBM; b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2KX h?2 KBM MHviB+ `2bmHib 7Q` i?2 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`@ iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K `2 bmKK`Bx2/ BM i?2 7QHHQrBM; i?2Q`2KX h?2Q`2K dX h?2`2 2tBbi  7mM+iBQM m = m(a, β) i?i T`QpB/2b b2KB`2;mH` #B+B`+mH` +2Mi`H +QM@ };m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K 7Q` bQK2 pHm2b Q7 a > 0 M/ β ∈ (π/2n, π/n)X AM T`iB+mH`- UV A7 n ≥ 3- i?2M 7Q` 2+? β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 π/n i?2`2 2tBbi i H2bi irQ +2Mi`H +QM};m`iBQMb- QM2 rBi? 0 < a < 1 M/ QM2 rBi? a > 1X JQ`2Qp2`- B7 n = 2- i?2M 7Q` 2+? β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 π/n i?2`2 2tBbib i H2bi QM2 +2Mi`H +QM};m`iBQM rBi? a > 1X h?2 pHm2b Q7 m i i?2b2 +2Mi`H +QM};m`iBQMb biBb7v i?i m → 0 b β → π/nX PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa d U#V A7 n ≥ 3- i?2M 7Q` 2+? β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 β = π/2n i?2`2 2tBbi i H2bi 7Qm` +2Mi`H +QM};m`iBQMb- QM2 rBi? a bm{+B2MiHv +HQb2 iQ i?2 Q`B;BM- QM2 rBi? 0 < a < 1 MQi M2+2bb`BHv +HQb2 iQ i?2 Q`B;BM- QM2 rBi? a bm{+B2MiHv H`;2 M/ QM2 rBi? a > 1 MQi M2+2bb`BHv H`;2X JQ`2Qp2`- B7 n = 2- i?2M 7Q` 2+? β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 β = π/2n i?2`2 2tBbib i H2bi i?`22 +2Mi`H +QM};m`iBQMb- QM2 rBi? a bm{+B2MiHv H`;2- QM2 rBi? a > 1 MQi M2+2bb`BHv H`;2 M/ QM2 rBi? 0 < a < 1X h?2 pHm2b Q7 m i i?2 +2Mi`H +QM};m`iBQMb rBi? 0 < a < 1 MQi +HQb2 iQ i?2 Q`B;BM M/ rBi? a > 1 MQi #2BM; H`;2 biBb7v i?i m → ∞ b β → π/2nX h?2 7mM+iBQM m(a, β) Bb ;Bp2M BM UReVX 7i2`  MmK2`B+H bim/v Q7 i?2 #2?pBQ` Q7 i?2 7KBHB2b Q7 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`@ iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K 7Q` n = 2, 3, 4 r2 ;2i i?2 7QHHQrBM; MmK2`B+H `2bmHibX _2bmHi R q?2M n = 2- r2 +M }M/ +QMiBMmQmb 7mM+iBQMb α1 (β) /2}M2/ 7Q` β ∈ (π/4, π/2)- M/ α2 (β) M/ α3 (β) /2}M2/ 7Q` β ∈ (π/4, b∗ ] rBi? b∗ = 0.9195936184.. M/  7mM+iBQM m = m(a, β) bm+? i?i i?2 7QHHQrBM; bii2K2Mib ?QH/ 7Q` i?2 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 6Ĝ#Q/v T`Q#H2K Ub22 6B;X NVX UV A7 β ∈ (π/4, b∗ )- i?2M m = m(β, a) T`QpB/2b i?`22 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb- QM2 rBi? a = α1 (β)- QM2 rBi? a = α2 (β) M/ QM2 rBi? a = α3 (β)X U#V A7 β = b∗ - i?2M m = m(β, a) T`QpB/2b irQ +2Mi`H +QM};m`iBQMb- QM2 rBi? a = α1 (b∗ ) M/ QM2 rBi? a = α2 (b∗ ) = α3 (b∗ ) Ui?2 7KBHB2b rBi? a = α2 (β) M/ a = α3 (β) +QBM+B/2 i i?Bb TQBMiVX U+V A7 β ∈ (b∗ , π/2)- i?2M m = m(β, a) T`QpB/2b  mMB[m2 7KBHv Q7 +2Mi`H +QM};m`iBQMb rBi? a = α1 (β)X JQ`2Qp2` m(α1 (β), β) → 0 r?2M β → π/2− c M/ m(α1 (β), β), m(α2 (β), β), m(α3 (β), β) → ∞ r?2M β → π/4+ X _2bmHi k q?2M n = 3- r2 +M }M/ +QMiBMmQmb 7mM+iBQMb α1 (β) M/ α2 (β) /2}M2/ 7Q` β ∈ (π/6, b∗ ]- 7mM+iBQMb α1 (β) M/ α2 (β) /2}M2/ 7Q` β ∈ (π/6, π/3)- rBi? b∗ = 0.7119233940.. M/  7mM+iBQM m = m(a, β) bm+? i?i i?2 7QHHQrBM; bii2K2Mib ?QH/ 7Q` i?2 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 9Ĝ#Q/v T`Q#H2K Ub22 6B;X N#VX UV A7 β ∈ (π/6, b∗ )- i?2M m = m(β, a) T`QpB/2b 7Qm` 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb- QM2 rBi? a = α1 (β)- QM2 rBi? a = α2 (β)- QM2 rBi? a = α3 (β)- M/ QM2 rBi? a = α4 (β)X U#V A7 β = b∗ - i?2M m = m(β, a) T`QpB/2b i?`22 +2Mi`H +QM};m`iBQMb- QM2 rBi? a = α1 (b∗ ) = α2 (b∗ ) Ui?2 7KBHB2b rBi? a = α1 (β) M/ a = α2 (β) +QBM+B/2 i i?Bb TQBMiV- QM2 rBi? α3 (b∗ ) M/ QM2 rBi? a = α4 (b∗ )X U+V A7 β ∈ (b∗ , π/3)- i?2M m = m(β, a) T`QpB/2b irQ 7KBHv Q7 +2Mi`H +QM};m`iBQMb- QM2 rBi? a = α3 (β) M/ QM2 rBi? a = α4 (β)X JQ`2Qp2` m(α3 (β), β), m(α4 (β), β) → 0 r?2M β → π/3− c m(α1 (β), β), m(α2 (β), β), m(α3 (β), β) → ∞ M/ m(α4 (β), β) → 0 r?2M β → π/6+ X _2bmHi j q?2M n = 4- r2 +M }M/ +QMiBMmQmb 7mM+iBQMb α1 (β) M/ α4 (β) /2}M2/ 7Q` β ∈ (π/8, b∗ ]- 7mM+iBQMb α2 (β) M/ α3 (β) /2}M2/ 7Q` β ∈ (π/8, π/4)- rBi? b∗ = 0.4665964724.. M/  7mM+iBQM m = m(a, β) bm+? i?i i?2 7QHHQrBM; bii2K2Mib ?QH/ 7Q` i?2 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 12Ĝ#Q/v T`Q#H2K Ub22 6B;X N+VX UV A7 β ∈ (π/8, b∗ )- i?2M m = m(β, a) T`QpB/2b 7Qm` 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb- QM2 rBi? a = α1 (β)- QM2 rBi? a = α2 (β)- QM2 rBi? a = α3 (β)- M/ QM2 rBi? a = α4 (β)X U#V A7 β = b∗ - i?2M m = m(β, a) T`QpB/2b i?`22 +2Mi`H +QM};m`iBQMb- QM2 rBi? a = α1 (b∗ ) = α4 (b∗ ) Ui?2 7KBHB2b rBi? a = α1 (β) M/ a = α4 (β) +QBM+B/2 i i?Bb TQBMiV- QM2 rBi? α2 (b∗ ) M/ QM2 rBi? a = α3 (b∗ )X U+V A7 β ∈ (b∗ , π/4)- i?2M m = m(β, a) T`QpB/2b irQ 7KBHv Q7 +2Mi`H +QM};m`iBQMb- QM2 rBi? a = α2 (β) M/ QM2 rBi? a = α3 (β)X JQ`2Qp2` m(α3 (β), β), m(α2 (β), β) → 0 r?2M β → π/4− c M/ m(α1 (β), β), m(α2 (β), β), m(α3 (β), β) → ∞ M/ m(α4 (β), β) → 0 r?2M β → π/8+ X 3 JX *P_"1_ L. *X oGGa q2 ?p2 HbQ bim/B2/ MmK2`B+HHv i?2 7KBHB2b Q7 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v 7Q` n = 5, . . . , 100 M/ r2 ?p2 Q#b2`p2/ i?i i?2 #2?pBQ` Bb i?2 bK2 b i?2 QM2 7Q` n = 4X aQ r2 bii2 i?2 7QHHQrBM; +QMD2+im`2X *QMD2+im`2 3X 6Q` HH n ≥ 4 r2 +M }M/  pHm2 β = b∗ M/ 7mM+iBQMb αi (β) 7Q` i = 1, 2, 3, 4 bm+? i?i UV 7Q` β ∈ (π/2n, b∗ ) i?2`2 2tBbi 2t+iHv 7Qm` /Bz2`2Mi 7KBHB2b Q7 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb- QM2 2KMiBM; 7`QK  TQBMi (a1 , π/2n) rBi? 0 < a1 < 1 Ui?2 7KBHv rBi? a = α1 (β)V- QM2 2KMiBM; 7`QK  TQBMi (a2 , π/2n) rBi? a2 > 1 Ui?2 7KBHv rBi? a = α1 (β)V- QM2 2KMiBM; 7`QK i?2 TQBMi (∞, 0) Ui?2 7KBHv rBi? a = α3 (β)V- M/ QM2 2KMiBM; 7`QK i?2 TQBMi (0, π/2n) Uα4 (β)Vc U#V 7Q` β = b∗ i?2`2 2tBbi 2t+iHv i?`22 /Bz2`2Mi +2Mi`H +QM};m`iBQMb- i?2 QM2 rBi? a = a1 (b∗ ) = a4 (b∗ )- i?2 QM2 rBi? a = a2 (b∗ ) M/ i?2 QM2 rBi? a = a3 (b∗ )c U+V 7Q` β ∈ (b∗ , π/n) i?2`2 2tBbi 2t+iHv irQ /Bz2`2Mi +2Mi`H +QM};m`iBQMb- QM2 i2M/BM; iQ  TQBMi (a∗1 , π/n) rBi? 0 < a∗1 < 1 r?2M β → π/n− Ui?2 7KBHv rBi? a = a2 (β)V M/ QM2 i2M/BM; iQ  TQBMi (a∗2 , π/n) rBi? a∗2 > 1 r?2M β → π/n− Ui?2 7KBHv rBi? a = a3 (β)VX JQ`2Qp2` i?2 Kbb2b bbQ+Bi2/ iQ i?2b2 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb m = m(a, β) biBb7v i?i m(α3 (β), β), m(α2 (β), β) → 0 r?2M β → π/2n− c M/ m(α1 (β), β), m(α2 (β), β), m(α3 (β), β) → ∞ M/ m(α3 (β), β) → 0 r?2M β → π/n+ X q2 MQi2 i?i i?2 `2HiBp2 H2M;i? Q7 i?2 BMi2`pH Q7 pHm2b Q7 i?2 T`K2i2` β T`QpB/BM; 7Qm` /Bz2`2Mi 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb ;2ib bKHH2` b n BM+`2b2bX qBi? `2HiBp2 H2M;?i r2 K2M `2HBip2 H2M;i rBi? `2bT2+i iQ i?2 iQiH H2M;i? Q7 i?2 BMi2`pH (π/2n, π/n)X jX _2;mH` #B+B`+mH` +2Mi`H +QM7B;m`iBQMb Q7 i?2 3nė#Q/v T`Q#H2K jXRX h?2 2[miBQMbX b r2 ?p2 b22M BM i?2 BMi`Q/m+iBQM i?2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`@ iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K +M #2 i?Qm;?i b i?2 HBKBi +b2 Q7  (3, n)Ĝ+`QrM Ub22 (k)V r?2`2 irQ irBbi2/ n@;QMb `2 BMb+`B#2/ BM  +B`+H2 Q7 i?2 bK2 `/BmbX 6`QK 2[miBQM UNV BM (k) i?2 2[miBQMb 7Q` +2Mi`H +QM};m`iBQMb Q7 Mv (3, n)Ĝ+`QrM rBi? n Kbb2b 2[mH iQ m1 = 1 i i?2 p2`iB+2b Q7  }`bi n@;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb a1 = 1 M/ irBbi2/ M M;H2 ̟1 = 0- n Kbb2b 2[mH iQ m2 i i?2 p2`iB+2b Q7  b2+QM/ n@;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb a2 M/ irBbi2/ M M;H2 ̟2 = 0- M/ n Kbb2b 2[mH iQ m3 i i?2 p2`iB+2b Q7  i?B`/ n@;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb a3 M/ irBbi2/ M M;H2 ̟3 = π/n `2     Sn C21 − Sn a2 + − a 2 C 12 m 2 + C 23 − a 2 C 13 m3 = 0, a22     UkV Sn C31 − Sn a3 + C32 − a3 C12 m2 + − a C 3 13 m 3 = 0, a23 r?2`2 n−1 X n−1 X 2πj  n−1 X 1 1 − cos n 1 πj  Sn = =  3/2 =  1/2 , 4 sin 2πj  2πj  j=1 n j=1 2 − 2 cos n j=1 2 2 − 2 cos n n X 2πj  ak − aℓ cos ̟k − ̟ℓ + n Ckℓ = Ckℓ (ak , aℓ ) =  3/2 , 2πj  j=1 a2k + a2ℓ − 2ak aℓ cos ̟k − ̟ℓ + n 7Q` k 6= ℓX h?mb i?2 2[miBQMb Q7 i?2 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K `2 ;Bp2M #v UkV rBi? a2 = a3 = a M/ i?2v +M #2 r`Bii2M b i?2 HBM2` bvbi2K BM i?2 p`B#H2b m2 - m3      a11 a12 m2 b1 = , UjV a21 a22 m3 b2 PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa N r?2`2 K1 K5 a11 = − K2 , a12 = − K3 , b 1 = K1 − K4 , a3 a3 K5 K1 a21 = 3 − K2 , a22 = 3 − K3 , b 2 = K1 − K6 , a a M/ K 1 = Sn , n X 2π j  1 − a cos n K2 = K2 (a) = C12 (1, a) =  3/2 , 2π j  j=1 1+ a2 − 2a cos n n 2π j  X 1 − a cos + π n n K3 = K3 (a) = C13 (1, a) =  3/2 , 2π j π j=1 1+ a2 − 2a cos n + n n 2π j  C21 (a, 1) X 1 − 1/a cos n K4 = K4 (a) = =  3/2 , a 2π j  j=1 1+ a2 − 2a cos n K5 = a2 C23 (a, a) = a2 C32 (a, a) n  n X 1 − cos 2πn j + πn X 1 =    3/2 =  1/2 , j=1 2 − 2 cos 2π j π j=1 2 2 − 2 cos 2π j π n + n n + n n  C31 (a, 1) X 1 − 1/a cos 2πn j + πn K6 = K6 (a) = =  3/2 . a j=1 1 + a2 − 2a cos 2π j + π n n aQHpBM; bvbi2K UjV r2 ;2i i?2 bQHmiBQM a22 b1 − a12 b2 a3 mN,2 m2 = m2 (a) = = , a11 a22 − a12 a21 mD U9V a11 b2 − a21 b1 a3 mN,3 m3 = m3 (a) = = a11 a22 − a12 a21 mD r?2`2 mN,2 = K12 − K1 K4 (a) − K1 K5 + K5 K6 (a) − a3 K3 (a)(K6 (a) − K4 (a)), mN,3 = K12 − K1 K5 + K4 (a)K5 − K1 K6 (a) + a3 K2 (a)(K6 (a) − K4 (a)), mD = (K5 − K1 )∆(a), M/ ∆(a) = −K1 − K5 + a3 (K2 (a) + K3 (a)). h?2 bQHmiBQM U9V Bb /2}M2/ r?2M i?2 /2MQKBMiQ` mD Bb /Bz2`2Mi 7`QK x2`QX "v G2KK j U+V BM TT2M/Bt R K5 − K1 6= 0 M/ #v G2KK 9 BM TT2M/Bt R i?2 7mM+iBQM ∆(a) ?b  mMB[m2 x2`Q M/ Bi #2HQM;b iQ i?2 BMi2`pH (0, 1)X aQ i?2 b2i r?2`2 mD Bb /Bz2`2Mi 7`QK x2`Q Bb MQi 2KTivX aQ BM r?i 7QHHQrb r2 rBHH bbmK2 i?i mD 6= 0X AM 7+i- i?Bb MQi b22Kb iQ #2 `2bi`B+iBp2 #2+mb2- b r2 rBHH b22 BM a2+iBQM jXj- r2 ?p2 MmK2`B+H 2pB/2M+2b i?i i?2`2 `2 MQ bQHmiBQMb Q7 UjV rBi? mD = 0 UmD - mN,2 - M/ mN,3 `2 MQi bBKmHiM2QmbHv yVX h?2b2 2tT`2bbBQMb ?p2 #22M Q#iBM2/ BM (Rj) #v mbBM;  /Bz2`2Mi TT`Q+?X LQi2 i?i a3 (K6 (a) − K4 (a)) m3 = m2 + . U8V K5 − K1 AM b?Q`i-  +QM};m`iBQM Q7 i?2 `2;mH` #B+B`+mH` 3nĜ#Q/v T`Q#H2K Bb +2Mi`H B7 m2 M/ m3 `2 ;Bp2M #v U9V M/ a Bb bQ i?i m2 , m3 > 0X Ry JX *P_"1_ L. *X oGGa jXkX S`QQ7 Q7 h?2Q`2K RX h?2 +2Mi`H +QM};m`iBQMb Q7 i?2 `2;mH` #B+B`+mH` 3nĜ#Q/v T`Q#H2K `2 ;Bp2M #v i?2 bQHmiBQMb Q7 UjV rBi? m2 , m3 > 0X G2i m2 = m2 (a) M/ m3 = m3 (a) #2 i?2 bQHmiBQMb Q7 UjV ;Bp2M BM U9VX q2 M22/ i?2 7QHHQrBM; mtBHB`v `2bmHi i?i ;Bp2 bQK2 T`QT2`iB2b Q7 m2 M/ m3 X S`QTQbBiBQM NX G2i m2 = m2 (a) M/ m3 = m3 (a) #2 i?2 7mM+iBQMb /2}M2/ BM U9VX h?2M i?2 7QHHQrBM; bii2K2Mib ?QH/ 7Q` n ≥ 2X UV m2 → 0+ r?2M a → 0+ 7Q` n ≥ 3 M/ m2 → 0− r?2M a → 0+ 7Q` n = 2c U#V m3 → 0+ r?2M a → 0+ c U+V m3 → −∞ r?2M a → 1+ c U/V m2 → ∞ M/ m3 → ∞ r?2M a → ∞c U2V m2 > m3 r?2M a > 1 M/ m2 < m3 r?2M a ∈ (0, 1)X h?2 T`QQ7 Q7 S`QTQbBiBQM N Bb ;Bp2M BM TT2M/Bt RX 6`QK S`QTQbBiBQM NUV M/ U#V- 7Q` 2+? n ≥ 3 i?2`2 2tBbi  bm{+B2MiHv bKHH BMi2`pH I1 = (0, a∗1 ) bm+? i?i m2 , m3 > 0X aBM+2 m2 < m3 r?2M 0 < a < 1 Ub22 S`QTQbBiBQM NU2VV i?Bb pHm2 a∗1 +Q``2bTQM/b 2Bi?2` iQ  x2`Q Q7 m2 Q` iQ  TQBMi r?2`2 i?2 /2MQKBMiQ` mD pMBb?2bX AM a2+iBQM jXj r2 T`Qp2- #v mbBM;  +QKTmi2` bbBbi2/ T`QQ7- i?i a∗1 Bb  x2`Q Q7 m2 r?2M n = 3, 4 M/ Bi Bb i?2 x2`Q Q7 mD r?2M n = 5X JQ`2Qp2` r2 ;Bp2 bi`QM; MmK2`B+H 2pB/2M+2b i?i a∗1 Bb i?2 x2`Q Q7 mD 7Q` n > 5X q2 +QMiBMm2 rBi? i?2 T`QQ7 Q7 h?2Q`2K RX 6`QK S`QTQbBiBQM NU/V 7Q` 2+? n ≥ 2 i?2`2 2tBbib  bm{+B2MiHv H`;2 pHm2 a∗2 > 1 bm+? i?i m2 , m3 > 0 7Q` HH a ∈ I2 = (a∗2 , ∞)X lbBM; S`QTQbB@ iBQM NU+V M/ U/V r2 ;2i i?i m3 ?b i H2bi  x2`Q 7Q` a > 1 M/ 7`QK G2KK 9 BM TT2M/Bt R r2 Q#iBM i?i mD ?b MQ x2`Qb 7Q` a > 1X aBM+2 m2 > m3 r?2M a > 1 Ub22 ;BM S`QTQbBiBQM NV- M/ +QMM2+iBM; HH #Qp2 +QKK2Mib iQ;2i?2` r2 +M bbm`2 i?i a∗2 Bb i?2 H`;2bi x2`Q Q7 m3 X AM a2+iBQM jXj mbBM;  +QKTmi2` bbBbi2/ T`QQ7 r2 T`Qp2- 7Q` n = 2, 3, 4, 5- i?i m3 ?b  mMB[m2 x2`Q- a∗2 - 7Q` a > 1 M/ r2 ;Bp2 bi`QM; MmK2`B+H 2pB/2M+2b i?i i?Bb HbQ ?TT2Mb 7Q` n > 5X h?2b2 `;mK2Mib- iQ;2i?2` rBi? S`QTQbBiBQM N- +QKTH2i2 i?2 T`QQ7 Q7 h?2Q`2K RX jXjX S`iB+mH` +b2b Q7 `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2KX jXjXRX *b2 n = 2X *QKTmiBM; i?2 pHm2b Q7 Ki rBi? i = 1, . . . , 6 7Q` n = 2 r2 ;2i 1 2 1 K1 = , K3 (a) = K6 (a) = , K5 = √ , 4 h22 2 1−a 1 1−a 1 UeV K2 (a) = + 2 , K4 (a) = − + , h12 (a + 1) a h12 a(a + 1)2 r?2`2 h12 = ((1 − a)2 )3/2 M/ h22 = (1 + a2 )3/2 X h?mb r?2M n = 2- i?2 bQHmiBQMb m2 = m2 (a) M/ m3 = m3 (a) Q7 bvbi2K UjV rBi? mD 6= 0 `2 ;Bp2M #v U9V rBi? Ki ;Bp2M #v UeVX LQi2 i?i BM Q`/2` i?i i?2b2 bQHmiBQMb T`QpB/2 +2Mi`H +QM};m`iBQMb #Qi? m2 M/ m3 ?p2 iQ #2 TQbBiBp2X L2ti- r2 }M/ i?2 b2i Q7 pHm2b Q7 a biBb7vBM; i?2b2 +QM/BiBQMbX aBM+2 a > 0- i?2 TQbbB#H2 +?M;2b Q7 bB;M Q7 m2 M/ m3 `2 ;Bp2M #v i?2 x2`Q2b Q7 mN,2 M/ mD - M/ mN,3 M/ mD `2bT2+iBp2Hv- b22 U9VX q2 bi`i +QKTmiBM; i?2 x2`Q2b Q7 mD X 6B`bi r2 i`Mb7Q`K 2[miBQM mD = 0 BMiQ  TQHvMQKBH 2[miBQM ?pBM; HH i?2 bQHmiBQMb Q7 2[miBQM mD = 0 M/ T`Q##Hv M2r QM2b BM i?2 7QHHQrBM; rvX .`QTTBM; i?2 /2MQKBMiQ`b r2 ;2i i?2 2[miBQM   √   g = (a + 1)2 4a3 ((a − 1)h22 − 2h12 ) + 1 + 2 2 h12 h22 − 4a3 h12 h22 = 0. AM Q`/2` iQ /`QT i?2 b[m`2 `QQib r2 +QMbB/2` 2[miBQM g = 0 b  TQHvMQKBH 2[miBQM BM i?2 p`B#H2b a- h12 M/ h22 X h?2M i?2 x2`Q2b Q7 mD +M #2 i?Qm;?i b bQHmiBQMb Q7 i?2 TQHvMQKBH bvbi2K g = 0- e1 = 0 M/ e2 = 0 rBi? e1 = h212 − ((1 − a)2 )3 , e2 = h222 − (1 + a2 )3 . q2 2HBKBMi2 i?2 p`B#H2 h12 #v K2Mb Q7 i?2 `2bmHiMi R1 = _2b[g, e1 , h12 ] M/ i?2 p`B#H2 h22 #v K2Mb Q7 i?2 `2bmHiMi R2 = _2b[R1 , e2 , h22 ]X h?2 `2bmHiBM; √ TQHvMQKBH Bb  TQHvMQKBH BM i?2 p`B#H2 a rBi? B``iBQMH +Q2{+B2Mib Ui?2 +Q2{+B2Mib /2T2M/ QM 2V i?i ?b HH i?2 x2`Q2b Q7 mD M/ PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa RR √ T`Q##Hv M2r QM2bX hQ pQB/ B``iBQMH +Q2{+B2Mib- r2 BMi`Q/m+2  M2r p`B#H2 h32 = 2 M/  M2r 2[miBQM e3 = h232 − 2X q2 2HBKBMi2 i?2 p`B#H2 h32 #v K2Mb Q7 i?2 `2bmHiMi R3 = _2b[R2 , e3 , h32 ] Q#iBMBM; BM i?Bb rv  TQHvMQKBH rBi? BMi2;2` +Q2{+B2Mib ?pBM; HH i?2 x2`Q2b Q7 mD M/ T`Q##Hv M2r QM2bX h?2 Q#iBM2/ TQHvMQKBH Bb (a − 1)8 P60 (a) r?2`2 P60 (a) Bb ;Bp2M #v (45041a28 + 471582a26 + 1263627a24 + 900044a22 + 1046137a20 + 172866a18 + 233227a16 +47784a14 + 27723a12 − 18750a10 + 6329a8 − 564a6 − 693a4 − 98a2 + 49)(4096a32 −2048a31 + 32128a30 − 12064a29 + 176561a28 − 42784a27 + 495902a26 − 50496a25 +1006987a24 + 27328a23 + 759372a22 + 93472a21 + 1118393a20 + 64800a19 + 259394a18 +11904a17 + 310283a16 − 31104a15 + 38184a14 − 41696a13 + 1675a12 − 15072a11 −10814a10 − 1344a9 + 3641a8 − 1344a7 − 1204a6 + 224a5 − 245a4 + 224a3 − 98a2 + 49) 6`QK MQr QM r2 rBHH /2MQi2 #v Pn (a)  TQHvMQKBH Q7 /2;`22 n BM i?2 p`B#H2 aX TTHvBM; aim`KǶb h?2Q`2K r2 ;2i i?i i?2 TQHvMQKBH 2[miBQM P60 (a) = 0 ?b 2t+iHv 7Qm` `2H bQHmiBQMb rBi? a > 0X q2 bQHp2 MmK2`B+HHv i?2 2[miBQM P60 (a) = 0 M/ r2 7QmM/ i?2 bQHmiBQMb a = a1 = 0.4656636054..- a = a2 = d2 = 0.5317860740..- a = a3 = 0.5390030006..- a = a4 = 0.5824356327..X "v bm#biBimiBM; i?2b2 bQHmiBQMb BMiQ i?2 BMBiBH 2[miBQM mD = 0 r2 b22 i?i QMHv i?2 bQHmiBQM a = d2 +M #2  bQHmiBQM Q7 i?2 BMBiBH 2[miBQMX h?Bb +M #2 T`Qp2/ BM  KQ`2 `B;Q`Qmb rv #v mbBM; BMi2`pH` `BiK2i?B+b Ub22 (R8)VX q2 ?p2 mb2/ Ji?2KiB+Ƕb +T#BHBiv Q7 QT2`iBM; QM BMi2`pH Q#D2+ib iQ ;2i M BMi2`pH 2M+HQbm`2 Q7 i?2 7mM+iBQM mD BM  bm{+B2MiHv bKHH BMi2`pH +QMiBMBM; i?2 TQbbB#H2 bQHmiBQMb Q7 i?2 2[miBQM mD = 0X q2 bi`i T`QpBM; i?i a1 ∈ a1 = [4656636054/1010 , 4656636055/1010 ] +MMQi #2  bQHmiBQM Q7 i?2 BMBiBH 2[miBQMX LQiB+2 i?i bBM+2 a1 < 1 r2 ?p2 i?i ((1 − a1 )2 )3/2 = (1 − a1 )3 X lbBM; BMi2`pH` `Bi?K2iB+b r2 ;2i 1 + a21 ∈ [h21 , h22 ] = [ 30421064834853172729 25·1018 , 4867370373949038521 4·1018 ], (1 − a1 )3 ∈ h1 = [ 1220490080420246024105505069 8·1027 , 19070157517273170971924514317 125·102 7 ]. JQ`2Qp2` r2 +M b22 i?i 3/2 3/2 h21 ∈ [ 13423062258 1010 , 13423062259 1010 ], h22 ∈ [ 13423062259 1010 , 13423062260 1010 ], √ bQ (1+a21 )3/2 ∈ h2 = [13423062258/1010 , 13423062260/1010 ]- KQ`2Qp2` 2 ∈ h3 = [14142135623/1010 , 14142135624/10 10 ]X "v bm#biBimiBM; BMiQ i?2 2tT`2bbBQM Q7 m i?2 pHm2b Q7 a- ((1−a)2 )3/2 - (1+a2 )3/2 √ D M/ 2 #v a1 - h1 - h2 M/ h3 `2bT2+iBp2Hv M/ /QBM; BMi2`pH` `Bi?K2iB+b ;BM r2 ;2i i?i mD ∈ [−0.1855803641, −0.1855803639]- bQ a1 /Q2b MQi biBb7v 2[miBQM mD = 0X _2T2iBM; i?Bb T`Q+2/m`2 7Q` i?2 `2KBMBM; bQHmiBQMb r2 ;2i i?i mD ∈ [−0.0000000001, 0.0000000004] 7Q` a2 ∈ [5317860740/1010 , 5317860741/1010 ]- mD ∈ [0.0271889606, 0.0271889613] 7Q` a3 ∈ [5390030006/1010 , 5390030007/1010 ] M/ mD ∈ [0.2330977572, 0.2330977582] 7Q` a4 ∈ [5824356327/1010 , 5824356328/1010 ]X h?Bb T`Qp2b i?i i?2 mMB[m2 bQHmiBQM Q7 P60 (a) = 0 T`QpB/BM;  bQHmiBQM Q7 i?2 BMBiBH 2[miBQMb Bb i?2 bQHmiBQM a = d2 X lbBM; i?2 bK2 T`Q+2/m`2 i?2 2[miBQM mN,2 = 0 +M #2 i`Mb7Q`K2/ BMiQ  TQHvMQKBH 2[miBQM Q7 i?2 7Q`K (a − 1)8 a4 P84 (a) = 0 r?2`2 i?2 TQHvMQKBH P84 (a) ?b 2t+iHv 3 `2H `QQib rBi? a > 0X KQM; i?2b2 bQHmiBQMb QMHv a = a12 = 0.6161447847.. M/ a = a22 = 2.8235222602.. `2 bQHmiBQMb Q7 i?2 BMBiBH 2[miBQM mN,2 = 0X .QBM; Bi ;BM r2 ;2i i?i mN,3 = 0 +M #2 i`Mb7Q`K2/ BMiQ  TQHvMQKBH 2[miBQM Q7 i?2 7Q`K (a−1)16 a4 P92 (a) = 0 r?2`2 i?2 TQHvMQKBH P92 (a) ?b 2t+iHv 3 `2H `QQib rBi? a > 0X KQM; i?2b2 bQHmiBQMb QMHv a = b12 = 0.5161941182.. M/ a = b22 = 3.5282322274.. `2 bQHmiBQMb Q7 i?2 BMBiBH 2[miBQM mN,3 = 0X LQi2 i?i mN,2 - mN,3 M/ mD `2 MQi bBKmHiM2QmbHv x2`QX h?Bb K2Mb i?i r?2M n = 2 i?2`2 +QmH/ MQi #2 bQHmiBQMb Q7 bvbi2K UjV rBi? mD = 0X 6BMHHv r2 MHvx2 i?2 bB;Mb Q7 m2 M/√ m3 X "v bm#biBimiBM; a = 4/10- (1 − (6/10)2 )3/2 ∈ [12493582352/1010 , 12493582353/1010 ] M/ 2 ∈ [14142135623/1010 , 14142135624/1010 ] BMiQ m2 M/ m3 M/ /QBM; BMi2`pH` `Bi?K2iB+b ;BM r2 ;2i i?i m2 ∈ [−0.3673154049, −0.3673154050] M/ m3 ∈ [0.6505307765, 0.6505307766]- bQ m2 Bb M2;iBp2 M/ m3 Bb TQbBiBp2 BM (0, b12 )X .QBM; i?2 bK2 7Q` a = 6/10- a = 2- a = 3 M/ a = 4 r2 +QM+Hm/2 i?i m2 > 0 7Q` a ∈ (d2 , a12 ) ∪ (a22 , ∞) M/ m3 > 0 7Q` a ∈ (0, b12 ) ∪ (d2 , 1) ∪ (b22 , ∞)X aQ i?2 `2;BQM r?2`2 i?2 Kbb2b m2 M/ m3 ;Bp2M BM U9V T`QpB/2 +2Mi`H +QM};m`iBQMb Bb a ∈ (d2 , a12 ) ∪ (b22 , ∞)- b22 6B;X k 7Q` i?2 THQib Q7 m2 M/ m3 X Rk JX *P_"1_ L. *X oGGa 15 15 m2 m2 10 m3 10 m3 5 5 a a 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 -5 -5 -10 -10 6B;Xk U0 < a < 1V 6B;Xk# Ua > 1V 6B;m`2 kX SHQi Q7 i?2 Kbb2b m2 U+QMiBMmQmb HBM2V M/ m3 U/b?2/ HBM2V 7Q` n = 2X 1tKBMBM; i?2 T`QT2`iB2b Q7 i?2 7mM+iBQMb m2 M/ m3 r2 ;2i i?i BM i?2 BMi2`pH (d2 , a12 ) i?2 7mM+@ iBQMb m2 , m3 → ∞ r?2M a → d+ 2 X aBM+2 m2 M/ m3 biBb7v i?2 `2HiBQM U8V M/ (K6 (a)−K4 (a))/(K5 − K1 ) Bb  }MBi2 MmK#2` 7Q` HH a 6= 1 r2 +M 2bBHv b22 i?i lima→d+ m2 /m3 = 1X "v +QKTmiBM; i?2 2 /2`BpiBp2b Q7 i?2 7mM+iBQMb m2 M/ m3 M/ MHvxBM; Bib x2`Qb b r2 ?p2 /QM2 rBi? i?2 x2`Q2b Q7 m2 M/ m3 - r2 b22 i?i i?2 7mM+iBQM m2 Bb /2+`2bBM; 7Q` a ∈ (0, d2 ) ∪ (d2 , 1) ∪ (1, 1.5015204804..) M/ BM+`2bBM; 7Q` a ∈ (1.5015204804.., ∞)c M/ i?2 7mM+iBQM m3 Bb BM+`2bBM; 7Q` a ∈ (0, 0.4812067311..) ∪ (ac2 , 1) ∪ (1, +∞) rBi? ac2 = 0.5670013389.. M/ /2+`2bBM; 7Q` a ∈ (0.4812067311.., d2 ) ∪ (d2 , ac2 ) Ub22 6B;X kVX JQ`2Qp2` m2 → 0 r?2M a → a− 12 - m3 (ac2 ) = 4.7014182338..