A edge cld expression
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Transcript of A edge cld expression
Normalization and Compactification of the EDGE CLD expressions
SetDirectory@"êUsersêsalvinoêDesktopêSTOCHASTICSêOCTAHEDRON"D;Directory@D
êUsersêsalvinoêDesktopêSTOCHASTICSêOCTAHEDRON
ü the final expressions for the edge case (i.e. gE" (r), or eqs. (44)-(47)) (they are worked out below this yellow block )
GEaa@r_D :=J2 2 - p + aN Csc@aD
18 p V-
J18 + J-9 + 7 3 N pN Csc@aD
216 2 p V* r;
GEbb@r_D :=1
36 3 V * r^3-
1
6 3 V * r+4 + 2 * Hp + aL
24 p V-
J18 - J9 - 13 3 N * pN * r
288 p V;
GEcc@r_D :=1
1296 p r3 V * Sin@aD
4 6 p + 144 2 r3 - 54 2 r4 + 27 2 p r4 - 30 6 p r4 + 36 2 -3 + 4 r2 - 150 2 r2
-3 + 4 r2 + 72 r3 a - 96 6 r2 ArcSinB1
2 -2 + 3 r2F - 8 6 ArcSinB
-7 + 9 r2
2 I-2 + 3 r2M3ê2F +
144 r3 ArcSinBr
-6 + 9 r2F + 18 6 r4 ArcSinB
3 I27 - 90 r2 + 96 r4 - 34 r6 + 2 r8M
2 r7 -2 + 3 r2F ;
GEdd@r_D := -1
432 p r3 V * Sin@aD-16 2 + 24 2 r2 + 8 6 p r2 - 12 p r3 -
9 2 p r4 + 12 6 p r4 + 8 2 -1 + r2 - 20 2 r2 -1 + r2 +
36 2 r4 ArcSinB-1 + r2
rF - 24 r3 ArcSinB
4 + 4 r2 - 7 r4
I2 - 3 r2M2F -
16 6 r2 ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F - 24 6 r4 ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F ;
CHECKS
LimitBGEdd@rD, r Ø 2 , Direction Ø 1F
FullSimplify@TrigToExp@HSimplify@Limit@GEcc@rD, r Ø 1, Direction Ø 1D - Limit@GEdd@rD, r Ø 1, Direction Ø -1DDL ê.8a Ø ArcCos@-1 ê 3D<DD
SimplifyBJSimplifyBLimitBGEcc@rD, r Ø 3 í 2, Direction Ø -1F -
LimitBGEbb@rD, r Ø 3 í 2, Direction Ø 1FFN ê. 8a Ø ArcCos@-1 ê 3D<F
SimplifyBJSimplifyBLimitBGEbb@rD, r Ø 2 ê 3 , Direction Ø -1F -
LimitBGEaa@rD, r Ø 2 ê 3 , Direction Ø 1FFN ê. 8a Ø ArcCos@-1 ê 3D<F
FullSimplifyBJ2 2 - p + aN Csc@aD
18 p Vê. :a Ø ArcCos@-1 ê 3D, V Ø 2 í 3>F
a = ArcCos@-1 ê 3D; V = 2 í 3;
plfnlaa = PlotBGEaa@rD, :r, 0, 2 ê 3 >, PlotRange Ø ::0, 2 + 0.05>, 8-0.003, 0.066<>,
AxesLabel Ø 9"r", "gE"HrL"=, PlotStyle Ø 8Blue, [email protected]<F;
plfnlbb = PlotBGEbb@rD, :r, 2 ê 3 , 3 í 2>, PlotStyle Ø 8Magenta, [email protected]<F;
plfnlcc = PlotBGEcc@rD, :r, 3 í 2, 1>, PlotStyle Ø 8Green, [email protected]<F;
plfnldd = PlotBGEdd@rD, :r, 1, 2 >, PlotStyle Ø 8Red, [email protected]<F;
plfnlE = Show@plfnlaa, plfnlbb, plfnlcc, plfnlddDClear@aD; Clear@VD;
0.2 0.4 0.6 0.8 1.0 1.2 1.4r
0.01
0.02
0.03
0.04
0.05
0.06
gE"HrL
ü normalization factor for the edge case
d1 = 2 ê 3 ; d2 = 3 í 2; d3 = 1; d4 = 2 ; V = 2 í 3; b = p ê 3; a = ArcCos@-1 ê 3D;
NormE := -1 ê Hp * V * Sin@aDL;
2 A-Edge_CLD_Expression.nb
ü some algebraic and trigonometric identities to be used later
: -2 + 3 r2 Ø D1 * 3 , -3 + 4 r2 Ø 2 * D2, -1 + r2 Ø D3>
:D1 Ø -2 + 3 r2 ì 3 , D2 Ø -3 + 4 r2 ì 2, D3 Ø -1 + r2 >
:ArcSec@-3D Ø a, ArcSecB 3 F Ø a ê 2, ArcCotB 2 F Ø Hp - aL ê 2, ArcTanB2 2 F Ø Hp - aL>
H* if 32
<r<1 *L
:H* if 3
2<r<1 *L
ArcCosB-1 + 3 -3 + 4 r2
4 -2 + 3 r2F Ø -ArcCosB
1 + 3 -3 + 4 r2
4 -2 + 3 r2F + ArcCosB-
D12 - 2 D22
D12F ,
ArcSecB -8 + 12 r2 F Ø ArcCosB1
2 * 3 * D1F>
: -1 + 2 r2 - -3 + 4 r2 Ø1 - -3 + 4 r2
2,
-I-3 + 4 r2M 1 - 2 r2 + -3 + 4 r2 Ø
-3 + 4 r2 * 1 - -3 + 4 r2
2,
-1 + 2 r2 + -3 + 4 r2 Ø1 + -3 + 4 r2
2,
I-3 + 4 r2M -1 + 2 r2 + -3 + 4 r2 Ø
-3 + 4 r2 * 1 + -3 + 4 r2
2,
-3 + 5 r2 + 2 r -3 + 4 r2 Ø r + -3 + 4 r2 ,
-3 + 5 r2 - 2 r -3 + 4 r2 Ø r - -3 + 4 r2 >
:ArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F Ø ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F + 2 * ArcSinB
1
23
-3 + 4 r2
-2 + 3 r2F ,
,
A-Edge_CLD_Expression.nb 3
ArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F Ø p - 2 ArcSinB
1
2 -2 + 3 r2F + ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F ,
ArcSinBr + 2 -3 + 4 r2
3 -2 + 3 r2F Ø ArcSinB
r - 2 -3 + 4 r2
3 -2 + 3 r2F + p - 2 * ArcSinB
r
3 I-2 + 3 r2M
F ,
ArcSinB3 + -3 + 4 r2
4 rF Ø ArcSinB
3 - -3 + 4 r2
4 rF + ArcSinB
3 -3 + 4 r2
2 r2F >
: ArcSinB1
2
9 - 12 r2
2 - 3 r2F - 2 * ArcSinB
1
2 -2 + 3 r2F Ø ArcSinB
-7 + 9 r2
2 I-2 + 3 r2M3ê2F - p ê 2 ,
-3 ArcSinB1
2
9 - 12 r2
2 - 3 r2F - 2 ArcSinB
1
2 -2 + 3 r2F + 5 ArcSinB
-3 + 4 r2
2 rF +
ArcSinB-9 + 12 r2
2 r2F Ø ArcSinB
3 I27 - 90 r2 + 96 r4 - 34 r6 + 2 r8M
2 r7 -2 + 3 r2F -
p
2>
:ArcCosB-1 + 3 -3 + 4 r2
4 -2 + 3 r2F Ø ArcCosB
5 - 6 r2
4 - 6 r2F - ArcCosB
1 + 3 -3 + 4 r2
4 -2 + 3 r2F,
ArcSecB -8 + 12 r2 F Ø ArcCosB1
2 -2 + 3 r2F >
:ArcTanB4 - 6 r2 + 3 r -3 + 4 r2
2F Ø
ArcTanB4 - 6 r2 - 3 r -3 + 4 r2
2F + p - 2 * ArcSinB
r
-6 + 9 r2F >
H* if 32
<r<1 *L
: H* if 3
2<r<1 *L
2 - 5 r2 + 3 r4 Ø r^2 - 1 * 3 * r^2 - 2 ,
,
4 A-Edge_CLD_Expression.nb
ArcTanB1 - 3 -3 + 4 r2
3 1 + -3 + 4 r2F Ø ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F,
ArcTanB1 + 3 -3 + 4 r2
3 1 - -3 + 4 r2F Ø ArcSinB
1 + 3 -3 + 4 r2
4 -2 + 3 r2F,
ArcSinBr + 2 -3 + 4 r2
3 -2 + 3 r2F Ø ArcSinB
r - 2 -3 + 4 r2
3 -2 + 3 r2F + 2 * ArcSinB
2
3
-3 + 4 r2
-2 + 3 r2F ,
ArcSinB3 - -3 + 4 r2
4 rF Ø ArcSinB
3 + -3 + 4 r2
4 rF + ArcSinB-
3 -3 + 4 r2
2 r2F >
H* 1 < r < 2 *L
: H* 1 < r < 2 *L ArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F Ø ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F +
2 * p
3,
ArcCscB -2 + 3 r2 F Ø -ArcSinB-1 + 3 -1 + r2
2 -2 + 3 r2F + p ê 3 ,
ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F Ø -ArcSinB
1
-2 + 3 r2F + 2 * p ê 3
H* attention to the factors in RED !! *L>
PlotBArcTanB1 - 3 -3 + 4 r2
3 1 + -3 + 4 r2F - ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F, :r, 3 í 2, 1>F
PlotBArcTanB1 + 3 -3 + 4 r2
3 1 - -3 + 4 r2F - ArcSinB
1 + 3 -3 + 4 r2
4 -2 + 3 r2F, :r, 3 í 2, 1>F
ü 0 < r < 2 ê3Simplify@GEAAaa@rD - Simplify@HNormE * HFaEcldAAold@rDL ê. 8ArcSec@-3D Ø a<LDD
GEAAaa@r_D :=H4 H1 + aL - 3 pL Csc@aD
36 p V-
J9 + 2 3 pN Csc@aD
216 2 p V* r;
A-Edge_CLD_Expression.nb 5
Clear@aD; FullSimplifyB
CoefficientListBJNormE * HFbEcldAold@rDL ê. :ArcSec@-3D Ø a, ArcSecB 3 F Ø a ê 2,
ArcCotB 2 F Ø Hp - aL ê 2, ArcTanB2 2 F Ø Hp - aL>N, rFF
SimplifyBJNormE * HFbEcldAold@rDL ê. :ArcSec@-3D Ø a, ArcSecB 3 F Ø a ê 2,
ArcCotB 2 F Ø Hp - aL ê 2, ArcTanB2 2 F Ø Hp - aL>N - GEBBaa@rDF
GEBBaa@r_D :=J-4 + 4 2 + p - 2 aN Csc@aD
36 p V-
J9 + J-9 + 5 3 N pN Csc@aD
216 2 p V* r;
Simplify@GEaa@rD - HGEAAaa@rD + GEBBaa@rDLD
GEaa@r_D :=J2 2 - p + aN Csc@aD
18 p V-
J18 + J-9 + 7 3 N pN Csc@aD
216 2 p V* r;
ü comparison with the Phys Rev result Pij(0+) = (-1/6pV)L[1-(p-b)Cotg(p-b)].This implies that Pii(0+)= - Pij(0+) = (1/6pV)L[1-(p-b)Cotg(p-b)].We have that L= 2 12 =24
This formula yields 246 p V
(1 - (p - a)Cotg(p-a)) = 246 p V
(1 - (p - a) Cos Hp-aLSinHp-aL
) = 246 p V
(1 + (p - a) CosHaLSinHaL
) =246 p V
(1 + (p - a) CosHaLSinHaL
) = 4p V
(1 + (p - a) CosHaLSinHaL
) = 4p V
(1 - (p - a) 1
2 2) = 4
p V (2 2 - (p - a)) 1
2 2 =
43 p V
(2 2 - (p - a)) 3
2 2= 43 p V
(2 2 - (p - a)) Csc@aD = 4 µ 18 p V3 p V
H2 Sqrt@2D- Hp - aLLCsc@aD18 p V
=24 GEaa[0].
But 24 is the factor present in eq. (2) of the ms. Hence the result is correct.
ü 2 ê3 < r < 3 í2SimplifyB
JJHNormE * FaEcldBBold@rDL ê. :ArcSec@-3D Ø a, ArcSecB 3 F Ø a ê 2, ArcCotB 2 F Ø Hp - aL ê 2,
ArcTanB2 2 F Ø Hp - aL, ArcCotB2 2 F Ø Ha - p ê 2L>N ê.
:Csc@aD Ø 3 í J2 * 2 N>N - GEAAbb@rDF
GEAAbb@r_D :=1
36 3 V r3-
1
6 3 V r+4 H1 + aL + p
24 2 p V-
J9 + 8 3 pN r
288 p V;
FullSimplify@TrigToExp@HSimplify@NormE * FbEcldBold@rD - GEBBbb@rDDL ê. 8a Ø ArcCos@-1 ê 3D<DD
GEBBbb@r_D :=J-4 + 4 2 + p - 2 aN Csc@aD
36 p V-
J9 + J-9 + 5 3 N pN Csc@aD
216 2 p V* r;
6 A-Edge_CLD_Expression.nb
SimplifyBHCoefficientList@FullSimplify@r^3 * HGEAAbb@rD + GEBBbb@rDLD, rDL ê.
:Csc@aD Ø 3 í J2 * 2 N>F
SimplifyBHGEAAbb@rD + GEBBbb@rD - GEbb@rDL ê. :Csc@aD Ø 3 í J2 * 2 N>F
GEbb@r_D :=1
36 3 V * r^3-
1
6 3 V * r+4 + 2 * Hp + aL
24 p V-
J18 - J9 - 13 3 N * pN * r
288 p V;
ü 3 í2 < r < 1FaEcldCCold@rD
H* 1st step *L
SimplifyBHFaEcldCCold@rDL ê. :ArcTanB1 - 3 -3 + 4 r2
3 1 + -3 + 4 r2F Ø ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F,
ArcTanB1 + 3 -3 + 4 r2
3 1 - -3 + 4 r2F Ø ArcSinB
1 + 3 -3 + 4 r2
4 -2 + 3 r2F>F
FactorBSimplifyB SimplifyB SimplifyB SimplifyB
HFaEcldCCold@rDL ê. :ArcTanB1 - 3 -3 + 4 r2
3 1 + -3 + 4 r2F Ø ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F,
ArcTanB1 + 3 -3 + 4 r2
3 1 - -3 + 4 r2F Ø ArcSinB
1 + 3 -3 + 4 r2
4 -2 + 3 r2F>F ê.
:ArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F Ø ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F +
2 * ArcSinB1
23
-3 + 4 r2
-2 + 3 r2F >F ê. :ArcSinB
r + 2 -3 + 4 r2
3 -2 + 3 r2F Ø
ArcSinBr - 2 -3 + 4 r2
3 -2 + 3 r2F + 2 * ArcSinB
2
3
-3 + 4 r2
-2 + 3 r2F >F ê.
:ArcSinB3 - -3 + 4 r2
4 rF Ø ArcSinB
3 + -3 + 4 r2
4 rF + ArcSinB-
3 -3 + 4 r2
2 r2F >FF
A-Edge_CLD_Expression.nb 7
1
5184 r3-32 6 p + 192 6 p r2 - 576 r3 - 675 p r3 + 108 2 r4 + 96 6 p r4 - 48 2 -3 + 4 r2 +
240 2 r2 -3 + 4 r2 + 396 r3 ArcCotB 2 F - 90 r3 ArcCotB2 2 F + 1152 r3 ArcSinB6 - 8 r2
6 - 9 r2F +
64 6 ArcSinB1
2
9 - 12 r2
2 - 3 r2F - 384 6 r2 ArcSinB
1
2
9 - 12 r2
2 - 3 r2F -
144 6 r4 ArcSinB1
2
9 - 12 r2
2 - 3 r2F -72 6 r4 ArcSinB
-9 + 12 r2
2 r2F + 288 r3 ArcTanB2 2 F
TogetherBFullSimplifyBTrigExpandBCosBArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F - ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2FFF,
Assumptions Ø : 3 í 2 < r < 1>FF
TogetherBFullSimplifyB
SinBArcCosB5 - 6 r2
2 I-2 + 3 r2MF ì 2F -
1
23
-3 + 4 r2
-2 + 3 r2, Assumptions Ø : 3 í 2 < r < 1>FF
R0 = 3 í 2; Step = H1 - R0L ê 21; DoBr = R0 + J * Step; val =
NBArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F - ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F + 2 * ArcSinB
1
23
-3 + 4 r2
-2 + 3 r2F , 30F;
Print@J, PaddedForm@val, 810, 8<DD;, 8J, 1, 20<F; Clear@rD;
R0 = 3 í 2; Step = H1 - R0L ê 21; DoBr = R0 + J * Step; val =
NBArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F - p - 2 ArcSinB
1
2 -2 + 3 r2F + ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F , 30F;
Print@J, PaddedForm@val, 810, 8<DD;, 8J, 1, 20<F; Clear@rD;
PlotBArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F -
p - 2 ArcSinB1
2 -2 + 3 r2F + ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F , :r, 3 í 2, 1>F
8 A-Edge_CLD_Expression.nb
PlotBArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F -
ArcSinB1 - 3 -3 + 4 r2
4 -2 + 3 r2F + 2 * ArcSinB
1
23
-3 + 4 r2
-2 + 3 r2F , :r, 3 í 2, 1>F
H* 2nd step *LSimplifyB
SimplifyBHFaEcldCCold@rDL ê. :ArcTanB1 - 3 -3 + 4 r2
3 1 + -3 + 4 r2F Ø ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F,
ArcTanB1 + 3 -3 + 4 r2
3 1 - -3 + 4 r2F Ø ArcSinB
1 + 3 -3 + 4 r2
4 -2 + 3 r2F>F ê.
:ArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F Ø p - 2 ArcSinB
1
2 -2 + 3 r2F + ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F >F
1
5184 r3
16 6 I-4 + 24 r2 + 9 r4M ArcSinB1
2 -2 + 3 r2F + 3 -192 r3 - 225 p r3 + 36 2 r4 + 8 6 p r4 -
16 -6 + 8 r2 + 80 r2 -6 + 8 r2 + 132 r3 ArcCotB 2 F - 30 r3 ArcCotB2 2 F -
192 r3 ArcSinBr - 2 -3 + 4 r2
3 -2 + 3 r2F + 24 6 r4 ArcSinB
3 - -3 + 4 r2
4 rF -
24 6 r4 ArcSinB3 + -3 + 4 r2
4 rF + 192 r3 ArcSinB
r + 2 -3 + 4 r2
3 -2 + 3 r2F + 96 r3 ArcTanB2 2 F
PlotB: ArcSinBr + 2 -3 + 4 r2
3 -2 + 3 r2F - ArcSinB
r - 2 -3 + 4 r2
3 -2 + 3 r2F ì p, 1 ê 2>, :r, 3 í 2, 1>F
FullSimplifyBExpandAllB
TrigExpandBCosBArcSinBr + 2 -3 + 4 r2
3 -2 + 3 r2F - ArcSinB
r - 2 -3 + 4 r2
3 -2 + 3 r2FFFF, : 3 í 2 < r < 1>F
TrigExpandBSinBArcCosB-6 - 7 r2
-6 + 9 r2F ì 2FF
TogetherB 1 - -6 - 7 r2
-6 + 9 r2ì 2F
A-Edge_CLD_Expression.nb 9
PlotBArcSinBr + 2 -3 + 4 r2
3 -2 + 3 r2F -
ArcSinBr - 2 -3 + 4 r2
3 -2 + 3 r2F + p - 2 * ArcSinB
r
3 I-2 + 3 r2M
F , :r, 3 í 2, 1>F
H* 3rd step *LSimplifyB SimplifyB SimplifyB
HFaEcldCCold@rDL ê. :ArcTanB1 - 3 -3 + 4 r2
3 1 + -3 + 4 r2F Ø ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F,
ArcTanB1 + 3 -3 + 4 r2
3 1 - -3 + 4 r2F Ø ArcSinB
1 + 3 -3 + 4 r2
4 -2 + 3 r2F>F ê.
:ArcSinB1 + 3 -3 + 4 r2
4 -2 + 3 r2F Ø p - 2 ArcSinB
1
2 -2 + 3 r2F + ArcSinB
1 - 3 -3 + 4 r2
4 -2 + 3 r2F >F ê.
:ArcSinBr + 2 -3 + 4 r2
3 -2 + 3 r2F Ø ArcSinB
r - 2 -3 + 4 r2
3 -2 + 3 r2F + p - 2 * ArcSinB
r
3 I-2 + 3 r2M
F >F
PlotB: ArcSinB3 + -3 + 4 r2
4 rF - ArcSinB
3 - -3 + 4 r2
4 rF ì p, 1 ê 2>, :r, 3 í 2, 1>F
SimplifyB SimplifyBTrigExpandBSinBArcSinB3 + -3 + 4 r2
4 rF - ArcSinB
3 - -3 + 4 r2
4 rFFF,
Assumptions Ø : 3 í 2 < r < 1>F ê.
: -1 + 2 r2 - -3 + 4 r2 Ø1 - -3 + 4 r2
2, -1 + 2 r2 + -3 + 4 r2 Ø
1 + -3 + 4 r2
2,
I-3 + 4 r2M -1 + 2 r2 + -3 + 4 r2 Ø
-3 + 4 r2 * 1 + -3 + 4 r2
2>F
10 A-Edge_CLD_Expression.nb
FactorBSimplifyB
3 -3 + 4 r2 + 7 -3 + 4 r2 + 2 -I-3 + 4 r2M 1 - 2 r2 + -3 + 4 r2
16 r2ê.
: -I-3 + 4 r2M 1 - 2 r2 + -3 + 4 r2 Ø
-3 + 4 r2 * 1 - -3 + 4 r2
2>FF
PlotBArcSinB3 + -3 + 4 r2
4 rF -
ArcSinB3 - -3 + 4 r2
4 rF + ArcSinB
3 -3 + 4 r2
2 r2F , :r, 3 í 2, 1>F
H* 4th step *LSimplifyB
1
5184 r316 6 I-4 + 24 r2 + 9 r4M ArcSinB
1
2 -2 + 3 r2F + 3 -192 r3 - 33 p r3 + 36 2 r4 +
8 6 p r4 - 16 -6 + 8 r2 + 80 r2 -6 + 8 r2 + 132 r3 ArcCotB 2 F -
30 r3 ArcCotB2 2 F - 384 r3 ArcSinBr
-6 + 9 r2F + 24 6 r4 ArcSinB
3 - -3 + 4 r2
4 rF -
24 6 r4 ArcSinB3 + -3 + 4 r2
4 rF + 96 r3 ArcTanB2 2 F ê.
:ArcSinB3 + -3 + 4 r2
4 rF Ø ArcSinB
3 - -3 + 4 r2
4 rF + ArcSinB
3 -3 + 4 r2
2 r2F >F
A-Edge_CLD_Expression.nb 11
H* final step *L
SimplifyB
FullSimplifyBSimplifyB1
5184 r316 6 I-4 + 24 r2 + 9 r4M ArcSinB
1
2 -2 + 3 r2F + 3 -192 r3 -
33 p r3 + 36 2 r4 + 8 6 p r4 - 16 -6 + 8 r2 + 80 r2 -6 + 8 r2 +
132 r3 ArcCotB 2 F - 30 r3 ArcCotB2 2 F - 384 r3 ArcSinBr
-6 + 9 r2F -
24 6 r4 ArcSinB-9 + 12 r2
2 r2F + 96 r3 ArcTanB2 2 F ê.
:ArcSec@-3D Ø a, ArcSecB 3 F Ø a ê 2, ArcCotB 2 F Ø Hp - aL ê 2,
ArcTanB2 2 F Ø Hp - aL, ArcCotB2 2 F Ø Ha - p ê 2L>FF ê.
:ArcCosBr
-6 + 9 r2F Ø
p
2- ArcSinB
r
-6 + 9 r2F ,
ArcCscB2 -2 + 3 r2 F Ø
ArcSinB1
2 -2 + 3 r2F,
ArcCscB2 r2
-9 + 12 r2F Ø ArcSinB
-9 + 12 r2
2 r2F>F
FaEcldCCSimpl@r_D :=1
1296 r3-12 -6 + 8 r2 + 60 r2 -6 + 8 r2 -
36 r3 H4 + p + 4 aL + 4 6 I-4 + 24 r2 + 9 r4M ArcSinB1
2 -2 + 3 r2F +
144 r3 p - 2 ArcSinBr
-6 + 9 r2F + 3 2 r4 9 + 2 3 p - 6 3 ArcSinB
-9 + 12 r2
2 r2F ;
checks
12 A-Edge_CLD_Expression.nb
a = ArcCos@-1 ê 3D; ParametricPlotB
88r, Re@FaEcldCCold@rD - FaEcldCCSimpl@rDD<, 8r, Im@FaEcldCCold@rD - FaEcldCCSimpl@rDD<<,
:r, 3 í 2 + 1 ê 1000, 1 - 1 ê 1000>, PlotRange Ø :: 3 í 2, 1>, 8-10^H-14L, 10^H-14L<>,
PlotStyle Ø 88Red, [email protected]<, 8Blue, [email protected]<<, AspectRatio Ø 1F
Clear@aD
FullSimplifyBLimitBFaEcldCCold@rD - FaEcldCCSimpl@rD,
r Ø 3 í 2, Direction Ø -1F ê. 8a Ø ArcCos@-1 ê 3D<F
N@FullSimplify@Limit@FaEcldCCold@rD - FaEcldCCSimpl@rD, r Ø 1, Direction Ø 1D ê.8a Ø ArcCos@-1 ê 3D<DD
R0 = 3 í 2; Step = H1 - R0L ê 21; a = ArcCos@-1 ê 3D;
Do@r = R0 + J * Step; val = N@FaEcldCCold@rD - FaEcldCCSimpl@rD, 100D;Print@J, PaddedForm@val, 810, 8<DD;, 8J, 1, 20<D; Clear@rD; Clear@aD;
NormE * FaEcldCCSimpl@rD
SimplifyB1
648 p r3 V * Sin@aD*
3 * r^3 * J-9 2 r - 2 p J18 + 6 rN + 48 H1 + aLN
2-
12 * -3 + 4 r2 I-1 + 5 r2M
2- 2 6 I-4 + 24 r2 + 9 r4M ArcSinB
1
2 -2 + 3 r2F -
9 r3 16 ArcSinBr
-6 + 9 r2F + 6 r ArcSinB
-9 + 12 r2
2 r2F - NormE * FaEcldCCSimpl@rDF
0
GEAAcc@r_D :=1
648 p r3 V * Sin@aD*
3 * r^3 * J-9 2 r - 2 p J18 + 6 rN + 48 H1 + aLN
2-
12 * -3 + 4 r2 I-1 + 5 r2M
2- 2 6 I-4 + 24 r2 + 9 r4M ArcSinB
1
2 -2 + 3 r2F -
9 r3 16 ArcSinBr
-6 + 9 r2F + 6 r ArcSinB
-9 + 12 r2
2 r2F ;
V * FullSimplifyBJLimitBHNormE * FaEcldCCold@rDL - GEAAcc@rD, r Ø 3 í 2, Direction Ø -1FN ê.