- m3 → 5.950134407.. r?2M a → a12 - m2 → 1.6921282709.. M/ m3 → 0 r?2M a → b+ − 22 X 6BMHHv r2 ;2i i?i m2 , m3 → ∞ r?2M a → ∞ rBi? lima→∞ m2 /m3 = 1 Ub22 6B;X k ;BMVX jXjXkX *b2 n = 3X q?2M n = 3 i?2 pHm2b Q7 Ki 7Q` i = 1, . . . 6 `2 1 5 K1 = √ , K5 = , 3 4 1−a 2+a 1 2−a K2 (a) = + , K3 (a) = + , UdV h13 h33 (1 + a)2 h23 1 − a 1 + 2a 1 1 − 2a K4 (a) = − + , K6 (a) = 2 − , a h13 a h33 a (1 + a) a h23 r?2`2 h13 = ((1 − a)2 )3/2 - h23 = (1 − a + a2 )3/2 - M/ h33 = (1 + a + a2 )3/2 X h?mb r?2M n = 3- i?2 bQHmiBQMb m2 = m2 (a) M/ m3 = m3 (a) Q7 bvbi2K UjV rBi? mD 6= 0 `2 ;Bp2M #v U9V rBi? Ki ;Bp2M #v UdVX L2ti r2 }M/ i?2 pHm2b Q7 a 7Q` r?B+? m2 = m2 (a) > 0 M/ m3 = m3 (a) > 0X 6B`bi r2 }M/ HH i?2 `2H bQHmiBQMb Q7 2[miBQMb mD = 0- mN,2 = 0 M/ mN,3 = 0 rBi? a > 0X 6QHHQrBM; √ bi2T #v bi2T i?2 T`Q+2/m`2 2tTHBM2/ BM a2+iBQM jXjXR #v BMi`Q/m+BM; i?2 M2r p`B#H2 h43 = 3 r2 i`Mb7Q`K 2[miBQMb mD = 0- mN,2 = 0 M/ mN,3 = 0 BMiQ TQHvMQKBH 2[miBQMb rBi? BMi2;2` +Q2{+B2Mib ?pBM; i?2 bK2 bQHmiBQMb b i?2 BMBiBH 2[miBQMb M/ T`Q##Hv M2r QM2bX h?2M r2 bQHp2 MmK2`B+HHv i?2 Q#iBM2/ TQHvMQKBH 2[miBQMb M/ r2 +?2+F r?B+? Q7 i?2B` bQHmiBQMb rBi? a > 0 T`QpB/2 bQHmiBQMb Q7 i?2 +Q``2bTQM/BM; BMBiBH 2[miBQMbX h?2 `2bmHib i?i r2 ?p2 Q#iBM2/ `2 bmKK`Bx2/ BM h#H2 kX LQi2 i?i mN,2 - mN,3 M/ mD `2 MQi bBKmHiM2QmbHv x2`QX h?Bb K2Mb i?i r?2M n = 3 i?2`2 +QmH/ MQi #2 bQHmiBQMb Q7 bvbi2K UjV rBi? mD = 0X S`Q+22/BM; ;BM b BM a2+iBQM jXjXR r2 MHvx2 i?2 bB;Mb Q7 m2 M/ m3 M/ r2 ;2i i?i m2 > 0 7Q` a ∈ (0, a13 ) ∪ (d3 , a23 ) ∪ (a33 , ∞) M/ m3 > 0 7Q` a ∈ (0, b13 ) ∪ (d3 , 1) ∪ (b23 , ∞)X AM b?Q`i- i?2 Kbb2b m2 M/ m3 ;Bp2M BM U9V +M T`QpB/2 +2Mi`H +QM};m`iBQMb r?2M a ∈ (0, a13 ) ∪ (d3 , a23 ) ∪ (b23 , ∞)- b22 6B;X j 7Q` i?2 THQib Q7 m2 M/ m3 X 1tKBMBM; i?2 T`QT2`iB2b Q7 i?2 7mM+iBQMb m2 M/ m3 r2 ;2i i?i BM i?2 BMi2`pH (0, a13 ) i?2 7mM+iBQMb m2 , m3 → 0 r?2M a → 0+ - KQ`2Qp2` lima→0+ m2 /m3 = 1c i?2 7mM+iBQM m2 Bb BM+`2bBM; 7Q` a ∈ (0, ac1 3 )- /2+`2bBM; 7Q` a ∈ (ac1 3 , a13 )- Bi ?b  KtBKmK i a = ac1 3 = 0.1030896914.. rBi? m2 (ac1 3 ) = 0.0003095830..- M/ m2 → 0 r?2M a → a− 13 c M/ i?2 7mM+iBQM m3 Bb BM+`2bBM; BM a ∈ (0, a13 ) M/ m3 → 0.0060680996.. r?2M a → a− 13 Ub22 6B;X jVX AM i?2 BMi2`pH (d3 , a23 ) i?2 7mM+iBQMb m2 , m3 → ∞ r?2M a → d+ 3 rBi? lim a→d + m2 /m3 = 1c i?2 7mM+iBQM m2 Bb /2+`2bBM; 3 M/ m2 → 0 r?2M a → a− 23 c i?2 7mM+iBQM m3 Bb /2+`2bBM; 7Q` a ∈ (d3 , ac2 3 )- Bb BM+`2bBM; 7Q` PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa Rj 1[miBQM SQHvMQKBH 2[miBQM ♯ aQHmiBQMb mD = 0 6561(−1 + a)16 P 170 (a) = 0 Rk a = d3 = 0.5823156170.. a = a13 = 0.1375329706.. mN,2 = 0 kd a = a23 = 0.6346038252.. (−1 + a)16 a4 P252 (a) = 0 a = a33 = 1.9309056653.. a = b13 = 0.5726308779.. mN,3 = 0 (−1 + a)32 a4 P252 (a) = 0 R3 a = b23 = 2.6361642533.. h#H2 kX _2H TQbBiBp2 bQHmiBQMb Q7 mD = 0- mN,2 = 0- M/ mN,3 = 0 7Q` n = 3X >2`2- ♯ K2Mb i?2 MmK#2` Q7 `2H `QQib Q7 i?2 TQHvMQKBH 2[miBQM Pn (a) = 0 7Q` a > 0X 15 15 0.01 m2 m2 m3 10 m3 m2 10 m3 5 0.005 5 a 0.4 0.8 a 2 3 4 5 6 a 0.1 0.2 -5 -5 -10 6B;X j# -10 6B;X j 6B;X j+ Ua > 1V U0 < a < 0.2V U0.2 < a < 1V 6B;m`2 jX SHQi Q7 i?2 Kbb2b m2 U+QMiBMmQmb HBM2V M/ m3 U/b?2/ HBM2V 7Q` n = 3X a ∈ (ac2 3 , a23 )- Bi ?b  KBMBKmK i a = ac2 3 = 0.6096781095.. rBi? m3 (ac2 3 ) = 4.7014182338..- M/ m3 → 4.4805332525.. r?2M a → a− 23 Ub22 6B;X j#VX AM i?2 BMi2`pH (b23 , ∞) #Qi? 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM;- m2 → 1.3553872894.. M/ m3 → 0 r?2M a → b+ 23 M/ m2 , m3 → ∞ r?2M a → ∞ rBi? lima→∞ m2 /m3 = 1 Ub22 6B;X j+VX jXjXjX *b2 n = 4X q?2M n = 4 i?2 pHm2b Q7 Ki 7Q` i = 1, . . . 6 `2 1 1 p √ K1 = + √ , K5 = 2 + 2, 4 2 √ √ 1−a 1 2 2 − 2a 2 + 2a K2 (a) = + + , K3 (a) = + , U3V h14 (1 + a)2 h24 h34 h44 √ √ 1−a 1 2 − 2 + 2a 2 + 2a K4 (a) = − + 2 + , K6 (a) = + , a h14 a(1 + a) h24 ah34 ah44 3/2 √ 3/2 √ 3/2 r?2`2 h14 = ((1 − a)2 )3/2 - h24 = a2 + 1 - h34 = a2 − 2a + 1 M/ h44 = a2 + 2a + 1 X h?mb r?2M n = 4- i?2 bQHmiBQMb m2 = m2 (a) M/ m3 = m3 (a) Q7 bvbi2K UjV rBi? mD 6= 0 `2 ;Bp2M #v U9V rBi? Ki ;Bp2M #v U3VX L2ti r2 }M/ i?2 pHm2b Q7 a 7Q` r?B+? m2 = m2 (a) > 0 M/ m3 = m3 (a) > 0X √ 6B`bi- p T`Q+22/BM; b BM a2+iBQMb jXjXR M/ jXjXk #v BMi`Q/m+BM; i?2 M2r p`B#H2b h54 = 2 M/ √ h64 = 2 + 2- r2 i`Mb7Q`K 2[miBQMb mD = 0- mN,2 = 0 M/ mN,3 = 0 BMiQ TQHvMQKBH 2[miBQMb rBi? BMi2;2` +Q2{+B2MibX q2 }M/ HH i?2 `2H bQHmiBQMb rBi? a > 0 Q7 i?2b2 2[miBQMb M/ i?2 `2bmHib i?i r2 ?p2 Q#iBM2/ `2 bmKK`Bx2/ BM h#H2 jX LQi2 i?i mN,2 - mN,3 M/ mD `2 MQi bBKmHiM2QmbHv x2`QX h?Bb K2Mb i?i r?2M n = 4 i?2`2 +QmH/ MQi #2 bQHmiBQMb Q7 bvbi2K UjV rBi? mD = 0X R9 JX *P_"1_ L. *X oGGa 1[miBQM SQHvMQKBH 2[miBQM ♯ aQHmiBQMb mD = 0 (−1 + a)64 P 880 (a) 8y a = d4 = 0.6118834277.. a = a14 = 0.3563101506.. mN,2 = 0 Ryd a = a24 = 0.6103522047.. (−1 + a)64 a16 P1264 (a) a = a34 = 1.4630863479.. a = b14 = 0.6121322975.. mN,3 = 0 (−1 + a)128 a16 P1328 (a) de a = b24 = 2.2685443458.. h#H2 jX _2H TQbBiBp2 bQHmiBQMb Q7 mD = 0- mN,2 = 0- M/ mN,3 = 0 7Q` n = 4X >2`2- ♯ K2Mb i?2 MmK#2` Q7 `2H `QQib Q7 i?2 TQHvMQKBH 2[miBQM Pn (a) = 0 7Q` a > 0X 15 15 15 0.4 m2 m2 m2 m2 m3 10 10 m3 10 m3 m3 0.2 5 5 5 a a a a 0.62 0.63 0.7 0.9 2 3 4 5 0.3 0.6 -5 -5 -5 -0.2 -10 -10 -10 6B;X 9 6B;X 9# U0.6 < 6B;X 9+ 6B;X 9/ U0 < a < 0.6V a < 0.63V U0.63 < a < 1V Ua > 1V 6B;m`2 9X SHQi Q7 i?2 Kbb2b m2 U+QMiBMmQmb HBM2V M/ m3 U/b?2/ HBM2V 7Q` n = 4X L2ti r2 MHvx2 i?2 bB;Mb Q7 m2 M/ m3 M/ r2 b22 i?2 `2;BQM r?2`2 i?2 Kbb2b m2 M/ m3 ;Bp2M BM U9V +M T`QpB/2 +2Mi`H +QM};m`iBQMb Bb a ∈ (0, a14 ) ∪ (a24 , d4 ) ∪ (b24 , ∞)X a22 6B;X 9 7Q` i?2 THQib Q7 m2 M/ m3 X 6BMHHv- r2 2tKBM2 i?2 T`QT2`iB2b Q7 i?2 7mM+iBQMb m2 M/ m3 r2 ;2i i?i BM i?2 BMi2`pH (0, a14 ) #Qi? 7mM+iBQMb m2 , m3 → 0 r?2M a → 0+ - KQ`2Qp2` lima→0+ m2 /m3 = 1c i?2 7mM+iBQM m2 Bb BM+`2bBM; 7Q` a ∈ (0, ac4 )- /2+`2bBM; 7Q` a ∈ (ac4 , a14 )- Bi ?b  KtBKmK i a = ac4 = 0.2746698699.. rBi? m2 (ac4 ) = 0.0085881109..- M/ m2 → 0 r?2M a → a− 14 c M/ i?2 7mM+iBQM m3 Bb BM+`2bBM; BM a ∈ (0, a14 ) M/ m3 → 0.1238514421.. r?2M a → a− 14 Ub22 6B;X 9VX AM i?2 BMi2`pH (a24 , d4 )- Ub22 6B;X 9#V #Qi? 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM;- m2 , m3 → ∞ r?2M a → d− 4 rBi? lima→d− m2 /m3 = 1- 4 M/ m2 → 0 M/ m3 → 2.3831374646.. r?2M a → a+ 24 X 6BMHHv BM i?2 BMi2`pH (b24 , ∞) #Qi? 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM; Ub22 6B;X 9/V- m2 → 1.0670996767.. M/ m3 → 0 r?2M a → b+ 24 M/ m2 , m3 → ∞ r?2M a → ∞ rBi? lima→∞ m2 /m3 = 1X PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa R8 *b2 0 < a < 1 1[miBQM SQHvMQKBH 2[miBQM ♯1 aQHmiBQMb mD = 0 κ (−1 + a)64 P 1084 (a) 8y a = d5 = 0.6313195684.. mN,2 = 0 κ (−1 + a)64 a4 P 1596 (a) RjR ě mN,3 = 0 κ (−1 + a) a P1596 (a) 38 a = b15 = 0.6425878402.. 128 4 *b2 a > 1 1[miBQM SQHvMQKBH 2[miBQM ♯2 aQHmiBQMb mD = 0 64 κ (−1 + a) P1072 (a) 8N ě mN,2 = 0 κ (−1 + a)64 a4 P1596 (a) Ry3 a = a15 = 1.2290630401.. mN,3 = 0 κ (−1 + a)128 a4 P1596 (a) Nd a = b25 = 2.0628290636.. h#H2 9X _2H TQbBiBp2 bQHmiBQMb Q7 mD = 0- mN,2 = 0- M/ mN,3 = 0 7Q` n = 5X >2`2 ♯1 U`2bT2+iBp2Hv- ♯2 V K2Mb i?2 MmK#2` Q7 `2H `QQib Q7 i?2 TQHvMQKBH 2[miBQM Pn (a) = 0 7Q` 0 < a < 1 U`2bT2+iBp2Hv- a > 1V- M/ κ Bb  +QMbiMiX jXjX9X *b2 n = 5X q?2M n = 5 i?2 pHm2b Q7 Ki 7Q` i = 1, . . . 6 `2 s 2 1 √ K1 = 1 + √ , K5 = + 5, 5 4 √ √ 1−a 2(4 + ηa) 2(4 + ξa) K2 (a) = + + , h15 h45 h55 √ √ K3 (a) = 1 + 2(4 − ξa) + 2(4 − ηa) , UNV (1 + a) 2 h25 h35 √ √ 1−a 2(η + 4a) 2(ξ + 4a) K4 (a) = − + + , ah15 ah ah √ 45 √ 55 1 2(4a − ξ) 2(4a − η) K6 (a) = + + , a(1 + a)2 ah25 ah35 3/2 3/2 3/2 r?2`2 h15 = ((1−a)2 )3/2 - h25 = 2a2 − aξ + 2 - h35 = 2a2 − aη + 2 - h45 = 2a2 + aη + 2 - 3/2 √ √ 2 h55 = 2a + aξ + 2 - ξ = 1 + 5- M/ η = 1 − 5X aQ r?2M n = 5- i?2 bQHmiBQMb m2 = m2 (a) M/ m3 = m3 (a) Q7 bvbi2K UjV rBi? mD 6= 0 `2 ;Bp2M #v U9V rBi? Ki 7Q` i = 1, . . . , 6 ;Bp2M #v UNVX q2 T`Q+22/ BM  bBKBH` rv i?M BM i?2 +b2b n = 2, 3, 4 iQ }M/ HH i?2 `2H bQHmiBQMb Q7 2[miBQMb mD = 0- mN,2 = 0 M/ mN,3 = 0 rBi? a > 0- #mi BM i?Bb +b2 iQ b?Q`i2M i?2 +QKTmiiBQMb r2 +QMbB/2` b2T`i2Hv i?2 +b2b a > 1 M/ 0 < a < 1 M/ r2 2HBKBMi2 i?2 b[m`2 `QQi +Q``2bTQM/BM; iQ h15 /B`2+iHv #v bBKTHB}+iBQM BMbi2/ Q7 2HBKBMiBM; Bi #v K2Mb Q7  `2bmHiMi rBi? `2bT2+i h15 X hQ i`Mb7Q`K 2[miBQMb mD = 0- mN,2 = 0 M/ mN,3 √ = 0 BMiQ TQHvMQKBH √ 2[miBQMb p rBi? BMi2;2` +Q2{+B2Mib- r2 BMi`Q/m+2 i?2 M2r p`B#H2b h65 = 2- h75 = 5 M/ h85 = 1 + 2/h75 X h?2 `2bmHib i?i r2 ?p2 Q#iBM2/ `2 bmKK`Bx2/ BM h#H2 9X LQi2 i?i mN,2 - mN,3 M/ mD `2 MQi bBKmHiM2QmbHv x2`QX h?Bb K2Mb i?i r?2M n = 5 i?2`2 +QmH/ MQi #2 bQHmiBQMb Q7 bvbi2K UjV rBi? mD = 0X MHvxBM; i?2 bB;Mb Q7 m2 M/ m3 r2 b22 i?i i?2 `2;BQM r?2`2 i?2 Kbb2b m2 M/ m3 ;Bp2M BM U9V +M T`QpB/2 +2Mi`H +QM};m`iBQMb Bb a ∈ (0, d5 ) ∪ (b25 , ∞)- b22 6B;X 8 7Q` i?2 THQib Q7 m2 M/ m3 X 1tKBMBM; i?2 T`QT2`iB2b Q7 i?2 7mM+iBQMb m2 M/ m3 r2 ;2i i?i BM i?2 BMi2`pH (0, d5 )- m2 M/ m3 `2 BM+`2bBM; Ub22 6B;X 8V- m2 , m3 → 0 r?2M a → 0+ rBi? lima→0+ m2 /m3 = 1- M/ m2 , m3 → ∞ r?2M a → d− 5 rBi? lima→d− m2 /m3 = 1X AM i?2 BMi2`pH (b25 , ∞)- m2 M/ m3 `2 BM+`2bBM; Ub22 5 6B;X 8#V- m2 → 0.8426164718.. M/ m3 → 0 r?2M a → b+ 25 - M/ m2 , m3 → ∞ r?2M a → ∞ rBi? lima→∞ m2 /m3 = 1X Re JX *P_"1_ L. *X oGGa 15 15 m2 m2 10 m3 10 m3 5 5 a a 0.2 0.4 0.8 1 2 3 4 5 -5 -5 -10 -10 6B;X 8 U0 < a < 1V 6B;X 8# Ua > 1V 6B;m`2 8X SHQi Q7 i?2 Kbb2b m2 U+QMiBMmQmb HBM2V M/ m3 U/b?2/ HBM2V 7Q` n = 5X 4 4 4 m2 m2 m2 m3 m3 2 m3 2 2 a a a 0.5 1.5 2 2.5 3 0.5 1.5 2 2.5 3 0.5 1.5 2 2.5 3 -2 -2 -2 -4 -4 -4 -6 6B;X e Un = 6V 6B;X e# Un = 10V 6B;X e+ Un = 20V 6B;m`2 eX SHQi Q7 i?2 Kbb2b m2 U+QMiBMmQmb HBM2V M/ m3 U/b?2/ HB2MV r?2M n = 6 BM 6B;X e- n = 10 BM 6B;X e# M/ n = 20 BM 6B;X e+X 1.45 0.78 1.40 0.76 0.74 1.35 0.72 1.30 n n 100 200 300 400 500 100 200 300 400 500 6B;X d 6B;X d# 6B;m`2 dX SHQi Q7 dn Ui?2 x2`Q Q7 mD V BM 6B;X d M/ THQi Q7 bn Ui?2 x2`Q Q7 mN,3 rBi? a > 1V BM 6B;X d#X jXjX8X LmK2`B+H bim/v 7Q` n > 5X q2 ?p2 MHvx2/ i?2 #2?pBQ` Q7 m2 M/ m3 b  7mM+iBQM Q7 a 7Q` n = 6, 7, . . . , 500 M/ r2 ?p2 b22M i?i Bi Bb 2bb2MiBHHv i?2 bK2 b i?2 QM2 7Q` n = 5 Ub22 6B;X e 7Q` n = 6, 10, 20VX JQ`2 T`2+Bb2Hv- 7Q` HH n = 5, . . . , 500- i?2 /2MQKBMiQ` mD ?b  mMB[m2 x2`Q dn < 1 Ui?2 2tBbi2M+2 Q7 bm+? x2`Q ?b #22M T`Qp2/ MHviB+HHv BM G2KK 9U/V BM TT2M/Bt RVX q2 ?p2 +QKTmi2/ MmK2`B+HHv i?2 pHm2 Q7 dn 7Q` n = 6, . . . , 500 M/ r2 ?p2 THQii2/ Bi BM 6B;X dUVX LQi2 i?i dn < dn+1 < 1 7Q` HH n = 6, . . . , 499X q2 Q#b2`p2 i?i 7Q` HH n = 6, . . . , 500 i?2 7mM+iBQMb m2 M/ m3 `2 BM+`2bBM; BM i?2 BMi2`pH (0, dn )X JQ`2Qp2` m2 < m3 BM i?Bb BMi2`pH Ui?Bb ?b H`2/v #22M T`Qp2/ MHviB+HHv BM S`QTQbBiBQM NU2VV- m2 , m3 → 0+ r?2M a → 0+ Ui?Bb ?b H`2/v #22M T`Qp2/ MHviB+HHv BM S`QTQbBiBQM NUV M/ U#VV M/ m2 , m3 → ∞ r?2M a → d− nX aQ 0 < m2 < m3 7Q` HH a ∈ (0, dn ) M/ i?2`27Q`2 i?2 BMi2`pH (0, dn ) T`QpB/2b +2Mi`H +QM};m`iBQMbX q2 HbQ Q#b2`p2 i?i 7Q` HH n = 6, . . . , 500 i?2 7mM+iBQM m2 Bb M2;iBp2 BM i?2 BMi2`pH (dn , 1)- i?mb i?Bb BMi2`pH /Q2b MQi T`QpB/2 +2Mi`H +QM};m`iBQMbX 6BMHHv- r2 Q#b2`p2 i?i 7Q` HH n = 6, . . . , 500- m3 → −∞ r?2M a → 1+ M/ m3 → ∞ r?2M a → ∞ Ui?Bb ?b H`2/v #22M T`Qp2/ MHviB+HHv BM S`QTQbBiBQM NU+V M/ U/VVVX JQ`2Qp2` r2 Q#b2`p2 i?i m3 Bb BM+`2bBM; BM i?2 BMi2`pH (1, ∞) 7Q` HH n = 6, . . . , 500 M/ i?2`27Q`2 m3 ?b  mMB[m2 x2`Q bn rBi? a > 1 U7`QK S`QTQbBiBQM NU+V M/ U/V PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa Rd 0.8 0.6 0.4 0.2 n 0 100 200 300 400 500 6B;m`2 3X SHQi Q7 m2 (bn )X ;BM r2 +M T`Qp2 MHviB+HHv i?2 2tBbi2M+2 Q7 i H2bi QM2 x2`Q Q7 m3 rBi? a > 1- MmK2`B+HHv r2 b22 i?i i?Bb x2`Q Bb mMB[m2VX q2 ?p2 +QKTmi2/ MmK2`B+HHv i?2 pHm2 Q7 bn 7Q` n = 6, . . . , 500 M/ r2 ?p2 THQii2/ Bi BM 6B;X d#X q2 b22 i?i bn > bn+1 > 1 7Q` HH n = 6, . . . , 499X aBM+2 m2 > m3 r?2M a > 1 Ui?Bb Bb T`Qp2/ MHviB+HHv BM S`QTQbBiBQM NU2VV- i?2 BMi2`pH (bn , ∞) T`QpB/2b +2Mi`H +QM};m`iBQMbX AM b?Q`i- i?2 b2i Q7 pHm2b Q7 a r?2`2 i?2 bQHmiBQMb m2 M/ m3 ;Bp2M #v U9V `2 TQbBiBp2 Bb a ∈ (0, dn ) ∪ (bn , ∞)- r?2`2 dn Bb i?2 x2`Q Q7 mD M/ bn Bb i?2 x2`Q Q7 i?2 MmK2`iQ` mN,3 rBi? a > 1X AM i?2 `2;BQM (0, dn ) #Qi? Kbb2b ;Q 7`QK x2`Q Ur?2M a i2M/b iQ yV iQ BM}MBiv Ur?2M a i2M/b iQ dn V- r?2`2b BM i?2 `2;BQM (bn , ∞) i?2 Kbb m3 ;Q2b 7`QK y Ur?2M a = bn V iQ BM}MBiv Ur?2M a → ∞V M/ i?2 Kbb m2 ;Q2b 7`QK i?2 TQbBiBp2 pHm2 m2 (bn ) iQ BM}MBiv Ur?2M a → ∞VX q2 ?p2 +QKTmi2/ MmK2`B+HHv i?2 pHm2 m2 (bn ) M/ r2 ?p2 THQii2/ Bi BM 6B;X 3X q2 Q#b2`p2 i?i i?2 pHm2 Q7 m2 (bn ) i2M/b `TB/Hv iQ y b n BM+`2b2bX 6Q` BMbiM+2- r?2M n = 5- m2 (bn ) = 0.8426164718..c r?2M n = 20- m2 (bn ) = 0.0236101462..c r?2M n = 40- m2 (bn ) = 0.0000894392..c r?2M n = 100- m2 (bn ) = 1.1846352161.. × 10−11 XXX HH MmK2`B+H +QKTmiiBQMb ?p2 #22M /QM2 rBi?  KBMBKmK Q7 Ryy /B;Bi T`2+BbBQM M/ r2 ?p2 2Mbm`2/ i?i HH i?2 /B;Bib ;Bp2M ?2`2 `2 2t+iX q2 Q#b2`p2 i?i b n BM+`2b2b i?2 /Bz2`2M+2 #2ir22M m2 M/ m3 /2+`2b2b `TB/Hv- b22 ;BM 6B;X eX h?mb b n BM+`2b2b- i?2 Kbb2b m2 M/ m3 BM  `2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQM Q7 i?2 3nĜ#Q/v T`Q#H2K i2M/ iQ #2 2[mHX 9X a2KB`2;mH` #B+B`+mH` +2Mi`H +QM7B;m`iBQMb Q7 i?2 3nė#Q/v T`Q#H2K 9XRX h?2 2[miBQMbX LQr r2 +QMbB/2` b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2K r?B+? +QMbBbib Q7 n #Q/B2b rBi? Kbb2b m1 = · · · = mn = 1 i i?2 p2`iB+2b Q7  `2;mH` nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb 1 M/ 2n #Q/B2b rBi? Kbb2b 2[mH mn+1 = · · · = m3n = m i i?2 p2`iB+2b Q7  b2KB`2;mH` 2nĜ;QM BMb+`B#2/ BM  +B`+H2 Q7 `/Bmb aX "v mbBM; +QKTH2t +QQ`/BMi2b i?2 TQbBiBQMb Q7 i?2 p2`iB+2b Q7 i?2 BMBiBH n@;QM +M #2 r`Bii2M b qj = eiβj rBi? βj = 2πj/n 7Q` j = 1, . . . , n M/ i?2 p2`iB+2b Q7 i?2 b2KB`2;mH` 2nĜ;QM bBimi2/ QM i?2 +B`+H2 Q7 `/Bmb a +M #2 r`Bii2M b qj+n = aei(βj −β) M/ qj+2n = aei(βj +β) rBi? β ∈ (0, π/n) M/ j = 1, . . . , n- b22 6B;X RU#VX Ai Bb 2bv iQ +?2+F i?i i?2 +2Mi2` Q7 Kbb Q7 i?2 bvbi2K Bb i i?2 Q`B;BMX lM/2` i?2b2 ?vTQi?2bBb- i?2 }`bi n 2[miBQMb Q7 URV #2+QK2 n X n X qk − qj+n X qk − qj+2n n qk − qj 3 +m 3 +m = λqk , URyV |qk − qj | |qk − qj+n | |qk − qj+2n |3 j=1,j6=k j=1 j=1 7Q` k = 1, . . . , n- i?2 7QHHQrBM; n 2[miBQMb #2+QK2 n X n X qk+n − qj qk+n − qj+n + m |qk+n − qj |3 |qk+n − qj+n |3 j=1 j=1,j6=k n URRV X qk+n − qj+2n +m = λqk+n , |qk+n − qj+2n |3 j=1 R3 JX *P_"1_ L. *X oGGa 7Q` k = 1, . . . , n- M/ i?2 Hbi n 2[miBQMb Q7 URV `2 Xn n X qk+2n − qj qk+2n − qj+n 3 + m |qk+2n − qj | |qk+2n − qj+n |3 j=1 j=1 n URkV X qk+2n − qj+2n +m = λqk+2n , |qk+2n − qj+2n |3 j=1,j6=k 7Q` k = 1, . . . , nX S`Q+22/BM; BM  bBKBH` rv i?M BM (9)- r2 /BpB/2 i?2 k@i? 2[miBQM Q7 URyV #v qk - k@i? 2[miBQM Q7 URRV #v qk+n M/ i?2 k@i? 2[miBQM Q7 URkV #v qk+2n M/ r2 ;2i bvbi2K Xn Xn 1 − ei(βj −βk ) 1 − a ei(βj −βk −β) + m |eiβk − eiβj |3 j=1 |eiβk − a ei(βj −β) |3 j=1,j6=k Xn 1 − a ei(βj −βk +β) +m = λ, j=1 |eiβk − a ei(βj +β) |3 n X n X 1 − 1/a ei(βj −βk +β) m 1 − ei(βj −βk ) + URjV |a ei(βk −β) − eiβj |3 a3 |ei(βk −β) − ei(βj −β) |3 j=1 j=1,j6=k n m X 1 − ei(βj −βk +2β) + 3 = λ, a j=1 |ei(βk −β) − ei(βj +β) |3 n X n 1 − 1/a ei(βj −βk −β) m X 1 − ei(βj −βk −2β) + |a ei(βk +β) − eiβj |3 a3 |ei(βk +β) − ei(βj −β) |3 j=1 j=1 n X m 1 − ei(βj −βk ) + 3 = λ, a j=1,j6=k |ei(βk +β) − ei(βj +β) |3 7Q` k = 1, . . . , nX >2`2 eiβk − eiβj = ei(βk ±β) − ei(βj ±β) = (2 − 2 cos(βj − βk ))1/2 , 1/2 eiβk − a ei(βj ±β) = 1 + a2 − 2a cos(βj − βk ± β) , 1/2 a ei(βk ±β) − eiβj = 1 + a2 − 2a cos(βj − βk ∓ β) , ei(βk ∓β) − ei(βj ±β) = (2 − 2 cos(βj − βk ± 2β))1/2 . aBM+2 7Q` HH k = 1, . . . , n i?2 b2i {βj − βk + ϕ}j=1,...,n KQ/mHmb 2π Bb 2[mH iQ i?2 b2i {2πj/n + ϕ}j=1,...,n 7Q` HH ϕ ∈ R- i?2 2[miBQMb Q7 bvbi2K URjV `2 BM/2T2M/2Mi Q7 kX aQ Bi Bb MQi `2bi`B+iBp2 iQ iF2 k = nX 7i2` bi`B;?i7Q`r`/ +QKTmiiBQMb r2 +M b22 i?i bvbi2K URjV Bb 2[mBpH2Mi iQ i?2 bvbi2K  K1 + m L2 + N2 i + m L3 + N3 i) = λ, m m  L4 + N4 i + 3 K1 + 3 L5 + N5 i = λ, a a UR9V m  m L6 + N6 i + 3 L7 + N7 i + 3 K1 = λ, a a r?2`2 K1 Bb /2}M2/ b BM a2+iBQM j Ub22 UjVV M/ n  X 1 − a cos 2πn j − β L2 = L2 (a, β) =  3/2 , 2 j=1 1 + a − 2a cos 2π j n −β n  X −a sin 2πn j − β N2 = N2 (a, β) =  3/2 , j=1 1 + a2 − 2a cos 2π j − β n PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa RN n 2π j  X 1 − 1/a cos +β n L4 = L4 (a, β) =  3/2 , 2π j j=1 1 + a2 − 2a cos n +β n 2π j  X −1/a sin +β n N4 = N4 (a, β) =  3/2 , 2π j j=1 1 + a2 − 2a cos n +β n  2π j X 1 − cos + 2β n L5 = L5 (β) =  3/2 , j=1 2 − 2 cos 2π j n + 2β n  X − sin 2πn j + 2β N5 = N5 (β) =  3/2 , j=1 2 − 2 cos 2π j n + 2β L3 = L3 (a, β) = L2 (a, −β), N3 = N3 (a, β) = N2 (a, −β), L6 = L6 (a, β) = L4 (a, −β), N6 = N6 (a, β) = N4 (a, −β), L7 = L7 (β) = L5 (−β), N7 = N7 (β) = N5 (−β). LQi2 i?i n−1 X 2π j  − sin n  3/2 = 0. 2π j  j=1 2 − 2 cos n aBM+2 7Q` HH ϕ ∈ R 2π j  2π (n−j)  2π j  2π (n−j)  cos n − ϕ = cos n +ϕ , sin n − ϕ = − sin n +ϕ , r2 b22 i?i L2 (a, β) = L3 (a, β), N2 (a, β) = −N3 (a, β), L4 (a, β) = L6 , N4 (a, β) = −N6 (a, β), L5 (β) = L7 (β), N5 (β) = −N7 (β). aQ bvbi2K UR9V Bb 2[mBpH2Mi iQ bvbi2K m m K1 + 2mL2 (a, β) = λ, L4 (a, β) + (K1 + L5 (β)) = λ, N4 (a, β) + N5 (β) = 0. UR8V a3 a3 aQHpBM; i?2 i?B`/ B/2MiBiv BM UR8V r2 ;2i N4 (a, β) m = −a3 UReV N5 (β) M/ 7`QK i?2 }`bi M/ b2+QM/ B/2MiBiB2b BM UR8V r2 Q#iBM (K1 − L4 (a, β)) m = a3 . URdV K1 − 2a3 L2 (a, β) + L5 (β) h?2`27Q`2- 7`QK UReV M/ URdV r2 ?p2 F (a, β) := N5 (β)(K1 − L4 (a, β)) + N4 (a, β)(K1 − 2a3 L2 (a, β) + L5 (β)) = 0. AM b?Q`i-  b2KB`2;mH` #B+B`+mH` +QM};m`iBQM Q7 i?2 3nĜ#Q/v T`Q#H2K Bb +2Mi`H B7 m = m(a, β) Bb ;Bp2M #v UReV M/ a, β `2 bm+? i?i F (a, β) = 0 M/ m(a, β) > 0X 9XkX /KBbbB#H2 pHm2b Q7 βX h?2 7QHHQrBM; T`QTQbBiBQM T`QpB/2b i?2 `M;2 Q7 pHm2b Q7 β i?i +M T`QpB/2 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2KX S`QTQbBiBQM RyX  M2+2bb`v +QM/BiBQM iQ ?p2  b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQM Q7 i?2 3nĜ#Q/v T`Q#H2K Bb i?i β ∈ (π/2n, π/n)X S`QQ7X q2 rBHH b22 i?i m = −a3 N4 (a, β)/N5 (β) > 0 B7 M/ QMHv B7 β ∈ (π/2n, π/n)X q2 `2+HH i?i n 1 d X 1 N4 (a, β) =  1/2 a2 dβ 2π j j=1 1 + a2 − 2a cos n +β ky JX *P_"1_ L. *X oGGa M/ i?i n 1 d X 1 N5 (β) = lim  1/2 . 2 a→1 dβ 2π j j=1 1 + a2 − 2a cos n + 2β *QMbB/2` }`bi i?2 +b2 BM r?B+? a ∈ [0, 1)X AM i?Bb +b2 mbBM; S`QTQbBiBQM R9 BM TT2M/Bt R rBi? α = 1/2 M/ u = β M/ iFBM; /2`BpiBp2b rBi? `2bT2+i iQ β r2 ;2i Z 1 ! # n d t−1/2 1 1 − (at)2n N4 (a, β) = 2 dt a π 0 dβ (1 − t)1/2 (1 − a2 t)1/2 B1 Z n sin(nβ) 1 −2n(at)n (1 − (at)2n ) = 1/2 (1 − t)1/2 (1 − a2 t)1/2 B 2 dt, a2 π 0 t 1 rBi? B1 = 1 + (at)2n − 2(at)n cos(nβ)- r?B+? Bb M2;iBp2 7Q` β ∈ (0, π/n) #2+mb2 i?2 BMi2;`M/ BM i?2 BMi2;`H Bb M2;iBp2X JQ`2Qp2`- mbBM; S`QTQbBiBQM R9 ;BM rBi? α = 1/2 M/ u = 2β M/ iFBM; /2`BpiBp2b rBi? `2bT2+i iQ β r2 ;2i Z ! # 1 n 1 d t−1/2 1 1 − (at)2n N5 (β) = lim dt 2 a→1− π 0 dβ (1 − t)1/2 (1 − a2 t)1/2 B2 Z 1 n sin(2nβ) −2n(at)n (1 − (at)2n ) = lim dt, π a→1− 0 t1/2 (1 − t)1/2 (1 − a2 t)1/2 B22 rBi? B2 = 1 + (at)2n − 2(at)n cos(2nβ)- r?B+? Bb M2;iBp2 B7 β ∈ (0, π/2n) M/ TQbBiBp2 B7 β ∈ (π/2n, π/n) #2+mb2 ;BM i?2 BMi2;`M/ Bb M2;iBp2X *QMbB/2` MQr i?2 +b2 a > 1X AM i?Bb +b2 mbBM; S`QTQbBiBQM R8 BM TT2M/Bt R rBi? α = 1/2 M/ u = β M/ iFBM; /2`BpiBp2b rBi? `2bT2+i iQ β r2 ;2i i?i Z 1   n d a2n − t2n N4 (a, β) = 2 dt a π 0 dβ t1/2 (1 − t)1/2 (a2 − t)1/2 B3 Z n sin(nβ) 1 −2n(at)n (a2n − t2n ) = 1/2 (1 − t)1/2 (a2 − t)1/2 B 2 dt, a2 π 0 t 3 rBi? B3 = a2n + t2n − 2an tn cos(nβ)- r?B+? Bb M2;iBp2 7Q` β ∈ (0, π/n)X JQ`2Qp2`- mbBM; S`QTQbBiBQM R8 ;BM rBi? α = 1/2 M/ u = 2β M/ iFBM; /2`BpiBp2b rBi? `2bT2+i iQ β r2 ;2i i?i Z   1 n 1 d a2n − t2n N5 (β) = lim dt 2 a→1+ π 0 dβ t1/2 (1 − t)1/2 (a2 − t)1/2 B4 Z 1 n sin(2nβ) −2n(at)n (a2n − t2n ) = lim dt, π a→1+ 0 t1/2 (1 − t)1/2 (1 − a2 t)1/2 B42 rBi? B4 = a2n + t2n − 2an tn cos(2nβ)- r?B+? Bb M2;iBp2 B7 β ∈ (0, π/2n) #2+mb2 i?2 BMi2;`M/ Bb M2;iBp2 M/ TQbBiBp2 B7 β ∈ (π/2n, π/n)- ;BM #2+mb2 i?2 BMi2;`M/ Bb M2;iBp2X h?2`27Q`2- N4 (a, β) Bb M2;iBp2 7Q` β ∈ (0, π/n) M/ N5 (β) Bb M2;iBp2 7Q` β ∈ (0, π/2n) M/ TQbBiBp2 7Q` β ∈ (π/2n, π/n)X AM b?Q`i- m = −a3 N4 (a, β)/N5 (β) Bb TQbBiBp2 B7 M/ QMHv B7 β ∈ (π/2n, π/n) M/ i?2 T`QTQbBiBQM Bb T`Qp2/X  9XjX S`QQ7 Q7 h?2Q`2K dUVX q2 bim/v i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb Q7 i?2 b2KB`2;mH` #B+B`+mH` 3nĜ#Q/v T`Q#H2K `QmM/ β = π/nX 6Q` T`QpBM; h?2Q`2K dUV r2 M22/ i?2 7QHHQrBM; mtBHB`v T`QTQbBiBQM +QM+2`MBM; i?2 bB;M Q7 i?2 7mM+iBQM F (a, β) b β → π/nX S`QTQbBiBQM RRX h?2 7QHHQrBM; ?QH/b 7Q` F̄ (a) = limβ→π/n F (a, β)X UV F̄ (a) > 0 r?2M a → ∞c U#V F̄ (a) < 0 r?2M a → 1c U+V F̄ (a) > 0 r?2M a → 0 M/ n ≥ 3 M/ F̄ (a) < 0 r?2M a → 0 M/ n = 2X q2 MQi2 i?i BM S`QTQbBiBQM RR r2 ?p2 i?i F̄ (a) +QmH/ #2 ±∞X S`QTQbBiBQM RR Bb T`Qp2/ BM TT2M/Bt kX PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa kR S`QQ7 Q7 h?2Q`2KdUVX Ai 7QHHQrb 7`QK S`QTQbBiBQM Ry i?i HH bQHmiBQMb Q7 F (a, β) = 0 7Q` β ∈ (π/2n, π/n) biBb7v m > 0- i?2`27Q`2 HH bQHmiBQMb Q7 F (a, β) = 0 T`QpB/2 +2Mi`H +QM};m`iBQMb Q7 i?2 b2KB`2;mH` #B+B`+mH` 3nĜ#Q/v T`Q#H2KX LQiB+2 i?i i?2 7mM+iBQM F Bb +QMiBMmQmb 7Q` a ∈ (0, ∞) M/ β ∈ (π/2n, π/n)- i?2`27Q`2 i?2 TQBMib r?2`2 i?2 bB;M Q7 F +?M;2b T`QpB/2b Hrvb bQHmiBQMb Q7 F (a, β) = 0X AM pB2r Q7 S`QTQbBiBQM RR r2 ?p2 i?i r?2M n ≥ 3- }t2/  pHm2 Q7 β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 π/n- i?2 7mM+iBQM F ?b i H2bi QM2 +?M;2 Q7 bB;M rBi? a > 1 M/ QM2 +?M;2 Q7 bB;M rBi? 0 < a < 1X aQ 7Q` 2+? β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 π/n i?2`2 `2 i H2bi irQ pHm2b Q7 a 7Q` r?B+? F (a, β) = 0- QM2 rBi? a > 1 M/ QM2 rBi? 0 < a < 1X q?2M n = 2 i?2 7mM+iBQM F ?b i H2bi QM2 +?M;2 Q7 bB;M rBi? a > 1X aQ 7Q` 2+? β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 π/n i?2`2 Bb i H2bi QM2 pHm2 Q7 a rBi? a > 1 7Q` r?B+? F (a, β) = 0X PM i?2 Qi?2` ?M/- 7`QK i?2 T`QQ7b Q7 G2KK 8 M/ S`QTQbBiBQM RR Ub22 TT2M/Bt kV r2 ?p2 i?i limβ→π/n N4 (a, β) = 0 M/ limβ→π/n N5 (β) = ∞- `2bT2+iBp2HvX h?2`27Q`2 7`QK UReV- m → 0 r?2M β → π/nX  9X9X S`QQ7 Q7 h?2Q`2K dU#VX LQr r2 bim/v i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb Q7 i?2 b2KB`2;@ mH` #B+B`+mH` 3nĜ#Q/v T`Q#H2K `QmM/ β = π/2nX 6Q` T`QpBM; h?2Q`2K dU#V r2 M22/ i?2 7QHHQrBM; irQ mtBHB`v T`QTQbBiBQMb +QM+2`MBM; i?2 bim/v Q7 i?2 bB;M Q7 i?2 7mM+iBQM F (a, β)X S`QTQbBiBQM RkX h?2 7QHHQrBM; ?QH/b 7Q` F̄ (a) = limβ→π/2n F (a, β)X UV F̄ (a) < 0 r?2M a → 0c U#V F̄ (a) < 0 r?2M a → ∞c U+V F̄ (a) > 0 r?2M a → 1X S`QTQbBiBQM RjX h?2 7QHHQrBM; bii2K2Mib ?QH/ 7Q` F̄ (β) = lima→0 F (a, β) M/ F̃ (β) = lima→∞ F (a, β)X UV 6Q` HH β ∈ (π/2n, π/n) r2 ?p2 F̄ (β) > 0 7Q` n ≥ 3 M/ F̄ (β) < 0 7Q` n = 2X U#V 6Q` HH β ∈ (π/2n, π/n) M/ n ≥ 2 r2 ?p2 F̃ (β) > 0X q2 MQi2 i?i BM S`QTQbBiBQM Rj- #Qi? F̄ (β) M/ F̃ (β) +QmH/ #2 ±∞X h?2 T`QQ7 Q7 S`QTQbBiBQMb Rk M/ Rj +M #2 7QmM/ BM TT2M/Bt jX S`QQ7 Q7 h?2Q`2K dU#VX b BM h?2Q`2K dUV `2+HH i?i HH bQHmiBQMb Q7 F (a, β) = 0 rBi? β ∈ (π/2n, π/n) biBb7v m > 0 M/ i?2`27Q`2 i?2v T`QpB/2 +2Mi`H +QM};m`iBQMb Q7 i?2 b2KB`2;mH` #B+B`+mH` 3nĜ#Q/v T`Q#H2KX JQ`2Qp2` i?2 TQBMib r?2`2 i?2 bB;M Q7 F +?M;2b T`QpB/2 bQHmiBQMb Q7 F (a, β) = 0X AM pB2r Q7 S`QTQbBiBQM Rk r2 ?p2 i?i BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 β = π/2n i?2 7mM+iBQM F ?b i H2bi QM2 +?M;2 Q7 bB;M rBi? a > 1 M/ QM2 +?M;2 Q7 bB;M rBi? 0 < a < 1X h?2`27Q`2 }t2/  pHm2 Q7 β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 β = π/2n- i?2`2 2tBbi i H2bi irQ x2`Qb Q7 F - QM2 rBi? a < 1 M/ QM2 rBi? 0 < a < 1X AM pB2r Q7 S`QTQbBiBQMb Rk M/ Rj r2 ?p2 i?i r?2M n ≥ 3 lim lim F (a, β) < 0, lim lim F (a, β) > 0. a→0 β→π/2n β→π/2n a→0 h?2`27Q`2 }t2/  pHm2 Q7 β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 π/2n- i?2`2 2tBbi i H2bi QM2 x2`Q Q7 F rBi? a bm{+B2MiHv +HQb2 iQ i?2 Q`B;BMX ;BM- BM pB2r Q7 S`QTQbBiBQMb Rk M/ Rj r2 ?p2 i?i 7Q` HH n ≥ 2 lim lim F (a, β) < 0, lim lim F (a, β) > 0 a→∞ β→π/2n β→π/2n a→∞ h?2`27Q`2 }t2/  pHm2 Q7 β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 π/2n- i?2`2 2tBbib i H2bi QM2 x2`Q Q7 F rBi? a bm{+B2MiHv H`;2X AM b?Q`i- }t2/  pHm2 Q7 β BM  bm{+B2MiHv bKHH M2B;?#Q`?QQ/ Q7 π/2n i?2`2 2tBbi i H2bi 7Qm` x2`Q2b Q7 F r?2M n ≥ 3- QM2 M2` a = 0- QM2 rBi? 0 < a < 1 MQi M2+2bb`BHv bKHH- QM2 rBi? a bm{+B2MiHv H`;2 M/ QM2 rBi? a > 1 MQi M2+2bb`BHv H`;2X q?2M n = 2 i?2`2 2tBbi i H2bi i?`22 bQHmiBQMb Q7 F QM2 rBi? a bm{+B2MiHv H`;2- QM2 rBi? a > 1 MQi M2+2bb`BHv H`;2- M/ QM2 rBi? 0 < a < 1X kk JX *P_"1_ L. *X oGGa 4 4 4 4 4 4 3 a3 (β) 3 3 a3 (β) 3 3 3 a3 (β) 2 2 2 2 2 2 a a a a2* a1 (β) a2* a2* a2 a2 (β) a2 (β) a2 (β) a2 a2 1 1 1 1 a1 (β) a1* a1 a1 a1 a1 (β) a1* a4 (β) a4 (β) p * p p * p p p b 1 1.2 1.4 0.6 b 0.8 b* 0.5 0.6 0.7 4 2 6 3 8 4 b b b 6B;X N n = 2 6B;X N# n = 3 6B;X N+ n = 4 4 4 4 4 3 3 3 3 2 2 2 2 a a a2* a2* a2 1* a2 1* a1 a1 a1 a1 p p p p b* 0.4 0.5 0.6 b* 0.3 0.4 0.5 10 5 12 6 b b 6B;X N/ n = 5 6B;X N2 n = 6 6B;m`2 NX aQHmiBQMb Q7 F (a, β) = 0X PM i?2 Qi?2` ?M/- BM i?2 T`QQ7 Q7 G2KK e Ub22 TT2M/Bt jV r2 ?p2 b22M i?i limβ→π/2n N5 (β) = 0X JQ`2Qp2` 7`QK U9XkV r2 ?p2 i?i a 6∈ {0; ∞} i?2M limβ→π/2n N4 (a, β) 6= 0X h?2`27Q`2 7`QK UReV r2 +M ;m`Mi22 i?i m → ∞ i i?2 +2Mi`H +QM};m`iBQMb +QKBM; 7`QK i?2 x2`Q2b Q7 F rBi? 0 < a < 1 MQi bKHH M/ a > 1 MQi H`;2X h?Bb +QKTH2i2b i?2 T`QQ7 Q7 h?2Q`2K dU#VX  8X S`iB+mH` +b2b Q7 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM7B;m`iBQMb Q7 i?2 3nė#Q/v T`Q#H2K q?2M n = 2 7`QK h?2Q`2K dU#V r2 ?p2 i H2bi i?`22 7KBHB2b U/2T2M/BM; QM βV Q7 +2Mi`H +QM};m`iBQMb BM  M2B;?#Q`?QQ/ Q7 β = π/4- QM2 2KMiBM; 7`QK  TQBMi (a1 , π/4)- QM2 2KMiBM; 7`QK (a2 , π/4) rBi? a2 > 1- M/ QM2 2KMiBM; 7`QK (∞, π/4)X 6`QK h?2Q`2K dUV r2 ?p2 i H2bi QM2 7KBHv Q7 +2Mi`H +QM};m`iBQMb BM  M2B;?#Q`?QQ/ Q7 β = π/2 i?i 2KMi2b 7`QK  TQBMi (a∗2 , π/2) rBi? a∗2 > 1X h?2v `2 ;Bp2M #v i?2 7KBHB2b Q7 x2`Q2b Q7 F 7Q` n = 2X q2 ?p2 bim/B2/ MmK2`B+HHv i?2b2 7KBHB2b Q7 x2`Q2b M/ r2 ?p2 Q#iBM2/ i?2 7QHHQrBM; Ub22 6B;X NV i?2 7KBHv Q7 +2Mi`H +QM};m`iBQMb 2KMiBM; √ 7`QK i?2 TQBMi (a1 , π/4) = (0.6240605991.., π/4) DQBMb i?2 7KBHv 2KMiBM; 7`QK (a2 , π/2) = ( 3, π/2) M/ i?2 7KBHv 2KMiBM; 7`QK (a2 , π/4) = ∗ (1.4339374069.., π/4) DQBMb i?2 7KBHv 2KMiBM; 7`QK (∞, π/4)X JQ`2Qp2`- i?2b2 `2 i?2 QMHv 7KB@ HB2b Q7 +2Mi`H +QM};m`iBQMbX AM T`iB+mH` B7 β ∈ (π/4, b∗ ) rBi? b∗ = 0.9195936184.. i?2 b2KB`2;mH` #B+B`+mH` 6Ĝ#Q/v T`Q#H2K ?b i?`22 /Bz2`2Mi +2Mi`H +QM};m`iBQMb- B7 b = b∗ Bi ?b irQ +2Mi`H +QM};m`iBQMb M/ B7 β > b∗ Bi ?b QMHv QM2 +2Mi`H +QM};m`iBQMX q?2M n ≥ 3 7`QK h?2Q`2K dUV r2 ?p2 i H2bi 7Qm` 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb Q7 i?2 b2KB`2;mH` #B+B`+mH` 3nĜ#Q/v T`Q#H2K BM  M2B;?#Q`?QQ/ Q7 β = π/2n- QM2 2KMiBM; 7`QK (0, 2π/n)- QM2 2KMiBM; 7`QK  TQBMi (a1 , π/2n) rBi? a1 ∈ (0, 1)- QM2 2KMiBM; 7`QK  TQBMi PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa kj (a2 , π/2n) rBi? a2 > 1 M/ QM2 2KMiBM; 7`QK (∞, π/2n)X JQ`2Qp2`- 7`QK h?2Q`2K dU#V r2 ?p2 i H2bi irQ 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb BM  M2B;?#Q`?QQ/ Q7 β = π/n- QM2 2KMiBM; 7`QK  TQBMi (a∗1 , π/n) rBi? a∗1 ∈ (0, 1) M/ QM2 2KMiBM; 7`QK  TQBMi (a∗2 , π/n) rBi? a∗2 > 1X b #Qp2 i?2v `2 ;Bp2M #v i?2 7KBHB2b Q7 x2`Q2b Q7 F X q2 ?p2 bim/B2/ MmK2`B+HHv i?2b2 7KBHB2b Q7 x2`Q2b 7Q` n = 3, 4, 5, 6 M/ r2 ?p2 Q#iBM2/ i?2 7QHHQrBM; Ub22 ;BM 6B;X NVX q?2M n = 3 i?2 7KBHv Q7 +2Mi`H +QM};m`iBQMb 2KMiBM; 7`QK i?2 TQBMi (0, π/6) DQBMb i?2 7KBHv 2KMiBM; 7`QK (a∗1 , π/3) = (0.4138879324.., π/3)- i?2 7KBHv 2KMiBM; 7`QK (a1 , π/6) = (0.6280478552.., π/6) DQBMb i?2 7KBHv 2KMiBM; 7`QK (a2 , π/6) = (1.1308109202.., π/6) M/ i?2 7KBHv 2KMiBM; 7`QK (∞, π/6) DQBMb i?2 7KBHv 2KMiBM; 7`QK (a∗2 , π/3) = (1.6197896088.., π/3)X JQ`2Qp2`- i?2b2 `2 i?2 QMHv 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb r?2M n = 3X AM T`iB+mH` B7 β ∈ (π/6, b∗ ) rBi? b∗ = 0.7119233840.. i?2 b2KB`2;mH` #B+B`+mH` 9Ĝ#Q/v T`Q#H2K ?b 7Qm` /Bz2`2Mi +2Mi`H +QM};m`iBQMb- B7 b = b∗ Bi ?b i?