8a Ø ArcCos@-1 ê 3D<F
V * FullSimplify@HLimit@HNormE * FaEcldCCold@rDL - GEAAcc@rD, r Ø 1, Direction Ø 1DL ê. 8a Ø ArcCos@-1 ê 3D<D
A-Edge_CLD_Expression.nb 13
ü the FB term
SimplifyB
HFbEcldCold@rDL ê. :ArcTanB4 - 6 r2 + 3 r -3 + 4 r2
2F Ø ArcTanB
4 - 6 r2 - 3 r -3 + 4 r2
2F +
p - 2 * ArcSinBr
-6 + 9 r2F > , Assumptions Ø : 3 í 2 < r < 1>F
SimplifyB
1
2592 r3-180 6 r4 ArcCscB
2 r
-3 + 4 r2F + 4 6 I4 + 27 r4M ArcSinBFactorB
1
2
9 - 12 r2
2 - 3 r2FF +
3 2 2 J9 - 9 p + 5 3 pN r4 + 96 r3 ArcSinBr
-6 + 9 r2F -
2 -3 + 4 r2 8 2 + 9 ArcSinB2
3F - 9 ArcTanB 2 F +
12 r2 -3 + 4 r2 5 2 - 4 ArcSinB2
3F + 4 ArcTanB 2 F - 3 r3 J-32 + 32 2 -
4 p + 72 ArcCotB 2 F + 11 ArcSec@-3D + 2 ArcSecB 3 F - 8 ArcTanB2 2 FN ê.
:ArcSec@-3D Ø a, ArcSecB 3 F Ø a ê 2, ArcCotB 2 F Ø Hp - aL ê 2,
ArcTanB2 2 F Ø Hp - aL,
ArcSinB2
3F Ø a ê 2,
ArcTanB 2 F Ø Ha ê 2L>F
FactorB3 -8 -6 + 8 r2 + 30 r2 -6 + 8 r2 F
SimplifyB3 J 2 J9 - 9 p + 5 3 pN r4 - 12 r3 J-4 + 4 2 + 3 p - 2 aNNF
FbEcldCSimpl@r_D :=1
1296 r33 r3 J 2 J9 + J-9 + 5 3 N pN r - 12 J-4 + 4 2 + 3 p - 2 aNN +
6 2 -3 + 4 r2 I-4 + 15 r2M - 90 6 r4 ArcSinB-3 + 4 r2
2 rF +
2 6 I4 + 27 r4M ArcSinB1
23
-3 + 4 r2
-2 + 3 r2F + 144 r3 ArcSinB
r
-6 + 9 r2F ;
Checks
14 A-Edge_CLD_Expression.nb
Checks
NBFullSimplifyBJLimitBFbEcldCSimpl@rD - FbEcldCold@rD, r Ø 3 í 2, Direction Ø -1FN ê.
8a Ø ArcCos@-1 ê 3D<F, 50F
N@FullSimplify@HLimit@FbEcldCSimpl@rD - FbEcldCold@rD, r Ø 1, Direction Ø 1DL ê.8a Ø ArcCos@-1 ê 3D<D, 50D
a = ArcCos@-1 ê 3D; PlotBFbEcldCSimpl@rD - FbEcldCold@rD, :r, 3 í 2, 1>F
Clear@aD
NormE *1
1296 r33 r3 J 2 J9 + J-9 + 5 3 N pN r - 12 J-4 + 4 2 + 3 p - 2 aNN +
6 2 -3 + 4 r2 I-4 + 15 r2M - 90 6 r4 ArcSinB-3 + 4 r2
2 rF +
2 6 I4 + 27 r4M ArcSinB1
23
-3 + 4 r2
-2 + 3 r2F + 144 r3 ArcSinB
r
-6 + 9 r2F ;
SimplifyBHNormE * FbEcldCSimpl@rD - GEBBcc@rDL ê. 8a Ø ArcCos@-1 ê 3D<,
Assumptions Ø : 3 í 2 < r < 1>F
0
GEBBcc@r_D := -1
1296 p r3 V * Sin@aD
6 2 -3 + 4 r2 I-4 + 15 r2M + 3 r3 J 2 J9 + J-9 + 5 3 N pN r - 12 J-4 + 4 2 + 3 p - 2 aNN -
90 6 r4 ArcSinB-3 + 4 r2
2 rF +
2 6 I4 + 27 r4M ArcSinB1
23
-3 + 4 r2
-2 + 3 r2F + 144 r3 ArcSinB
r
-6 + 9 r2F ;
SimplifyBGEAAcc@rD + GEBBcc@rD, Assumptions Ø : 3 í 2 < r < 1>F
A-Edge_CLD_Expression.nb 15
GeccNotSimpl@r_D :=1
1296 p r3 V * Sin@aD
-2 6 I4 + 27 r4M ArcSinB1
2
9 - 12 r2
2 - 3 r2F - 4 6 I-4 + 24 r2 + 9 r4M ArcSinB
1
2 -2 + 3 r2F +
3 48 2 r3 - 18 2 r4 + 9 2 p r4 - 7 6 p r4 + 12 -6 + 8 r2 -
50 r2 -6 + 8 r2 + 24 r3 a + 30 6 r4 ArcSinB-3 + 4 r2
2 rF +
48 r3 ArcSinBr
-6 + 9 r2F + 6 6 r4 ArcSinB
-9 + 12 r2
2 r2F ;
FullSimplifyBJLimitBGEbb@rD, r Ø 3 í 2, Direction Ø 1F -
LimitBGeccNotSimpl@rD, r Ø 3 í 2, Direction Ø -1FN ê. 8a Ø ArcCos@-1 ê 3D<F
0
PlotBArcTanB4 - 6 r2 + 3 r -3 + 4 r2
2F -
ArcTanB4 - 6 r2 - 3 r -3 + 4 r2
2F + p - 2 * ArcSinB
r
-6 + 9 r2F , :r, 3 í 2, 1>F
16 A-Edge_CLD_Expression.nb
ü simplification of GEccNotSimpl[r]
aaaaa = 144 2 r3 - 54 2 r4 + 27 2 p r4 - 21 6 p r4 + 36 -6 + 8 r2 - 150 r2 -6 + 8 r2 + 72 r3 a;
ExpandB -2 6 I4 + 27 r4M ArcSinB1
2
9 - 12 r2
2 - 3 r2F -
4 6 I-4 + 24 r2 + 9 r4M ArcSinB1
2 -2 + 3 r2F + 3 48 2 r3 - 18 2 r4 + 9 2 p r4 -
7 6 p r4 + 12 -6 + 8 r2 - 50 r2 -6 + 8 r2 + 24 r3 a + 30 6 r4 ArcSinB-3 + 4 r2
2 rF +
48 r3 ArcSinBr
-6 + 9 r2F + 6 6 r4 ArcSinB
-9 + 12 r2
2 r2F - aaaaaF;
bbbbb@r_D := -8 6 * ArcSinB1
2
9 - 12 r2
2 - 3 r2F - 2 * ArcSinB
1
2 -2 + 3 r2F -
96 6 r2 ArcSinB1
2 -2 + 3 r2F + 144 r3 ArcSinB
r
-6 + 9 r2F +
18 6 r4 ArcSinB-9 + 12 r2
2 r2F - 3 * ArcSinB
1
2
9 - 12 r2
2 - 3 r2F -
2 * ArcSinB1
2 -2 + 3 r2F + 5 * ArcSinB
-3 + 4 r2
2 rF ;
SimplifyB1
1296 p r3 V * Sin@aD* Hbbbbb@rD + aaaaaL - GeccNotSimpl@rDF
0
A-Edge_CLD_Expression.nb 17
bbb11@r_D := ArcSinB1
2
9 - 12 r2
2 - 3 r2F - 2 * ArcSinB
1
2 -2 + 3 r2F;
bbb22@r_D := ArcSinB-9 + 12 r2
2 r2F -
3 * ArcSinB1
2
9 - 12 r2
2 - 3 r2F - 2 * ArcSinB
1
2 -2 + 3 r2F + 5 * ArcSinB
-3 + 4 r2
2 rF;
bbb22aa@r_D := -3 * ArcSinB1
2
9 - 12 r2
2 - 3 r2F + 3 * ArcSinB
-3 + 4 r2
2 rF;
bbb22bb@r_D := ArcSinB-9 + 12 r2
2 r2F - 2 * ArcSinB
1
2 -2 + 3 r2F + 2 * ArcSinB
-3 + 4 r2
2 rF;
Simplify@bbb22@rD - bbb22aa@rD - bbb22bb@rDD
SimplifyBbbbbb@rD - -8 6 * bbb11@rD -
96 6 r2 ArcSinB1
2 -2 + 3 r2F + 144 r3 ArcSinB
r
-6 + 9 r2F + 18 6 r4 * bbb22@rD F
ü Simplification of bbb11[r] and bbb22[r]
ü IDENTITY bbb11@rD =
ArcSinB 12
9-12 r2
2-3 r2F - 2 * ArcSinB 1
2 -2+3 r2F Ø ArcSinB -7+9 r2
2 I-2+3 r2M3ë2
F - p ê 2
or
{ArcSinB 12
9-12 r2
2-3 r2F Ø 2 * ArcSinB 1
2 -2+3 r2F + ArcSinB -7+9 r2
2 I-2+3 r2M3ë2
F - p ê 2 }
FullSimplifyBTrigExpand@Sin@bbb11@rD + p ê 2DD, Assumptions Ø : 3 í 2 < r < 1>F
PlotB:bbb11@rD - ArcSinB-7 + 9 r2
2 I-2 + 3 r2M3ê2F - p ê 2 >, :r, 3 í 2, 1>F
LimitBbbb11@rD - ArcSinB-7 + 9 r2
2 I-2 + 3 r2M3ê2F - p ê 2 , r Ø 3 í 2, Direction Ø -1F
LimitBbbb11@rD - ArcSinB-7 + 9 r2
2 I-2 + 3 r2M3ê2F - p ê 2 , r Ø 1, Direction Ø 1F
ü IDENTITY {bbb22[r] Æ (ArcSin[ 3 I27-90 r2+96 r4-34 r6+2 r8M
2 r7 -2+3 r2] - p
2) }
or
{ ArcSinB -9+12 r2
2 r2FÆ ( ArcSin[ 3 I27-90 r2+96 r4-34 r6+2 r8M
2 r7 -2+3 r2] - p
2
+3 * ArcSinB 12
9-12 r2
2-3 r2F + 2 * ArcSinB 1
2 -2+3 r2F - 5 * ArcSinB -3+4 r2
2 rF) }
18 A-Edge_CLD_Expression.nb
ü
IDENTITY {bbb22[r] Æ (ArcSin[ 3 I27-90 r2+96 r4-34 r6+2 r8M
2 r7 -2+3 r2] - p
2) }
or
{ ArcSinB -9+12 r2
2 r2FÆ ( ArcSin[ 3 I27-90 r2+96 r4-34 r6+2 r8M
2 r7 -2+3 r2] - p
2
+3 * ArcSinB 12
9-12 r2
2-3 r2F + 2 * ArcSinB 1
2 -2+3 r2F - 5 * ArcSinB -3+4 r2
2 rF) }
bbb22@rD
-3 ArcSinB1
2
9 - 12 r2
2 - 3 r2F - 2 ArcSinB
1
2 -2 + 3 r2F + 5 ArcSinB
-3 + 4 r2
2 rF + ArcSinB
-9 + 12 r2
2 r2F
FullSimplifyBTrigExpand@Sin@bbb22@rD + p ê 2DD, Assumptions Ø : 3 í 2 < r < 1>F
LimitBbbb22@rD - ArcSinB3 I27 - 90 r2 + 96 r4 - 34 r6 + 2 r8M
2 r7 -2 + 3 r2F -
p
2,
r Ø 3 í 2, Direction Ø -1F
LimitBbbb22@rD - ArcSinB3 I27 - 90 r2 + 96 r4 - 34 r6 + 2 r8M
2 r7 -2 + 3 r2F -
p
2,
r Ø 1, Direction Ø 1F
PlotBNBbbb22@rD - ArcSinB3 I27 - 90 r2 + 96 r4 - 34 r6 + 2 r8M
2 r7 -2 + 3 r2F -
p
2, 20F,
:r, 3 í 2, 1>F
PlotBArcSinB-9 + 12 r2
2 r2F -
ArcSinB3 I27 - 90 r2 + 96 r4 - 34 r6 + 2 r8M
2 r7 -2 + 3 r2F -
p
2+ 3 * ArcSinB
1
2
9 - 12 r2
2 - 3 r2F +
2 * ArcSinB1
2 -2 + 3 r2F - 5 * ArcSinB
-3 + 4 r2
2 rF , :r, 3 í 2, 1>F
A-Edge_CLD_Expression.nb 19
ü compactification of GEccNotSimpl[r]
1
1296 p r3 V * Sin@aDCancelB
TogetherB -8 6 * bbb11@rD - 96 6 r2 ArcSinB1