`22 +2Mi`H +QM};m`iBQMb M/ B7 β > b∗ Bi ?b irQ +2Mi`H +QM};m`iBQMbX q?2M n = 4 i?2 7KBHv Q7 +2Mi`H +QM};m`iBQMb 2KMiBM; 7`QK (0, π/8) DQBMb i?2 7KBHv 2KMi@ BM; 7`QK (a1 , π/8) = (0.6351161391.., π/8)- i?2 7KBHv 2KMiBM; 7`QK (a2 , π/8) = (1.0636734282.., π/8) DQBMb i?2 7KBHv 2KMiBM; 7`QK (a∗1 , π/4) = (0.697380509.., π/4) M/ i?2 7KBHv 2KMi@ BM; 7`QK (∞, π/8) DQBMb i?2 7KBHv 2KMiBM; 7`QK (a∗2 , π/4) = (1.6024084862.., π/4)X JQ`2Qp2`- i?2b2 `2 i?2 QMHv 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb r?2M n = 4X AM T`iB+mH` B7 β ∈ (π/2n, b∗ ) rBi? b∗ = 0.4665964724.. i?2 b2KB`2;mH` #B+B`+mH` 12Ĝ#Q/v T`Q#H2K ?b 7Qm` /Bz2`2Mi +2Mi`H +QM};m`iBQMb- B7 b = b∗ Bi ?b i?`22 +2Mi`H +QM};m`iBQMb M/ B7 β > b∗ Bi ?b irQ +2Mi`H +QM};m`iBQMbX h?2 bK2 #2?pBQ` Q++m`b 7Q` n = 5, . . . , 100 Ub22 6B;X N 7Q` n = 5, 6V- bQ r2 +QMD2+im`2 i?i i?Bb ?TT2Mb 7Q` HH n > 6X q2 MQi2 i?i r?2M n = 5- a1 = 0.6434495204..- a2 = 1.0379259369..- a∗1 = 0.822828699..- a∗2 = 1.5979217289.. M/ b∗ = 0.3406546931..c M/ r?2M n = 6- a1 = 0.6515248377..- a2 = 1.0252694202..- a∗1 = 0.8843211381..- a∗2 = 1.5922353553.. M/ b∗ = 0.2733239284..X 6BMHHv ?p2 +QKTmi2 i?2 pHm2b Q7 i?2 Kbb2b 7Q` i?2 7KBHB2b Q7 +2Mi`H +QM};m`iBQMb rBi? n = 2, 3, 4 M/ r2 ?p2 THQii2/ i?2K BM 6B;X RyX TT2M/Bt RX S`QQ7 Q7 S`QTQbBiBQM N q2 bii2 M/ T`Qp2 bQK2 mtBHB`v `2bmHib i?i rBHH #2 mb2/ BM i?2 T`QQ7 Q7 S`QTQbBiBQM NX q2 M22/ i?2 7QHHQrBM; irQ T`QTQbBiBQMb iF2M 7`QK (R)X S`QTQbBiBQM R9 U(R- S`QTQbBiBQM d)VX 6Q` 0 6 a < 1- α ∈ (0, 1) M/ u ∈ [0, 2π) r2 ?p2 Xn 1 α j=1 1 + a − 2a cos 2πn j + u 2 Z n sin(πα) 1 tα−1 1 1 − (at)2n = α 2 α 2n − 2(at)n cos(nu) dt. π 0 (1 − t) (1 − a t) 1 + (at) S`QTQbBiBQM R8 U(R- S`QTQbBiBQM 3)VX 6Q` a > 1- α ∈ (0, 1) M/ u ∈ [0, 2π) r2 ?p2 Xn 1 α j=1 1 + a − 2a cos 2πn j + u 2 Z n sin(πα) 1 tα−1 1 a2n − t2n = α 2 α 2n + t2n − 2(at)n cos(nu) dt. π 0 (1 − t) (a − t) a q2 M22/ i?2 7QHHQrBM; mtBHB`v `2bmHiX G2KK RX G2i u ∈ R M/ H2i γ = 2π j/n + uX UV h?2 7QHHQrBM; B/2MiBi2b ?QH/ 7Q` HH ℓ ∈ N Xn n X 2ℓπ j  2ℓπ j  cos + u = 0, sin + u = 0, UR3V n n j=1 j=1 k9 JX *P_"1_ L. *X oGGa m m 8 m 15 6 0.015 m(a2 (β), β) m(a2 (β), β) 10 4 0.01 m(a1 (β), β) 5 m(a3 (β), β) ! ! 2 0.005 m(a1 (β), β) m(a4 (β), β) m(a3 (β), β) ! ! b p p p b p p b b* 1 b* 1 1.2 1.4 b* 1 3 6B;X Ry# h?2 THQi Q7 m(a4 (β), β) 4 2 6 3 6B;X Ry n = 2 6B;X Ry# n = 3 8 m 7Q` n = 3 6 m(a2 (β), β) 4 m(a1 (β), β) 2 m(a4 (β), β) ! m(a3 (β), β) ! ! ! ! p ! p b b* 8 4 6B;X N+ n = 4 6B;m`2 RyX oHm2b Q7 m i i?2 7KBHB2b Q7 b2KB`2;mH` #B+B`+mH` +2Mi`H +QM};m`iBQMb Q7 i?2 3nĜ#Q/v T`Q#H2KX r?2M n ≥ 2 M/ n 6= ℓX U#V 6Q` n ≥ 3 r2 ?p2 n X n cos2 γ = , URNV 2 j=1 M/ 7Q` n = 2 r2 ?p2 2 X  cos2 π j + u = 2 cos2 u. UkyV j=1 U+V 6Q` HH n ≥ 1 r2 ;2i n X n a − cos γ d X 1  3/2 =−  1/2 . da j=1 1 + a2 − 2a cos γ j=1 1 + a2 − 2a cos γ U/V G2i n X 1 − 1/a cos γ L(a, u) =  3/2 . j=1 1+ a2 − 2a cos γ q2 ?p2 L(a, u) = −n/2 + O(a) r?2M n ≥ 3 M/ L(a, u) = 2 − 6 cos2 u + O(a) r?2M n = 2X PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa k8 S`QQ7X lbBM; i?2 bmK Q7 i?2 }`bi n i2`Kb Q7  ;2QK2i`B+ b2`B2b r2 ;2i Xn ei(2ℓπj/n+u) = 0, j=1 √ 7Q` HH ℓ ∈ Z rBi? ℓ 6= n U?2`2 i = −1VX h?Bb T`Qp2b bii2K2Mi UVX 6`QK i?2 7Q`KmH Q7 i?2 +QbBMmb Q7 irB+2 M M;H2 M TTHvBM; bii2K2Mi UV rBi? ℓ = 2 M/ 2u BMbi2/ Q7 u r2 ;2i Xn   n      2 2π j 1X 2π j n cos +u = cos 2 +u +1 = . n 2 n 2 j=1 j=1 h?Bb T`Qp2b bii2K2Mi U#V 7Q` n > 2X aii2K2Mi U#V 7Q` n = 2 M/ bii2K2Mi U+V 7QHHQrb 7`QK /B`2+i +QKTmiiBQMbX 1tTM/BM; i?2 7mM+iBQM L BM Gm`2Mi b2`B2b `QmM/ a = 0 r2 ?p2 n n n 1X 2π j  X X 2π j  L(a, u) = − cos +u + 1−3 cos2 + u + O(a). a n n j=1 j=1 j=1 h?2M- r?2M n ≥ 3 BM pB2r Q7 UR3V rBi? ℓ = 1 iQ;2i?2` rBi? URNV r2 Q#iBM 3n n L(a, u) = n − + O(a) = − + O(a) 2 2 M/ r?2M n = 2 BM pB2r Q7 UR3V rBi? ℓ = 1 M/ UkyV r2 Q#iBM L(a, u) = 2 − 6 cos2 u + O(a). h?Bb +QKTH2i2b i?2 T`QQ7 Q7 bii2K2Mi U/VX  q2 M22/ i?2 7QHHQrBM; i2+?MB+H H2KKX G2KK kX q2 ?p2 f (v) = (1 + v)(1 − v 2n ) − 4nv n (1 − v) > 0 7Q` n ≥ 2 M/ v ∈ (0, 1)X S`QQ7X LQi2 i?i f (v) =(1 + v)(1 − v n )(1 + v n ) − 4nv n (1 − v)  =(1 − v) (1 + v)(1 + v + v 2 + . . . + v n−1 )(1 + v n ) − 4nv n :=(1 − v)g(v) r?2`2 g(v) = (1 + v)(1 + v + v 2 + . . . + v n−1 )(1 + v n ) − 4nv n = 1 + 2v + 2v 2 + 2v 3 + . . . + 2v n−1 + 2v n + 2v n + 2v n+1 + . . . UkRV + 2v 2n−1 + v 2n − 4nv n = 1 + 2v + . . . + 2v n−1 + (2 − 4n)v n + 2v n+1 + . . . + 2v 2n−1 + v 2n . q2 b22 i?i g(1) = 0 M/ g ′ (1) = 0. AM/22/- g(1) = 1 + 2(n − 1) + 2 − 4n + 2(n − 1) + 1 = 0, M/ bBM+2 g ′ (v) = 2 + 4v + 6v 2 + . . . + 2(n − 1)v n−2 + (2 − 4n)nv n−1 + 2(n + 1)v n + . . . + 2(2n − 1)v 2n−2 + 2nv 2n−1 , i?2M g ′ (1) = 2(1 + 2 + 3 + . . . + (n − 1)) + n(2 − 4n) + 2(n + 1 + n + 2 + . . . + 2n − 1) + 2n  2n−1 X   2n−1 X  =2 j − n + (1 − 2n)n + n = 2 j + n(1 − 2n) = 0. j=1 j=1 ke JX *P_"1_ L. *X oGGa h?2`27Q`2- 2n−2 X g(v) = (1 − v)2 h(v), h(v) = cj v j j=0 7Q` bQK2 +Q2{+B2Mib cj X q2 rBHH b?Qr #v BM/m+iBQM i?i ( (j + 1)2 7Q` j = 0, . . . , n − 1, cj = (2n − j − 1)2 7Q` j = n, . . . , 2n − 2. AM pB2r Q7 UkRV r2 ?p2 i?i i?2 +Q2{+B2Mib cj biBb7v c0 = 1- c1 = 2 + 2c0 = 4- ck + ck−2 − 2ck−1 = 2 7Q` k = 2, . . . , n − 1, n + 1, . . . , 2n − 2 , cn + cn−2 − 2cn−1 = 2 − 4n, c2n−3 − 2c2n−2 = 2 M/ c2n−2 = 1 UM/ bQ c2n−3 = 4VX q2 T`Qp2 #v BM/m+iBQM i?2 +b2b 7Q` j = 0, . . . , n − 1X Ai Bb +H2` 7Q` j = 0, 1 M/ r2 rBHH T`Qp2 Bi 7Q` bQK2 2 ≤ j < n − 1X LQi2 i?i #v i?2 BM/m+iBQM ?vTQi?2b2b cj+1 = −cj−1 + 2cj + 2 = −j 2 + 2j 2 + 4j + 2 + 2 = j 2 + 4j + 4 = (j + 2)2 M/ bQ i?2 BM/m+iBQM Bb biBb}2/ 7Q` j = 0, . . . , n − 1X 6Q` j = n r2 ?p2 cn = 2 − 4n − (n − 1)2 + 2n2 = 2 − 4n − n2 + 2n − 1 + 2n2 = n2 − 2n + 1 = (n − 1)2 , M/ 7Q` j = n + 1 r2 ?p2 cn+1 = −cn−1 + 2cn + 2 = −(n − 1 + 1)2 + 2(n − 1)2 + 2 = (2n − (n + 1) − 1)2 . 6Q` j = n, . . . , 2n − 2 i?2 BM/m+iBQM ?vTQi?2b2b vB2H/b cj = (2n − j − 1)2 X LQi2 i?i Bi Bb +H2` 7Q` j = n M/ j = n + 1 M/ r2 rBHH b?Qr Bi 7Q` n + 1 < j ≤ 2n − 2X AM T`iB+mH` i?2 +b2b j = 2n − 3 M/ j = 2n − 2 `2 HbQ i`BpBHHv biBb}2/X aQ- r2 QMHv M22/ iQ b?Qr Bi 7Q` n + 1 < j < 2n − 3X "v i?2 BM/m+iBQM ?vTQi?2b2b 7Q` Mv n + 1 < j < 2n − 3 r2 ?p2 cj = 2 − cj−2 + 2cj−1 = 2 − (2n − (j − 2) − 1)2 + 2(2n − (j − 1) − 1)2 = 2 − (2n − j + 1)2 + 2(2n − j)2 = (2n − j − 1)2 M/ i?2 BM/m+iBQM ?vTQi?2b2b ?QH/bX AM b?Q`i i?2 H2KK Bb T`Qp2/X  h?2 M2ti `2bmHi +QM+2`M T`QT2`iB2b Q7 i?2 7mM+iBQMb Ki (a) BMi`Q/m+2/ BM a2+iBQM jX G2KK jX h?2 7QHHQrBM; bii2K2Mib ?QH/ 7Q` HH n ≥ 2, UV K6 (a) > K4 (a) 7Q` a ∈ (0, 1)c U#V K6 (a) < K4 (a) 7Q` a > 1c U+V K5 > K1 > 0X U/V a3 K4 (a) = K2 (1/a) M/ a3 K6 (a) = K3 (1/a)X S`QQ7X LQi2 i?i n 2πj  X a − cos + π n n aK6 (a) =  , π 3/2 j=1 1+ a2 − 2a cos 2πj n + n n  X a − cos 2πj n aK4 (a) = 3/2 . j=1 1+ a2 − 2a cos 2πj n q?2M a ∈ (0, 1)- mbBM; G2KK RU+V M/ S`QTQbBiBQM R9 rBi? α = 1/2 M/ u = 0 r2 ;2i Z ! # n 1 d t−1/2 1 1 − (at)2n aK4 (a) = − dt. π 0 da (1 − t)1/2 (1 − a2 t)1/2 1 + (at)2n − 2(at)n M/ mbBM; G2KK RU+V M/ S`QTQbBiBQM R9 rBi? α = 1/2 M/ u = π/n r2 ;2i Z ! # n 1 d t−1/2 1 1 − (at)2n aK6 (a) = − dt. π 0 da (1 − t)1/2 (1 − a2 t)1/2 1 + (at)2n + 2(at)n PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa kd >2M+2- 7Q` a ∈ (0, 1)- Z 1 ! # n d t−1/2 1 4(at)n K6 (a) − K4 (a) = − dt πa 0 da (1 − t)1/2 (1 − a2 t)1/2 ((at)2n − 1) Z 1 n 4(ta)n (ta2 (−1 + (ta)2n ) + n(ta2 − 1)(1 + (ta)2n )) =− dt > 0, πa 0 a(1 − t)1/2 t1/2 (1 − a2 t)3/2 ((ta)2n − 1)2 #2+mb2 i?2 BMi2;`M/ Bb M2;iBp2 7Q` a ∈ (0, 1)X h?2`27Q`2- K6 (a) > K4 (a) 7Q` 0 < a < 1 M/ bQ bii2K2Mi UV Bb T`Qp2/X 6Q` a > 1 mbBM; G2KK RU+V M/ S`QTQbBiBQM R8 rBi? α = 1/2 M/ u = 0 r2 ;2i Z ! # n 1 d t−1/2 1 a2n − t2n aK4 (a) = − dt. π 0 da (1 − t)1/2 (a2 − t)1/2 a2n + t2n − 2an tn M/ mbBM; G2KK RU+V M/ S`QTQbBiBQM R8 rBi? α = 1/2 M/ u = π/n M/ iFBM; /2`BpiBp2b rBi? `2bT2+i iQ a r2 ;2i Z ! # n 1 d t−1/2 1 a2n − t2n aK6 (a) = − dt. π 0 da (1 − t)1/2 (a2 − t)1/2 a2n + t2n + 2an tn >2M+2- 7Q` a > 1- Z ! # 1 n d t−1/2 1 4(at)n K6 (a) − K4 (a) = − dt πa 0 da (1 − t)1/2 (a2 − t)1/2 t2n − a2n Z 1 n 4(ta)n (a2 (a2n − t2n ) + n(a2 − t)(a2n + t2n )) =− dt < 0, πa 0 a(1 − t)1/2 t1/2 (a2 − t)3/2 (t2n − a2n )2 #2+mb2 i?2 BMi2;`M/ Bb TQbBiBp2 7Q` a > 1X h?2`27Q`2- K6 (a) < K4 (a) 7Q` a > 1 M/ bii2K2Mi U#V Bb T`Qp2/X hQ T`Qp2 bii2K2Mi U+V r2 T`Q+22/ b 7QHHQrbX LQi2 i?i n 1X 1 1 K5 = lim  1/2 := lim A0 , a→1 2 2 2πj π a→1 2 j=1 1 + a − 2a cos n + n n−1 1X 1 K1 = lim a→1 2 2 2πj 1/2 j=1 1 + a − 2a cos n   Xn 1 X 1 1 1 = lim  −  a→1 2 2 2πj 1/2 (1 + a 2 − 2a cos(2πj))1/2 j=11 + a − 2a cos n j=1 1 := lim (A1 − A2 ), a→1 2 h?mb 1 K5 − K1 = lim (A0 − A1 + A2 ) UkkV a→1 2 r?2`2 A0 - A1 - A2 `2 i?2 bmKKiBQMb /2}M2/ #Qp2X TTHvBM; S`QTQbBiBQM R9 r2 ?p2 i?i A0 − A1 + A2 r?2M a ∈ (0, 1) Bb ;Bp2M #v Z n 1 1 − (ta)n A= dt π 0 (1 − t)1/2 t1/2 (1 − a2 t)1/2 (1 + (ta)n ) Z n 1 1 + (ta)n − dt π 0 (1 − t)1/2 