2 -2 + 3 r2F + 144 r3 ArcSinB
r
-6 + 9 r2F +
18 6 r4 * bbb22@rD ê. :bbb11@rD Ø ArcSinB-7 + 9 r2
2 I-2 + 3 r2M3ê2F - p ê 2 ,
bbb22@rD Ø ArcSinB3 I27 - 90 r2 + 96 r4 - 34 r6 + 2 r8M
2 r7 -2 + 3 r2F -
p
2>F + aaaaa F
1
1296 p r3 V
4 6 p + 144 2 r3 - 54 2 r4 + 27 2 p r4 - 30 6 p r4 + 36 2 -3 + 4 r2 - 150 2 r2
-3 + 4 r2 + 72 r3 a - 96 6 r2 ArcSinB1
2 -2 + 3 r2F - 8 6 ArcSinB
-7 + 9 r2
2 I-2 + 3 r2M3ê2F +
144 r3 ArcSinBr
-6 + 9 r2F + 18 6 r4 ArcSinB
3 I27 - 90 r2 + 96 r4 - 34 r6 + 2 r8M
2 r7 -2 + 3 r2F Csc@aD
Gecc@r_D :=1
1296 p r3 V * Sin@aD
4 6 p + 144 2 r3 - 54 2 r4 + 27 2 p r4 - 30 6 p r4 + 36 2 -3 + 4 r2 - 150 2 r2
-3 + 4 r2 + 72 r3 a - 96 6 r2 ArcSinB1
2 -2 + 3 r2F - 8 6 ArcSinB
-7 + 9 r2
2 I-2 + 3 r2M3ê2F +
144 r3 ArcSinBr
-6 + 9 r2F + 18 6 r4 ArcSinB
3 I27 - 90 r2 + 96 r4 - 34 r6 + 2 r8M
2 r7 -2 + 3 r2F ;
Checks
LimitBGeccNotSimpl@rD - Gecc@rD, r Ø 3 í 2, Direction Ø -1F
Limit@GeccNotSimpl@rD - Gecc@rD, r Ø 1, Direction Ø 1D
a = ArcCos@-1 ê 3D; V = 2 í 3;
PlotBHGeccNotSimpl@rD - Gecc@rDL, :r, 3 í 2, 1>, PlotStyle Ø 8Red, [email protected]<,
PlotRange Ø :: 3 í 2, 1>, 8-10^H-14L, 10^H-14L<>, PlotPoints Ø 500F
Clear@aD; Clear@VD;
ü 1 < r < 2
20 A-Edge_CLD_Expression.nb
ü
1 < r < 2ü the FA case
A-Edge_CLD_Expression.nb 21
:ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F Ø arctanAA,
ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F Ø arctanBB, ArcTanB
1 - -1 + r2
1 + -1 + r2F Ø arctanCC,
ArcCscB3 r
2 2 - -2 + 3 r2F Ø -arcsinDD, ArcSinB
-1 + -1 + r2
2 rF Ø arcsinAA,
ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F Ø arcsinBB, ArcSinB
2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F Ø arcsinCC,
ArcSinB2 -2 + -2 + 3 r2
3 rF Ø arcsinDD,
ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F Ø arcsinEE,
arctanCC Ø -arcsinAA ,
arctanCC Ø -ArcSinB-1 + -1 + r2
2 rF ,
ArcTanB1 - -1 + r2
1 + -1 + r2F Ø -ArcSinB
-1 + -1 + r2
2 rF ,
arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F Ø ArcSinB
2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F,
ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F Ø ArcSinB
2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F,
>
FaEcldDDold@rD
22 A-Edge_CLD_Expression.nb
HFaEcldDDold@rDL ê. :ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F Ø arctanAA,
ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F Ø arctanBB, ArcTanB
1 - -1 + r2
1 + -1 + r2F Ø arctanCC,
ArcCscB3 r
2 2 - -2 + 3 r2F Ø -arcsinDD, ArcSinB
-1 + -1 + r2
2 rF Ø arcsinAA,
ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F Ø arcsinBB, ArcSinB
2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F Ø
arcsinCC, ArcSinB2 -2 + -2 + 3 r2
3 rF Ø arcsinDD,
ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F Ø arcsinEE>
A-Edge_CLD_Expression.nb 23
trigcontr@r_D :=
-64 6 arcsinBB + 384 6 arcsinBB r2 - 576 arcsinCC r3 + 576 arcsinEE r3 - 576 arctanAA r3 -
576 arctanBB r3 + 162 2 arcsinAA r4 + 144 6 arcsinBB r4 - 216 arcsinCC r4 -
216 arcsinEE r4 + 216 arctanAA r4 - 216 arctanBB r4 + 162 2 arctanCC r4 +
32 6 arcsinBB -2 + 3 r2 - 192 6 arcsinBB r2 -2 + 3 r2 + 288 arcsinCC r3 -2 + 3 r2 -
288 arcsinEE r3 -2 + 3 r2 + 288 arctanAA r3 -2 + 3 r2 + 288 arctanBB r3 -2 + 3 r2 -
81 2 arcsinAA r4 -2 + 3 r2 - 72 6 arcsinBB r4 -2 + 3 r2 +
108 arcsinCC r4 -2 + 3 r2 + 108 arcsinEE r4 -2 + 3 r2 - 108 arctanAA r4 -2 + 3 r2 +
108 arctanBB r4 -2 + 3 r2 - 81 2 arctanCC r4 -2 + 3 r2 ì 5184 r3 -2 + -2 + 3 r2 ;
ratnlcontr@r_D := 2 p J-4 6 + 24 6 r2 - 36 r3 + 9 J-3 + 6 N r4N -2 + -2 + 3 r2 +
3 -16 2 -2 + -2 + 3 r2 - 96 r3 -2 + -2 + 3 r2 +
3 r4 -12 2 + 6 -4 + 6 r2 - -4 + 3 r2 + 8 -2 + 3 r2 -
r2 96 2 + 40 2 -1 + r2 - 48 -4 + 6 r2 - 20 4 - 10 r2 + 6 r4 -
14 -4 + 3 r2 + 8 -2 + 3 r2 + 4 I-2 + 3 r2M -4 + 3 r2 + 8 -2 + 3 r2 +
6 9 r4 + 6 r2 -17 + 2 -2 + 3 r2 + 8 7 + 4 -2 + 3 r2 +
I-2 + 3 r2M 9 r4 + 6 r2 -17 + 2 -2 + 3 r2 + 8 7 + 4 -2 + 3 r2 ì
2592 r3 -2 + -2 + 3 r2 ;
Simplify@HHtrigcontr@rDL ê. 8arctanCC Ø -arcsinAA<LD
trigcontrAA@r_D :=1
1296 r3J-2 6 arcsinBB I-4 + 24 r2 + 9 r4MN;
trigcontrBB@r_D :=72
1296* HarcsinCC - arcsinEE + arctanAA + arctanBBL;
trigcontrCC@r_D :=27 * r
1296* HarcsinCC + arcsinEE - arctanAA + arctanBBL;
FullSimplify@trigcontrAA@rD + trigcontrBB@rD +trigcontrCC@rD - HHtrigcontr@rDL ê. 8arctanCC Ø -arcsinAA<LD
0
24 A-Edge_CLD_Expression.nb
HtrigcontrAA@rDL ê. :arcsinBB Ø ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F>
-
I-4 + 24 r2 + 9 r4M ArcSinB 1+3 -1+r2
2 -2+3 r2F
108 6 r3
ü conversion of arctanAA==ArcTanB2 -1+r+ -1+r2
1+ -1+r2F into ArcSin[
2 -1+r+ -1+r2
-1+2 r+ -1+r2]
FullSimplifyBTrigExpandBSinBArcTanB2 -1 + r + -1 + r2
1 + -1 + r2FFF,
Assumptions Ø :1 < r < 2 >F ;
PlotBArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F - ArcSinB
2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F, :r, 1, 2 >F
ü conversion of arctanBB Æ ArcTanB2 1+r- -1+r2
1+ -1+r2F into ArcSinB
2 -1-r+ -1+r2
-1-2 r+ -1+r2F
FullSimplifyBTrigExpandBSinB ArcTanB2 1 + r - -1 + r2
1 + -1 + r2FFF,
Assumptions Ø :1 < r < 2 >F
PlotBArcTanB2 1 + r - -1 + r2
1 + -1 + r2F - ArcSinB
2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F, :r, 1, 2 >F
ü Identity: {arctanCC Æ - arcsinAA}
arctanCC Ø ArcTanB1 - -1 + r2
1 + -1 + r2F is equal to -ArcSinB
-1 + -1 + r2
2 rF Ø H-arcsinAAL
Thus 8arctanCC Ø -arcsinAA<
PlotBArcTanB1 - -1 + r2
1 + -1 + r2F - -ArcSinB
-1 + -1 + r2
2 rF , :r, 1, 2 >F
A-Edge_CLD_Expression.nb 25
ü simplification of arcsinCC - arcsinEE + arctanAA + arctanBB (trigcontrBB@rD)
ü it is proved that (arcsinCC - arcsinEE) Æ ArcSinB2 2 r 4-8 r2 1+ -1+r2 +r4 1+6 -1+r2
I-2+3 r2M -4+9 r4+4 r2 1+3 -1+r2F
H* first pair arcsinCC-arcsinEE *L
FullSimplifyB FullSimplifyBTrigExpandBSinB HarcsinCC - arcsinEEL ê.
:arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F, arcsinEE Ø ArcSinB
r + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F> FF, Assumptions Ø :1 < r < 2 >F ê.
:
H1 + rL2 r2 - 2 -1 + r2
I-2 + 3 r2M -1 - 2 r + -1 + r22
Ø
H1 + rL * 1 - -1 + r2
-2 + 3 r2 * 1 + 2 r - -1 + r2,
H-1 + rL H1 + rL3 r2 - 2 -1 + r2
I-2 + 3 r2M -1 - 2 r + -1 + r22
Ø
H1 + rL * 1 - -1 + r2 * -1 + r2
-2 + 3 r2 * 1 + 2 r - -1 + r2,
H-1 + rL3 H1 + rL r2 - 2 -1 + r2
I-2 + 3 r2M -1 + 2 r + -1 + r22
Ø
H-1 + rL * 1 - -1 + r2 * -1 + r2
-2 + 3 r2 * -1 + 2 r + -1 + r2,
r2 - 2 -1 + r2
-2 + 3 r2Ø
1 - -1 + r2
-2 + 3 r2>, Assumptions Ø :1 < r < 2 >F
PlotB2 2 r 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r2
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2, :r, 1, 2 >F
26 A-Edge_CLD_Expression.nb
PlotBArcSinB2 2 r 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r2
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2F -
HarcsinCC - arcsinEEL ê. :arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F> , :r, 1, 2 >F
FullSimplifyBLimitB ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F -
ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F -
ArcSinB2 2 r 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r2
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2F, r Ø 1, Direction Ø -1F F
FullSimplifyBLimitB ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F -
ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F -
ArcSinB2 2 r 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r2
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2F, r Ø 2 , Direction Ø 1FF
FullSimplifyBSqrtB1
3+2 Â 2
3*
1
3-2 Â 2
3FF
FullSimplifyBSqrtBÂ
3+2 2
3* -
Â
3+2 2
3FF
FullSimplifyB -p
2- Â LogB
1
3+2 Â 2
3F - Â LogB
Â
3+2 2
3F ê.
:LogB1
3+2 Â 2
3F Ø Â * ArcSinB
2 * 2
3F, LogB
Â
3+2 2
3F Ø Â * ArcSinB
1
3F>F
A-Edge_CLD_Expression.nb 27
PlotB:-1 - 2 r + -1 + r2 , -1 + 2 r + -1 + r2 >, :r, 1, 2 >F
ü --------------------------------- end ------
ü it is proved that (arctanAA + arctanBB) Æ ArcSinB2 2 r 2 -1+ -1+r2 +r2 5+3 -1+r2
-4+9 r4+4 r2 1+3 -1+r2F
ArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F Ø ArcSinB
2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F
H* simplification of the second pair *L
FullSimplifyB FullSimplifyBTrigExpandBSinB HarctanAA + arctanBBL ê. :arctanAA Ø
ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F, arctanBB Ø ArcSinB
2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> FF,
Assumptions Ø :1 < r < 2 >F ê. :
I-1 + r2M r2 + 2 -1 + r2
-1 - 2 r + -1 + r22
Ø
-1 + r2 * 1 + -1 + r2
1 + 2 r - -1 + r2,
I-1 + r2M r2 + 2 -1 + r2
-1 + 2 r + -1 + r22
Ø
-1 + r2 * 1 + -1 + r2
-1 + 2 r + -1 + r2,
r2 + 2 -1 + r2
-1 - 2 r + -1 + r22
Ø
1 + -1 + r2
1 + 2 r - -1 + r2,
r2 + 2 -1 + r2 Ø 1 + -1 + r2 > , Assumptions Ø :1 < r < 2 >F
2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2
ArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F Ø ArcSinB
2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F
28 A-Edge_CLD_Expression.nb
PlotB:2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2,2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2>,
:r, 1, 2 >, PlotRange Ø ::1, 2 >, 8-1, 1<>F
1.1 1.2 1.3 1.4
-1.0
-0.5
0.0
0.5
1.0
PlotBArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F -
HarctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F, arctanBB Ø
ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> , :r, 1, 2 >, PlotPoints Ø 1000F
PlotBArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F -
HarctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
> ,
A-Edge_CLD_Expression.nb 29
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> ,
:r, 1, 2 >, PlotRange Ø ::1, 2 >, 8-1, 1<>, PlotPoints Ø 1000,
PlotStyle Ø 8Red, [email protected]<F
FullSimplifyBLimitBArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F -
HarctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> , r Ø 1, Direction Ø -1FF
FullSimplifyBLimitBArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F -
HarctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> , r Ø 2 , Direction Ø 1FF
30 A-Edge_CLD_Expression.nb
PlotBArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F -
HarctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> , :r, 1, 2 >, PlotPoints Ø 1000F
PlotBArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F -
HarctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> ,
:r, 1, 2 >, PlotRange Ø ::1, 2 >, 8-1, 1<>, PlotPoints Ø 1000,
PlotStyle Ø 8Red, [email protected]<F
FullSimplifyBLimitBArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F -
HarctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
> , , FF
A-Edge_CLD_Expression.nb 31
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> , r Ø 1, Direction Ø -1FF
FullSimplifyBLimitBArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F -
HarctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> , r Ø 2 , Direction Ø 1FF
ü --------------------------------- end ------
ü the two results just obtained are summed and simplified. In fact one obtains
arcsinCC - arcsinEE + arctanAA + arctanBB Æ ArcSinB 2 2 r -1+r2
-2+3 r2F
HarcsinCC - arcsinEEL Ø ArcSinB2 2 r 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r2
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2F
HarctanAA + arctanBBL Ø ArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F
FullSimplifyBTrigExpandBSinBArcSinB2 2 r 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r2
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2F +
ArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2FFF, Assumptions Ø :1 < r < 2 >F
32 A-Edge_CLD_Expression.nb
FullSimplifyB 2 2 r 4 1 -
8 r2 2 -1 + -1 + r2 + r2 5 + 3 -1 + r22
-4 + 9 r4 + 4 r2 1 + 3 -1 + r22
-
8 r2 1 -
8 r2 2 -1 + -1 + r2 + r2 5 + 3 -1 + r22
-4 + 9 r4 + 4 r2 1 + 3 -1 + r22
+
r4 1 -
8 r2 2 -1 + -1 + r2 + r2 5 + 3 -1 + r22
-4 + 9 r4 + 4 r2 1 + 3 -1 + r22
-
8 r2 I-1 + r2M 1 -
8 r2 2 -1 + -1 + r2 + r2 5 + 3 -1 + r22
-4 + 9 r4 + 4 r2 1 + 3 -1 + r22
+
6 r4 I-1 + r2M 1 -
8 r2 2 -1 + -1 + r2 + r2 5 + 3 -1 + r22
-4 + 9 r4 + 4 r2 1 + 3 -1 + r22
+
4 1 -
8 r2 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r22
I2 - 3 r2M2 -4 + 9 r4 + 4 r2 1 + 3 -1 + r22
-
16 r2 1 -
8 r2 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r22
I2 - 3 r2M2 -4 + 9 r4 + 4 r2 1 + 3 -1 + r22
+
-
A-Edge_CLD_Expression.nb 33
15 r4 1 -
8 r2 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r22
I2 - 3 r2M2 -4 + 9 r4 + 4 r2 1 + 3 -1 + r22
-
4 I-1 + r2M 1 -
8 r2 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r22
I2 - 3 r2M2 -4 + 9 r4 + 4 r2 1 + 3 -1 + r22
+
9 r4 I-1 + r2M 1 -
8 r2 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r22
I2 - 3 r2M2 -4 + 9 r4 + 4 r2 1 + 3 -1 + r22
ì
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2 ê.