t1/2 (1 − a2 t)1/2 (1 − (ta)n ) Z 1 1 1 + ta + dt π 0 (1 − t) t (1 − a2 t)1/2 (1 − ta) 1/2 1/2 Z   n 1 4(ta)n 1 + ta 1 = 2n + dt. π 0 −1 + (ta) n(1 − ta) (1 − t) t (1 − a2 t)1/2 1/2 1/2 k3 JX *P_"1_ L. *X oGGa PM i?2 Qi?2` ?M/- TTHvBM; S`QTQbBiBQM R8 r2 ?p2 i?i A0 − A1 + A2 r?2M a > 1 Bb ;Bp2M #v Z   n 1 4(ta)n a+t 1 A= − 2n + dt. π 0 a − t2n n(a − t) (1 − t)1/2 t1/2 (a2 − t)1/2 7i2` /QBM; i?2 bm#biBimiBQM a → 1/a- i?2 2tT`2bbBQM A +M #2 r`Bii2M b a AX h?mb lim A = lim A = lim A0 − A1 + A2 . UkjV a→1− a→1+ a→1 LQr r2 b?Qr i?i A > 0X LQi2 i?i iFBM; v = ta r2 ;2i 4(ta)n 1 + ta 4v n 1 1+v + = + . −1 + (ta)2n n(1 − ta) −1 + v 2n n 1 − v lbBM; G2KK k r2 ;2i i?i 4v n 1 1+v (1 + v)(1 − v 2n ) − 4nv n (1 − v) + = > 0, −1 + v 2n n 1 − v n(1 − v)(1 − v 2n ) 7Q` v < 1X aQ A > 0 M/ iFBM; i?2 HBKBi r?2M a → 1− iQ;2i?2` rBi? UkjV M/ UkkV r2 ;2i lima→1− A = lima→1+ A = 2(K5 − K1 ) > 0 7Q` HH n ≥ 2X JQ`2Qp2` K1 Bb TQbBiBp2 #v /2}MBiBQMX aQ- r2 ?p2 T`Qp2/ bii2K2Mi U+VX aii2K2Mi U/V 7QHHQrb 7`QK /B`2+i +QKTmiiBQMbX  AM i?2 7QHHQrBM; H2KK r2 T`QpB/2 T`QT2`iB2b Q7 i?2 7mM+iBQM ∆(a) = −K1 −K5 +a3 (K2 (a)+K3 (a)) i?i TT2`b BM i?2 /2MQKBMiQ` mD BM U9VX G2KK 9X 6Q` HH n ≥ 2 i?2 7QHHQrBM; bii2K2Mib ?QH/X UV ∆(a) Bb BM+`2bBM; 7Q` HH a ∈ (0, 1)c U#V ∆(0) < 0- ∆(a) → ∞ r?2M a → 1− M/ ∆(a) → −∞ r?2M a → 1+ c U+V ∆(a) < 0 7Q` HH a > 1c U/V ∆(a) ?b  mMB[m2 x2`Q M/ Bi #2HQM;b iQ i?2 BMi2`pH (0, 1)X S`QQ7X q2 }`bi MQi2 i?i b2iiBM; b = 1/a r2 ?p2 n  3 1X b − cos 2πj n 1 a K2 (a) = 3/2 := b K̄2 (b), b 2 2πj j=1 1 + b − 2b cos Uk9V n n 2πj  3 1X b − cos n + πn 1 a K3 (a) =  3/2 := K̄3 (b). b 1 + b2 − 2b cos 2πj + π b j=1 n n G2i ∆(b) = (1/bK̄2 (b) + 1/bK̄3 (b))X q2 rMi iQ b?Qr i?i 7Q` a ∈ (0, 1)   ′ d∆(b) db ′ 1 ∆ (a) = · = (∆(b)) b=1/a · − 2 > 0. db b=1/a da a aQ- Bi Bb bm{+B2Mi iQ b?Qr i?i ∆(b)′ < 0 7Q` b > 1X G2i Xn Xn 1 1 T = 1/2 + 1/2 . 2 2πj 2 2πj π j=1 1 + b − 2b cos n j=1 1 + b − 2b cos n + n lbBM; G2KK RU+V r2 ;2i i?i K̄2 (b) + K̄3 (b) = −dT /db Uk8V M/ bQ (∆(b)) ¯ ′ = 1 T b2 1 − 1b T2 r?2`2 dT d2 T T1 = , T2 = . db db2 PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa kN aBM+2 7`QK S`QTQbBiBQM R8 r2 ?p2 Z   2n 1 d 1 t2n + b2n T1 = − dt π 0 db t1/2 (1 − t)1/2 (b2 − t)1/2 t2n − b2n Z   n 1 1 2b(t2n + b2n ) 8nt2n b2n−1 = − dt, π 0 t1/2 (1 − t)1/2 (b2 − t)3/2 (t2n − b2n ) (b2 − t)1/2 (t2n − b2n )2 iFBM; MQi?2` /2`BpiBp2 rBi? `2bT2+i iQ b r2 ;2i Z  n 1 1 −8nt2n b2n−2 ((2n − 1)t2n + (2n + 1)b2n ) T2 = π 0 t1/2 (1 − t)1/2 (b2 − t)1/2 (t2n − b2n )3    # 2 8nb2n b2 − t t2n + 2b2 + t b4n − t4n + dt. (b2 − t)5/2 (t2n − b2n )2 aBM+2 T1 < 0 Ui?2 BMi2;`M/ Bb M2;iBp2V M/ T2 > 0 Ui?2 BMi2;`M/ Bb TQbBiBp2V r2 `2/BHv Q#iBM i?i ¯ (∆(b)) ′ < 0 M/ bQ ∆′ (a) > 0 7Q` a ∈ (0, 1)X AM b?Q`i bii2K2Mi UV Bb T`Qp2/X Ai Bb +H2` i?i ∆(0) = −K1 − K5 < 0- b22 G2KK jU+VX JQ`2Qp2` n−1  X 1 − cos 2πjn 1−a lim K2 (a) =  3/2 + lim  , a→1 2πj a→1 1 + a2 − 2a 3/2 j=1 2 − 2 cos n bQ lim K2 (a) = ∞ M/ lim K2 (a) = −∞. UkeV a→1− a→1+ 6m`i?2`KQ`2 lima→1 K3 (a) = K5 r?B+? Bb /Bz2`2Mi 7`QK x2`Q M/ 7`QK BM}MBivX >2M+2- mbBM; UkeV r2 ;2i lim ∆(a) = ∞ M/ lim ∆(a) = −∞. a→1− a→1+ h?Bb +QKTH2i2b i?2 T`QQ7 Q7 bii2K2Mi U#VX LQr r2 b?Qr i?i ∆(a) < 0 7Q` a > 1X hQ /Q bQ- r2 rBHH b?Qr i?i a3 K2 (a) + a3 K3 (a) < 0X LQi2 i?i i?Bb Bb bm{+B2Mi #2+mb2 −K1 − K5 < 0X *H2`Hv BM pB2r Q7 Uk9V M/ Uk8V r2 ?p2 a3 K2 (a) + a3 K3 (a) = 1b (K̄2 (b) + K̄3 (b)) = − 1b dTdb rBi? b < 1X aQ- TTHvBM; S`QTQbBiBQM R9 r2 ;2i Z 1   1 dT 2n d 1 1 + (bt)2n − =− dt b db πb 0 db t1/2 (1 − t)1/2 (1 − b2 t)1/2 1 − (bt)2n Z  2n 1 4n(tb)2n =− πb 0 b t1/2 (1 − t)1/2 (1 − b2 t)1/2 (1 − (bt)2n )2  bt(1 + (bt)2n ) + 1/2 dt t (1 − t)1/2 (1 − b2 t)3/2 (1 − (bt)2n ) M/ bBM+2 i?2 BMi2;`M/ Bb TQbBiBp2 r2 `2/BHv ?p2 i?i a3 K2 (a) + a3 K3 (a) < 0 7Q` a > 1 M/ bQ ∆(a) < 0 7Q` a > 1X AM b?Q`i- bii2K2Mi U+V Bb T`Qp2/X h?2 T`QQ7 Q7 bii2K2Mi U/V Bb  /B`2+i +QMb2[m2M+2 Q7 i?2 bii2K2Mib UVĜU+V iQ;2i?2` rBi? i?2 "QHxMQ@*m+?v i?2Q`2KX  S`QQ7 Q7 S`QTQbBiBQM NX q2 2tTM/ m2 = a3 mN,2 /mD BM Gm`2Mi b2`B2b `QmM/ a = 0- b22 U9VX h?2 2tTMbBQM Q7 mD `QmM/ a = 0 Bb ;Bp2M #v mD = (K5 − K1 )(−K1 − K5 + O(a3 )). lbBM; G2KK RU/V rBi? u = 0 U`2bT2+iBp2Hv- u = π/nV iQ 2tTM/ K4 U`2bT2+iBp2Hv- K6 V BM Gm`2Mi b2`B2b `QmM/ a = 0 r2 ;2i n n K4 (a) = − + O(a) M/ K6 (a) = − + O(a) UkdV 2 2 r?2M n ≥ 3 M/ K4 (a) = −4 + O(a) M/ K6 (a) = 2 + O(a) Uk3V jy JX *P_"1_ L. *X oGGa r?2M n = 2X h?2`27Q`2- r?2M n ≥ 3 r2 ?p2 K12 + n2 K1 − K1 K5 − n2 K5 K1 + n2 m2 = a 3 + O(a4 ) = a3 + O(a4 ) (K5 − K1 )(−K1 − K5 ) K5 + K1 M/ bBM+2 K1 , K5 > 0 r2 Q#iBM lima→0+ m2 = 0+ X √ q?2M n = 2 r2 +QKTmi2 /B`2+iHv i?2 [mMiBiB2b K1 M/ K5 M/ r2 ?p2 K1 = 1/4- K5 = 1/ 2X aQ 2tTM/BM; m3 BM Gm`2Mi b2`B2b `QmM/ a = 0 r2 ;2i  17 √  m2 = −a3 + 2 2 + O(a4 ). 7 >2M+2- lima→0+ m2 = 0− X h?Bb +QKTH2i2b i?2 T`QQ7 Q7 bii2K2Mi UVX 6Q` bii2K2Mi U#V r2 MQi2 i?i #v U8V a3 (K6 (a) − K4 (a)) m3 = m2 + . K5 − K1 lbBM; i?2 Gm`2Mi b2`B2b Q7 K4 M/ K6 `QmM/ a = 0 ;Bp2M BM UkdV r2 ;2i a3 (K6 − K4 (a)) = O(a4 ) K5 − K1 r?2M n ≥ 3 M/ bQ K1 + n2 m3 = a3 + O(a4 ), K5 + K 1 r?B+? vB2H/b lima→0+ m3 = 0+ r?2M n ≥ 3X PM i?2 Qi?2` ?M/- 7Q` n = 2- mbBM; i?2 Gm`2Mi b2`B2b Q7 K4 M/ K6 `QmM/ a = 0 ;Bp2M BM Uk3V M/ i?2 pHm2b Q7 K1 M/ K5 7Q` n = 2 +QKTmi2/ #Qp2 r2 ;2i a3 (K6 − K4 (a)) a3 = (6 + O(a)) K5 − K1 K5 − K1 r?B+? vB2H/b a3 √ (7 + 34 2) + O(a4 ), m3 = 7 M/ bQ lima→0+ m3 = 0 . h?Bb +QKTH2i2b i?2 T`QQ7 Q7 bii2K2Mi U#VX + q2 2tTM/ m3 = a3 mN,3 /mD BM Gm`2Mi b2`B2b `QmM/ a = 1 UrBi? a > 1V- b22 U9VX LQi2 i?i 1 K2 (a) = K1 − + O(a − 1), K3 (a) = K5 + O(a − 1), (a − 1)2 1 1 K4 (a) = K1 + 2 − + 1 + O(a − 1), K6 = K5 + O(a − 1), (a − 1) a−1 M/ a3 = 1 + O(a − 1)X >2M+2- 1 (K5 − K1 ) mN,3 = + O((a − 1)−3 ) M/ mD = − + O((a − 1)−1 ). (a − 1)4 (a − 1)2 h?2`27Q`2- 1 m3 = − + O((a − 1)−1 ). (a − 1)2 (K5 − K1 ) aBM+2 BM pB2r Q7 G2KK jU+V r2 ?p2 K5 > K1 > 0- i?2M lima→1+ m3 = −∞X r?B+? +QKTH2i2b i?2 T`QQ7 Q7 bii2K2Mi U+VX 6Q` bii2K2Mi U/V- MQi2 i?i iFBM; b = 1/a M/ mbBM; G2KK jU/V r2 ?p2 K2 (a) = K2 (1/b) = b3 K4 (b)- K3 (a) = K3 (1/b) = b3 K6 (b)- K4 (a) = K2 (1/a)/a3 = b3 K2 (b) M/ K6 (a) = K3 (1/a)/a3 = b3 K3 (b)X h?mb- mbBM; G2KK RU/V 7Q` n ≥ 3 r2 ?p2 n n K4 (b) = − + O(b), K6 (b) = − + O(b). 2 2 JQ`2Qp2` 2tTM/BM; BM TQr2` b2`B2b `QmM/ b = 0 r2 ;2i K2 (b) = n + O(b), K3 (b) = n + O(b). PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa jR >2M+2 7Q` n ≥ 3 n n K2 (b) = − b3 + O(b4 ), K3 (b) = − b3 + O(b4 ), 2 2 UkNV K4 (b) = nb3 + O(b4 ), K6 (b) = nb + O(b4 ), 3 S`Q+22/BM; BM i?2 bK2 rv 7Q` n = 2 r2 ;2i K2 (b) = −4b3 + O(b4 ), K3 (b) = 2b3 + O(b4 ), UjyV K4 (b) = 2b3 + O(b4 ), K6 (b) = 2b3 + O(b4 ). lbBM; UkNV M/ UjyV i?2 MmK2`iQ`b Q7 m2 M/ m3 Ub22 U9VV +M #2 r`Bii2M b 1 (K 2 − K1 K5 ) + O(1), b3 1 M/ i?2 /2MQKBMiQ`b Q7 m2 M/ m3 Ub22 ;BM U9VV +M #2 r`Bii2M b (K5 − K1 )(−K1 − K5 − n) + O(b) 7Q` HH n ≥ 2X h?mb i?2 Gm`2Mi 2tTMbBQM Q7 m2 M/ m3 `QmM/ b = 0 #2+QK2b U7i2` bBKTHB7vBM; K5 − K 1 V 1 K1 3 + O(b−2 ). b K1 + K5 + n h?2M iFBM; BMiQ ++QmMi i?i BM pB2r Q7 G2KK jU+V r2 ?p2 K5 > K1 > 0- r2 +QM+Hm/2 i?i lim m2 = lim m2 = ∞, lim m3 = lim m3 = ∞. a→∞ b→0+ a→∞ b→0+ h?Bb T`Qp2b bii2K2Mi U/VX 6`QK G2KK jUV r2 ;2i K6 − K4 (a) > 0 7Q` a ∈ (0, 1)X 6`QK G2KK jU#V r2 ;2i K6 − K4 (a) < 0 7Q` a > 1 M/ 7`QK G2KK jU+V r2 ;2i K5 − K1 > 0X h?mb 7`QK U8V r2 ;2i m2 < m3 r?2M a ∈ (0, 1) M/ m2 > m3 r?2M a > 1- r?B+? T`Qp2b bii2K2Mi U2VX  TT2M/Bt kX S`QQ7 Q7 S`QTQbBiBQM RR q2 M22/ i?2 7QHHQrBM; mtBHB`v H2KKX G2KK 8X G2i [n/2]−1   X −2(1 + a2 ) cos 2π j + π + 5a − a cos 4π j + 2π n n n n E(a) :=   5/2 . j=0 a 1 + a2 − 2a cos 2πn j + πn h?2M 1 lim N4 (a, β)(K1 − 2a3 L2 (a, β) + L5 (β)) = E(a) β→π/n 4 B7 n Bb 2p2M M/   1 1 lim N4 (a, β)(K1 − 2a3 L2 (a, β) + L5 (β)) = E(a) + β→π/n 4 a(1 + a)3 B7 n Bb Q//X S`QQ7X 6B`bi MQi2 i?i lim N4 (a, β)(K1 − 2a3 L2 (a, β)) = 0 β→π/n #2+mb2  n X 2π j π −1/a sin n + n lim N4 (a, β) =  3/2 = 0 β→π/n 2π j π j=1 1+ a2 − 2a cos n + n M/  n X 2π j π 1 − a cos n − n lim L2 (a, β) =  3/2 β→π/n 2π j π j=1 1 + a2 − 2a cos n − n jk JX *P_"1_ L. *X oGGa Bb }MBi2 7Q` HH a > 0X PM i?2 Qi?2` ?M/-   n−2 2π j  2π (n−1)   X 1 − cos n + 2β  1 − cos + 2β n L5 (β) =    3/2  +  3/2 2π (n−1) j=0 2 − 2 cos 2π j + 2β 2 − 2 cos + 2β n n := L5,1 (β) + L5,2 (β). *H2`Hv limβ→π/n N4 (a, β)L5,1 (β) = 0X h?2`27Q`2- r2 M22/ iQ bim/v limβ→π/n N4 (a, β)L5,2 (β)X LQi2 i?i 2tTM/BM; L5,2 `QmM/ β = π/n r2 ?p2   1 − cos 2π(n−1) n + 2β 1 1 π π 2  L5,2 (β) =   =  + β − + O β − .  3/2 4 β − πn 24 n n 2 − 2 cos 2π(n−1) n + 2β PM i?2 Qi?2` ?M/- r2 +M `2r`Bi2 N4 (a, β) b   [n/2]−1 2π j  2π (−j−1)  X  −1/a sin n + β 1/a sin +β  n N̄4 (a, β) =    3/2 −   3/2  j=0 1 + a2 − 2a cos 2πn j + β 1 + a2 − 2a cos 2π (−j−1) n +β B7 n Bb 2p2M M/ b N̄4 (a, β) + N̄4∗ (a, β) rBi? 2π [n/2]  −1/a sin +β N̄4∗ (a, β) =  n 3/2 2π [n/2] 1 + a2 − 2a cos n +β B7 n Bb Q//X 1tTM/BM; N̄4 `QmM/ β = π/n r2 ;2i N̄4 (a, β) = E(a)(β − π/n) + O((β − π/n)2 ), M/ 2tTM/BM; N̄4∗ HbQ `QmM/ β = π/n r2 ;2i 1 N̄4∗ (a, β) = (β − π/n) + O((β − π/n)3 ). a(1 + a)3 h?mb- 2tTM/BM; N4 (a, β)L5,2 (β) `QmM/ β = π/n r2 Q#iBM 1 N4 (a, β)L5,2 (β) = E(a) + O((β − π/n)) 4 B7 n Bb 2p2M M/   1 1 N4 (a, β)L5,2 (β) = E(a) + + O((β − π/n)) 4 a(1 + a)3 B7 n Bb Q//X hFBM; i?2 HBKBi b β → π/n r2 Q#iBM i?2 `2bmHi i?i r2 rMi2/ iQ T`Qp2X  S`QQ7 Q7 S`QTQbBiBQM RRX LQi2 i?i n X 2π (j+1)  − sin n lim N5 (β) =  3/2 β→π/n 2π (j+1)  j=1,j6=n−1 2 − 2 cos n 2π (n−1)  − sin n + 2β + lim  β→π/n 2π (n−1) 3/2 = ∞. 