: I-1 + r2M 1 -
8 r2 2 -1 + -1 + r2 + r2 5 + 3 -1 + r22
-4 + 9 r4 + 4 r2 1 + 3 -1 + r22
Ø
-1 + r2 * 4 - 4 r2 + 3 r4 - 4 r2 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2,
1 -
8 r2 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r22
I2 - 3 r2M2 -4 + 9 r4 + 4 r2 1 + 3 -1 + r22
Ø
8 - 28 r2 + 2 r4 + 21 r6 - 24 r2 -1 + r2 + 44 r4 -1 + r2
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2,
34 A-Edge_CLD_Expression.nb
I-1 + r2M 1 -
8 r2 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r22
I2 - 3 r2M2 -4 + 9 r4 + 4 r2 1 + 3 -1 + r22
Ø
8 - 28 r2 + 2 r4 + 21 r6 - 24 r2 -1 + r2 + 44 r4 -1 + r2 * -1 + r2
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2,
1 -
8 r2 2 -1 + -1 + r2 + r2 5 + 3 -1 + r22
-4 + 9 r4 + 4 r2 1 + 3 -1 + r22
Ø
4 - 4 r2 + 3 r4 - 4 r2 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2> , Assumptions Ø :1 < r < 2 >F
A-Edge_CLD_Expression.nb 35
PlotB: I-1 + r2M 1 -
8 r2 2 -1 + -1 + r2 + r2 5 + 3 -1 + r22
-4 + 9 r4 + 4 r2 1 + 3 -1 + r22
-
-1 + r2 * 4 - 4 r2 + 3 r4 - 4 r2 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2>, :r, 1, 2 >F
PlotB: 1 -
8 r2 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r22
I2 - 3 r2M2 -4 + 9 r4 + 4 r2 1 + 3 -1 + r22
-
8 - 28 r2 + 2 r4 + 21 r6 - 24 r2 -1 + r2 + 44 r4 -1 + r2
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2>, :r, 1, 2 >F
PlotB: I-1 + r2M 1 -
8 r2 4 - 8 r2 1 + -1 + r2 + r4 1 + 6 -1 + r22
I2 - 3 r2M2 -4 + 9 r4 + 4 r2 1 + 3 -1 + r22
-
8 - 28 r2 + 2 r4 + 21 r6 - 24 r2 -1 + r2 + 44 r4 -1 + r2 * -1 + r2
I-2 + 3 r2M -4 + 9 r4 + 4 r2 1 + 3 -1 + r2>, :r, 1, 2 >F
PlotB: 1 -
8 r2 2 -1 + -1 + r2 + r2 5 + 3 -1 + r22
-4 + 9 r4 + 4 r2 1 + 3 -1 + r22
-4 - 4 r2 + 3 r4 - 4 r2 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2>,
:r, 1, 2 >F
PlotB2 2 r -1 + r2
-2 + 3 r2, :r, 1, 2 >, PlotRange Ø ::1, 2 >, 8-0.5, 1.5<>F
36 A-Edge_CLD_Expression.nb
PlotB HarcsinCC - arcsinEE + arctanAA + arctanBBL ê.
:arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F, arctanAA Ø
ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F, arctanBB Ø ArcSinB
2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> -
ArcSinB2 2 r -1 + r2
-2 + 3 r2F, :r, 1, 2 >F
FullSimplifyBLimitB HarcsinCC - arcsinEE + arctanAA + arctanBBL ê.
:arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F, arctanAA Ø
ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F, arctanBB Ø ArcSinB
2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> -
ArcSinB2 2 r -1 + r2
-2 + 3 r2F, r Ø 1, Direction Ø -1FF
A-Edge_CLD_Expression.nb 37
FullSimplifyBLimitB HarcsinCC - arcsinEE + arctanAA + arctanBBL ê.
:arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F, arctanAA Ø
ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F, arctanBB Ø ArcSinB
2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> -
ArcSinB2 2 r -1 + r2
-2 + 3 r2F, r Ø 2 , Direction Ø 1FF
ü --------------------------------- end ------
ü Proof that arcsinCC + arcsinEE - arctanAA + arctanBB == p
8arcsinCC + arcsinEE - arctanAA + arctanBB Ø p<
PlotB HarcsinCC + arcsinEE - arctanAA + arctanBBL ê.
:arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F,
,
38 A-Edge_CLD_Expression.nb
arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> - p, :r, 1, 2 >F
LimitB HarcsinCC + arcsinEE - arctanAA + arctanBBL ê.
:arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F,
arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> - p, r Ø 1, Direction Ø -1F
LimitB HarcsinCC + arcsinEE - arctanAA + arctanBBL ê.
:arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
,
A-Edge_CLD_Expression.nb 39
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F,
arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> - p, r Ø 2 , Direction Ø 1F
FullSimplifyBSqrtB1
3+2 Â 2
3*
1
3-2 Â 2
3FF
FullSimplifyBSqrtBÂ
3+2 2
3* -
Â
3+2 2
3FF
FullSimplifyB -p
2- Â LogB
1
3+2 Â 2
3F - Â LogB
Â
3+2 2
3F ê.
:LogB1
3+2 Â 2
3F Ø Â * ArcSinB
2 * 2
3F, LogB
Â
3+2 2
3F Ø Â * ArcSinB
1
3F>F
40 A-Edge_CLD_Expression.nb
ü proof of the identity arcsinCC + arcsinEE =
ArcSinB 2+r-2 r2+2 -1+r2 +3 r -1+r2
-2+3 r2 1+2 r- -1+r2F+ ArcSinB
r+2 r2+3 r -1+r2 -2 1+ -1+r2
-2+3 r2 -1+2 r+ -1+r2F = = p - ArcSinB- 2 2 I-2+r2M
3 r2+2 -1+r2F
FullSimplifyB FullSimplifyBTrigExpandB
SinB HarcsinCC + arcsinEEL ê. :arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F> FF,
Assumptions Ø :1 < r < 2 >F ê. :
H1 + rL2 r2 - 2 -1 + r2
I-2 + 3 r2M -1 - 2 r + -1 + r22
Ø
H1 + rL * 1 - -1 + r2
-2 + 3 r2 * 1 + 2 r - -1 + r2,
H-1 + rL H1 + rL3 r2 - 2 -1 + r2
I-2 + 3 r2M -1 - 2 r + -1 + r22
Ø
H1 + rL * 1 - -1 + r2 * -1 + r2
-2 + 3 r2 * 1 + 2 r - -1 + r2,
H-1 + rL3 H1 + rL r2 - 2 -1 + r2
I-2 + 3 r2M -1 + 2 r + -1 + r22
Ø
H-1 + rL * 1 - -1 + r2 * -1 + r2
-2 + 3 r2 * -1 + 2 r + -1 + r2,
r2 - 2 -1 + r2
-2 + 3 r2Ø
1 - -1 + r2
-2 + 3 r2>, Assumptions Ø :1 < r < 2 >F
PlotB-2 2 I-2 + r2M
3 r2 + 2 -1 + r2, :r, 1, 2 >F
A-Edge_CLD_Expression.nb 41
PlotB: 1 - ArcSinB-2 2 I-2 + r2M
3 r2 + 2 -1 + r2F ì p -
HarcsinCC + arcsinEEL ê. :arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F> ì p >, :r, 1, 2 >F
42 A-Edge_CLD_Expression.nb
ü ...... end ........