2 − 2 cos n + 2β AM pB2r Q7 G2KK 8- Bi Bb +H2` i?i i?2 bB;M Q7 F (a, β) b β → π/n Bb /2i2`KBM2/ #v i?2 bB;M Q7 i?2 /Bz2`2M+2 G ;Bp2M #v G = lim (K1 − L4 (a, β)) = K1 − L̄4 (a), β→π/n r?2`2  n X 2π j π 1 − 1/a cos n + n L̄4 (a) = lim L4 (a, β) =  3/2 . β→π/n 2π j π j=1 1+ a2 − 2a cos n + n PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa jj q?2M a → ∞ r2 ?p2 lim G = K1 − lim L̄4 (a) = K1 > 0. a→∞ a→∞ h?Bb T`Qp2b bii2K2Mi UV Q7 i?2 T`QTQbBiBQMX 6m`i?2`KQ`2- r?2M a → 1- mbBM; G2KK jU+V- r2 Q#iBM n  X 1 − cos 2πn j + πn lim G = K1 −  3/2 = K1 − K5 < 0. a→1 2π j π j=1 2 − 2 cos n + n aQ- bii2K2Mi U#V Bb T`Qp2/X lbBM; G2KK RU/V rBi? u = π/n- i?2 2tTMbBQM Q7 L̄4 `QmM/ a = 0 Bb ;Bp2M #v L̄4 (a) = −n/2 + O(a), r?2M n ≥ 3 M/ L̄4 (a) = 2 + O(a), r?2M n = 2X >2M+2- r?2M n ≥ 3 i?2 2tTMbBQM Q7 G `QmM/ a = 0 Bb G = K1 + n/2 + O(a) M/ bQ lima→0 G > 0. 6Q` n = 2- +QKTmiBM; i?2 pHm2 Q7 K1 - i?2 2tTMbBQM Q7 G `QmM/ a = 0 Bb G = 1/4 − 2 + O(a) M/ bQ lima→0 G = −7/4 < 0. h?Bb vB2H/b bii2K2Mi U+V Q7 i?2 T`QTQbBiBQMX  TT2M/Bt jX S`QQ7 Q7 S`QTQbBiBQMb Rk M/ Rj q2 M22/ i?2 7QHHQrBM; mtBHB`v H2KKX G2KK eX q2 ?p2 lim N5 (β)(K1 − L4 (a, β)) = 0. β→π/2n S`QQ7X 6B`bi MQi2 i?i n 2π j  X 1 − 1/a cos + π n 2n lim L4 (a, β) =  3/2 , β→π/2n 2π j π j=1 1+ a2 − 2a cos n + 2n r?B+? Bb }MBi2 7Q` HH a > 0X JQ`2Qp2`- Bi Bb 2bv iQ b22 i?i n  X − sin 2πn j + πn lim N5 (β) =  3/2 = 0 β→π/2n 2π j π j=1 2 − 2 cos n + n M/ bQ limβ→π/2n N5 (β)(K1 − L4 (a, β)) = 0, b r2 rMi2/ iQ T`Qp2X  S`QQ7 Q7 S`QTQbBiBQM RkX AM pB2r Q7 G2KK e i?2 bB;M Q7 F (a, β) `QmM/ β = π/2n Bb /2i2`KBM2/ #v i?2 bB;M Q7 N4 (a, β)(K1 −2a3 L2 (a, β)+L5 (β)) mMH2bb limβ→π/2n N4 (a, β)(K1 −2a3 L2 (a, β)+L5 (β)) = 0X 6`QK i?2 MHvbBb Q7 i?2 bB;M Q7 N4 (a, β)(K1 − 2a3 L2 (a, β) + L5 (β)) `QmM/ β → π/2n r2 rBHH b22 i?i limβ→π/2n N4 (a, β)(K1 − 2a3 L2 (a, β) + L5 (β)) 6= 0X AM pB2r Q7 S`QTQbBiBQM Ry r2 ?p2 i?i N4 (a, β) < 0 7Q` α ∈ (0, π/n) M/ bQ N4 (a, β) < 0X aQ- r2 M22/ iQ bim/v i?2 bB;M Q7 K1 − 2a3 L2 (a, β) + L5 (β) b β → π/2nX >2M+2- lim K1 − 2a3 L2 (a, β) + L5 (β) = K1 − 2a3 L̄2 (a) + K5 := H, β→π/2n r?2`2  n X 2π j π 1 − a cos n − 2n L̄2 (a) = lim L2 (a, β) =  3/2 . β→π/2n 2π j π j=1 1+ a2 − 2a cos n − 2n q?2M a → 0 r2 ?p2 i?i L̄2 (a) → nX aQ lima→0 H = K1 + K5 > 0 BM pB2r Q7 G2KK jU+VX h?Bb T`Qp2b bii2K2Mi UV Q7 i?2 T`QTQbBiBQMX hQ bim/v i?2 #2?pBQ` r?2M a → ∞ r2 }`bi Q#b2`p2 i?i KFBM; i?2 +?M;2 b = 1/a r2 ;2i a3 L̄2 (a) = L(b, π/2n) r?2`2 L Bb i?2 7mM+iBQM /2}M2/ BM G2KK RU/VX h?mb TTHvBM; G2KK RU/V rBi? u = π/2n M/ mbBM; G2KK jU+V r2 ?p2 lim H = K1 + n + K5 > 0, a→∞ r?2M n ≥ 3 M/  lim H = K1 − 2 2 − 6 cos2 π 4 + K5 = K1 + 2 + K5 > 0, a→∞ j9 JX *P_"1_ L. *X oGGa r?2M n = 2X h?Bb T`Qp2b bii2K2Mi U#VX 6BMHHv- iQ T`Qp2 bii2K2Mi U+V r2 M22/ iQ bim/v i?2 bBM; Q7 H r?2M a → 1X LQi2 i?i n−1 n 1X 1 X 1 lim H =  1/2 −  1/2 a→1 2 2π j 2π j π j=1 2 − 2 cos n j=1 2 − 2 cos n − 2n n 1X 1 +  1/2 . 2 2π j π j=1 2 − 2 cos n + n AM Q`/2` iQ bim/v i?2 bB;M Q7 lima→1 H r2 `2r`Bi2 lima→1 H b lima→1 H1 r?2`2 n 1 1X 1 1X 1 H1 =  1/2 − 2π j  (1 + a − 2a cos(2πj))1/2 2 2 2 j=1 1 + a2 − 2a cos n j=1 n X n 1 1X 1 −  1/2 +  1/2 . 2π j π 2 2π j π j=1 1 + a2 − 2a cos n − 2n j=1 1 + a2 − 2a cos n + n LQr TTHvBM; S`QTQbBiBQM R9 rBi? α = 1/2 M/ u = 0c n = 1- α = 1/2 M/ u = 0c α = 1/2 M/ u = −π/2nc M/ α = 1/2 M/ u = π/n `2bT2+iBp2Hv r2 ;2i Z   1 1 1 4(at)2n n 1 + at H1 = − dt π 0 t1/2 (1 − t)1/2 (1 − a2 t)1/2 1 − (at)4n 2(1 − at) a2iiBM; v = at M/ N = 2n M/ mbBM; G2KK k r2 ;2i 4(at)2n n 1 + at 4v N N2 1+v 4n − = 2N − 1 − (at) 2(1 − at) 1−v 2(1 − v) 4v N N (1 − v) − (1 + v)(1 − v 2N ) = < 0. (1 − v 2N )2(1 − v) h?2`27Q`2 lima→1− H1 < 0X TTHvBM; S`QTQbBiBQM R8 iQ H1 rBi? α = 1/2 M/ u = 0c n = 1- α = 1/2 M/ u = 0c α = 1/2 M/ u = −π/2nc M/ α = 1/2 M/ u = π/n `2bT2+iBp2Hv r2 ;2i Z   1 1 1 4(at)2n n a+t H̄1 = − dt. π 0 t1/2 (1 − t)1/2 (a2 − t)1/2 a4n − t4n 2(a − t) 7i2` /QBM; i?2 bm#biBimiBQM a → 1/a r2 ;2i i?i H̄1 +M #2 r`Bii2M b a H1 X h?mb lima→1+ H1 < 0X AM b?Q`i r2 ?p2 i?i H < 0 r?2M a → 1 M/ bii2K2Mi U+V Bb T`Qp2/X  S`QQ7 Q7 S`QTQbBiBQM RjX q2 bi`i T`QpBM; bii2Ki UV 7Q` n = 2X *QKTmiBM; /B`2+iHv i?2 [mMiBiB2b K1 - L2 (a, β)- L4 (a, β)- N4 (a, β)- L5 (β)- M/ N5 (β) 7Q`  }t2/ β M/ 2tTM/BM; F `QmM/ a = 0 r2 ;2i 1  F (a, β) = − tan β sec β 18 cos β + 6 cos(3β) + 17 cot3 β + 7 + O(a2 ). 16 aQ F̄ (β) < 0 7Q` HH β ∈ (π/2n, π/n) M/ n = 2X LQr r2 rBHH b?Qr i?i 7Q` Mv β ∈ (π/2n, π/n) M/ n ≥ 3 r2 ?p2 F̄ (β) > 0X q2 }t β M/ r2 2tTM/ F `QmM/ a = 0X 6B`bi- 2tTM/BM; N4 `QmM/ a = 0 r2 ;2i n n 1X 2π j  3X 4π j  N4 (a, β) = − sin +β − sin + 2β + O(a) a n 2 n j=1 j=1 M/ BM pB2r Q7 UR3V r2 ?p2 N4 (a, β) = O(a)X aQ 2tTM/BM; N4 (a, β)(K1 − 2a3 L2 (a, β) + L5 (β)) `QmM/ a = 0 r2 Q#iBM N4 (a, β)(K1 − 2a3 L2 (a, β) + L5 (β)) = O(a). PL auJJ1h_A* "A*A_*lG_ *1Lh_G *PL6A:l_hAPLa j8 PM i?2 Qi?2` ?M/- mbBM; G2KK RU/V rBi? u = β M/ n ≥ 3- i?2 2tTMbBQM Q7 L4 `QmM/ a = 0 Bb ;Bp2M #v L4 (a, β) = −n/2 + O(a)X h?2`27Q`2- `QmM/ a = 0 r2 ;2i n F (a, β) = (K1 + )N5 (β) + O(a). UjRV 2 h?2 bB;M Q7 N5 (β) 7Q` Mv β ∈ (0, π/n) rb bim/B2/ BM S`QTQbBiBQM Ry M/ r2 Q#iBM2/ i?i N5 (β) Bb M2;iBp2 B7 β ∈ (0, π/2n) M/ TQbBiBp2 B7 β ∈ (π/2n, π/n)X h?2`27Q`2- 7`QK UjRV r2 ?p2 i?i 7Q` Mv β ∈ (π/2n, π/n) M/ n ≥ 3- F̄ (β) > 0 r?B+? +QKTH2i2b i?2 T`QQ7 Q7 bii2K2Mi UVX LQr r2 +QMbB/2` i?2 +b2 BM r?B+? a → ∞X 6Bt2/ β ∈ (π/2n, π/n) r2 ?p2 i?i N5 (β) Bb TQbBiBp2X JQ`2Qp2`- lim L4 (a, β) = lim N4 (a, β) = 0. a→∞ a→∞ JFBM; i?2 +?M;2 b = 1/a r2 ;2i a3 L2 (a, β) = L(b, β)- r?2`2 L Bb i?2 7mM+iBQM /2}M2/ BM G2KK RU/VX h?mb- TTHvBM; G2KK RU/V rBi? u = β r2 ;2i i?i lima→∞ a3 L2 (a, β) = limb→0 −n/2+ O(b) r?2M n ≥ 3 M/ lima→∞ a3 L2 (a, β) = limb→0 2 − 6 cos2 (β) + O(b) r?2M n = 2X aQ- lim (K1 − 2a3 L2 (a, β) + L5 (β))N4 (a, β) = 0, a→∞ lim N5 (β)(K1 − L4 (a, β)) = N5 (β)K1 > 0, a→∞ 7Q` HH n ≥ 2X AM b?Q`i- 7Q` Mv β ∈ (π/2n, π) M/ n ≥ 2- F̃ (β) > 0X h?Bb +QM+Hm/2b i?2 T`QQ7 Q7 i?2 T`QTQbBiBQMX  +FMQrH2/;2K2Mib h?2 mi?Q`b rQmH/ HBF2 iQ i?MF i?2 MQMvKQmb `2pB2r2`b 7Q` i?2B` ?2HT7mH +QKK2Mib M/ bm;;2b@ iBQMbX h?2 }`bi mi?Q` Bb T`iBHHv bmTTQ`i2/ #v 61.1_fJAL1*P ;`Mi MmK#2`b JhJkyRe@ddkd3@S M/ SA.kyRN@Ry9e83:"@AyyX h?2 b2+QM/ mi?Q` Bb T`iBHHv bmTTQ`i2/ #v 6*hfSQ`im;H i?`Qm;? lA.fJhfy998NfkyRNX _272`2M+2b (R) "M;- .X- 1HK#bQmi- "X, _2T`2b2MiiBQMb Q7 +QKTH2t 7mM+iBQMb- K2Mb QM i?2 `2;mH` n@;QM M/ TTHB+iBQMb iQ ;`pBiiBQMH TQi2MiBHX CX S?vbX , Ji?X :2MX je- RR9j8ĜRR98y UkyyjVX (k) "``#ûb- 1X- *Q`b- CXJX, PM +2Mi`H +QM};m`iBQMb Q7 i?2 κn@#Q/v T`Q#H2KX CX Ji?XMHXTTHX9de- dkyĜdje UkyRNVX (j) GHB#`2- CX- J2HHQ- GX6X , h`BTH2 M/ [m/`mTH2 M2bi2/ +2Mi`H +QM};m`iBQMb 7Q` i?2 THM` M@#Q/v T`Q#H2KX S?vbX . kj3- 8ejĜ8dR UkyyNVX (9) *Q`#2`- JX- .2H;/Q- CX- GHB#`2- CX, PM i?2 2tBbi2M+2 Q7 +2Mi`H +QM};m`iBQMb Q7 T M2bi2/ M@;QMbX ZmHX h?2Q`v .vMX avbiX 3- k88Ĝke8 UkyyNVX (8) >;B?`- uX, *2H2biBH J2+?MB+b- pQHX R- +?TX jX h?2 JAh S`2bb- *K#`B/;2 URNdyV (e) >ûMQi- PX- >X- _Qmbb2m- *X, aTB/2`r2# +2Mi`H +QM};m`iBQMbX ZmHX h?2Q`v .vMX avbiX R3 RRj8ĜRRey UkyRNVX (d) >QTT2- _X, 1`r2Bi2`mM; /2` #2FMMi2M aT2+BHHbmM; /2b .`2BFT2`T`Q#H2KbX `+?BpX /2` Ji?X S?vbX e9- kR3Ĝkkj UR3dNVX (3) EH2KT2`2`- qX"X, aQK2 T`QT2`iB2b Q7 `Qb2ii2 +QMƉ;m`iBQMb Q7 ;`pBiiBM; #Q/B2b BM ?QKQ;`T?B+ 2[mBHB#`BmKX bi`QMX CX ed- Rek@Red URNekVX (N) GQM;H2v- qX_X, aQK2 T`iB+mH` bQHmiBQMb BM i?2 T`Q#H2K Q7 nĜ#Q/B2bX K2`X Ji?X aQ+X Rj- jk9Ĝjj8 URNydVX (Ry) J`+?2bBM- JX,  7KBHv Q7 i?`22 M2bi2/ `2;mH` TQHv;QM +2Mi`H +QM};m`iBQMbX bi`X aT+2 a+B2M+2 je9- Rey UkyRNVX (RR) JQ2+F2H- _X- aBKQ- *X, "B7m`+iBQMb Q7 bTiBH +2Mi`H +QM};m`iBQMb 7`QK THM` QM2bX aAJ CX Ji?X MHX ke- Nd3ĜNN3 URNN8VX (Rk) JQMiH/B- CX, 1tBbi2M+2 Q7 bvKK2i`B+ +2Mi`H +QM};m`iBQMbX *2H2biX J2+?X .vMX bi`QMX Rkk- 9y8Ĝ9R3 UkyR8VX (Rj) aBHmbxvF- X,  M2r +2Mi`H +QM};m`iBQM BM i?2 THM` N Ĝ#Q/v T`Q#H2KX *`Ti?BM CX Ji?X jy- 9yRĜ9y3 UkyR9VX (R9) aBHmbxvF- X, PM  +Hbb Q7 +2Mi`H +QM};m`iBQMb BM i?2 THM` jM@#Q/v T`Q#H2KX Ji?X *QKTmiX a+BX RR- 98dĜ9ed UkyRdVX (R8) hm+F2`- q, oHB/i2/ MmK2`B+b,  b?Q`i BMi`Q/m+iBQM iQ `B;Q`Qmb +QKTmiiBQMbX S`BM+2iQM lMBp2`bBiv S`2bb- S`BM+2@ iQM- LCX UkyRRVX (Re) qBMiM2`- X, h?2 MHviB+H 7QmM/iBQMb Q7 +2H2biBH K2+?MB+bX S`BM+2iQM Ji?X a2`B2b 8- S`BM+2iQM lMBp2`bBiv S`2bb- S`BM+2iQM- LC URN9RVX (Rd) um- sX- w?M;- aX, hrBbi2/ M;H2b 7Q` +2Mi`H +QM};m`iBQMb 7Q`K2/ #v irQ irBbi2/ `2;mH` TQHv;QMbX CX .BzX 1[X k8j- kRyeĜkRkk UkyRkVX je JX *P_"1_ L. *X oGGa (R3) um- sX- w?M;- aX, *2Mi`H +QM};m`iBQMb 7Q`K2/ #v irQ irBbi2/ `2;mH` TQHv;QMbX CX Ji?X MHX TTHX 9k8- jdkĜj3y UkyR8VX (RN) w?Q- 6X- *?2M- CX, *2Mi`H +QM};m`iBQMb 7Q` (pN + gN )@#Q/v T`Q#H2KbX *2H2biBH J2+?X .vMKX bi`QMQKX RkR- RyRĜRye UkyR8VX (ky) w?M;- aX- w?Qm- ZX, S2`BQ/B+ bQHmiBQMb 7Q` i?2 kL@#Q/v T`Q#H2KbX S`Q+X KX Ji?X aQ+X RjR- kReRĜkRdy UkyyjVX 6+mHii /2 *BĐM+B2b B h2+MQHQ;BX lMBp2`bBii /2 oB+ @ lMBp2`bBii *2Mi`H /2 *iHmMv UloB+@ l**V- *X /2 H Gm`-Rj- y38yy oB+- aTBM 1KBH //`2bb, KQMib2``iX+Q`#2`!mpB+X+i .2T`iK2MiQ /2 Ji2K£iB+- AMbiBimiQ amT2`BQ` hö+MB+Q- lMBp2`bB//2 /2 GBb#Q- Ry9N@yyR GBb#Q- SQ`im;H 1KBH //`2bb, +pHHb!Ki?XBbiXmiHXTi