ü proof of the identity -arctanAA + arctanBB =
- ArcSinB2 -1+r+ -1+r2
-1+2 r+ -1+r2F + ArcSinB
2 -1-r+ -1+r2
-1-2 r+ -1+r2F = = ArcSinB- 2 2 I-2+r2M
3 r2+2 -1+r2F
i.e.
ArcSinB2 -1-r+ -1+r2
-1-2 r+ -1+r2F Æ( ArcSinB
2 -1+r+ -1+r2
-1+2 r+ -1+r2F+ ArcSinB- 2 2 I-2+r2M
3 r2+2 -1+r2F)
FullSimplifyB FullSimplifyB
TrigExpandBSinB H-arctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> FF, Assumptions Ø :1 < r < 2 >F ê.
:
I-1 + r2M r2 + 2 -1 + r2
-1 - 2 r + -1 + r22
Ø
-1 + r2 * 1 + -1 + r2
1 + 2 r - -1 + r2,
I-1 + r2M r2 + 2 -1 + r2
-1 + 2 r + -1 + r22
Ø
-1 + r2 * 1 + -1 + r2
-1 + 2 r + -1 + r2,
r2 + 2 -1 + r2
-1 - 2 r + -1 + r22
Ø
1 + -1 + r2
1 + 2 r - -1 + r2,
r2 + 2 -1 + r2 Ø 1 + -1 + r2 > , Assumptions Ø :1 < r < 2 >F
PlotB-2 2 I-2 + r2M
3 r2 + 2 -1 + r2, :r, 1, 2 >F
A-Edge_CLD_Expression.nb 43
PlotB ArcSinB-2 2 I-2 + r2M
3 r2 + 2 -1 + r2F ì p -
H-arctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F> ì p, :r, 1, 2 >F
ü ...... end ........
ü proof of the identity arcsinCC + arcsinEE - arctanAA + arctanBB = = porarcsinCC + arcsinEE - arctanAA + arctanBB = = p or
ArcSinB 2+r-2 r2+2 -1+r2 +3 r -1+r2
-2+3 r2 1+2 r- -1+r2F + ArcSinB
r+2 r2+3 r -1+r2 -2 1+ -1+r2
-2+3 r2 -1+2 r+ -1+r2F -
ArcSinB2 -1+r+ -1+r2
-1+2 r+ -1+r2F + ArcSinB
2 -1-r+ -1+r2
-1-2 r+ -1+r2F = = p
HarcsinCC + arcsinEEL ê. :arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F> +
H-arctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F,
arctanBB Ø ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F>
44 A-Edge_CLD_Expression.nb
PlotB ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F -
ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F + ArcSinB
2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F +
ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F - p, :r, 1, 2 >F
FullSimplifyBLimitB ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F -
ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F + ArcSinB
2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F +
ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F - p, r Ø 1, Direction Ø -1FF
FullSimplifyBLimitB ArcSinB2 -1 - r + -1 + r2
-1 - 2 r + -1 + r2F -
ArcSinB2 -1 + r + -1 + r2
-1 + 2 r + -1 + r2F + ArcSinB
2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F +
ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F - p, r Ø 2 , Direction Ø 1FF
ü ...... end ........
ü simplification of trigcontrAA[r] trigcontrBB[r] , trigcontrCC[r]
HtrigcontrAA@rDL ê. :arcsinBB Ø ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F>
A-Edge_CLD_Expression.nb 45
SimplifyBHtrigcontrBB@rDL ê.
: arcsinCC Ø -H- arcsinEE + arctanAA + arctanBB L + ArcSinB2 2 r -1 + r2
-2 + 3 r2F> F
H trigcontrCC@rDL ê. 8arcsinCC Ø -HarcsinEE - arctanAA + arctanBBL + p<
trigcontrSimpl@r_D :=
-
I-4 + 24 r2 + 9 r4M ArcSinB 1+3 -1+r2
2 -2+3 r2F
108 6 r3+
1
18ArcSinB
2 2 r -1 + r2
-2 + 3 r2F +
p r
48;
PlotBtrigcontrSimpl@rD - Htrigcontr@rDL ê. :arctanAA Ø ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F,
arctanBB Ø ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F, arctanCC Ø ArcTanB
1 - -1 + r2
1 + -1 + r2F,
arcsinDD Ø -ArcCscB3 r
2 2 - -2 + 3 r2F, arcsinAA Ø ArcSinB
-1 + -1 + r2
2 rF,
arcsinBB Ø ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F,
arcsinCC Ø ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F,
arcsinDD Ø ArcSinB2 -2 + -2 + 3 r2
3 rF,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F> , :r, 1, 2 >F
FullSimplifyB
46 A-Edge_CLD_Expression.nb
FullSimplifyB
LimitBtrigcontrSimpl@rD - Htrigcontr@rDL ê. :arctanAA Ø ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F,
arctanBB Ø ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F, arctanCC Ø ArcTanB
1 - -1 + r2
1 + -1 + r2F,
arcsinDD Ø -ArcCscB3 r
2 2 - -2 + 3 r2F, arcsinAA Ø ArcSinB
-1 + -1 + r2
2 rF,
arcsinBB Ø ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F, arcsinCC Ø ArcSinB
2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F, arcsinDD Ø ArcSinB
2 -2 + -2 + 3 r2
3 rF,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F> , r Ø 1, Direction Ø -1FF
FullSimplifyB
LimitBtrigcontrSimpl@rD - Htrigcontr@rDL ê. :arctanAA Ø ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F,
arctanBB Ø ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F, arctanCC Ø ArcTanB
1 - -1 + r2
1 + -1 + r2F,
arcsinDD Ø -ArcCscB3 r
2 2 - -2 + 3 r2F, arcsinAA Ø ArcSinB
-1 + -1 + r2
2 rF,
,
A-Edge_CLD_Expression.nb 47
arcsinBB Ø ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F, arcsinCC Ø ArcSinB
2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F, arcsinDD Ø ArcSinB
2 -2 + -2 + 3 r2
3 rF,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F> , r Ø 2 , Direction Ø 1FF
NBp
96-
1
96Â LogB1 - 2 Â 2 F +
1
96Â LogB1 + 2 Â 2 F +
1
48Â LogB
Â
3+2 2
3F, 50F
NB11 p
288-
11
288Â LogB1 - 2 Â 2 F +
11
288Â LogB1 + 2 Â 2 F +
11
144Â LogB
Â
3+2 2
3F, 50F
NB-7
72 2+
1
963 - 2 2 -
1
36
3
2- 2 +
1
36
3
2+ 2 +
1
963 + 2 2 , 50F
fcnaus@r_D :=
trigcontrSimpl@rD - Htrigcontr@rDL ê. :arctanAA Ø ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F,
arctanBB Ø ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F, arctanCC Ø ArcTanB
1 - -1 + r2
1 + -1 + r2F,
arcsinDD Ø -ArcCscB3 r
2 2 - -2 + 3 r2F, arcsinAA Ø ArcSinB
-1 + -1 + r2
2 rF,
arcsinBB Ø ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F, arcsinCC Ø ArcSinB
2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F, arcsinDD Ø ArcSinB
2 -2 + -2 + 3 r2
3 rF,
arcsinEE Ø ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F> ;
R0 = 1; RF = 2 ; Step = HRF - R0L ê 101;Do@R = R0 + J * Step; val = N@fcnaus@RD, 30D;Print@J, ", ", PaddedForm@val, 810, 8<DD;, 8J, 1, 100<D
ü ...... end ........
48 A-Edge_CLD_Expression.nb
ü
...... end ........
ü simplification of the rational contribution
ü definitions/identity
:rdcndAA Ø -1 + r2 , rdcndBB Ø -2 + 3 r2 , rdcndCC Ø 2 - 5 r2 + 3 r4 ,
rdcndDD Ø -4 + 3 r2 + 8 -2 + 3 r2 ,
rdcndEE Ø 56 - 102 r2 + 9 r4 + 32 -2 + 3 r2 + 12 r2 -2 + 3 r2 >
8rdcndCC Ø rdcndAA * rdcndBB,rdcndEE Ø rdcndDD * H2 - rdcndBBL<
: 56 - 102 r2 + 9 r4 + 32 -2 + 3 r2 + 12 r2 -2 + 3 r2 Ø
-4 + 3 r2 + 8 -2 + 3 r2 * 2 - -2 + 3 r2 >
ü proof of the identities
ExpandBHrdcndCC^2 - HrdcndAA * rdcndBBL^2L ê.
:rdcndAA Ø -1 + r2 , rdcndBB Ø -2 + 3 r2 , rdcndCC Ø 2 - 5 r2 + 3 r4 ,
rdcndDD Ø -4 + 3 r2 + 8 -2 + 3 r2 ,
rdcndEE Ø 56 - 102 r2 + 9 r4 + 32 -2 + 3 r2 + 12 r2 -2 + 3 r2 >F
Identity 56 - 102 r2 + 9 r4 + 32 -2 + 3 r2 + 12 r2 -2 + 3 r2 Ø -4 + 3 r2 + 8 -2 + 3 r2
* 2 - -2 + 3 r2
or rdcndEE Ø rdcndDD * H2 - rdcndBB)
ExpandB56 - 102 r2 + 9 r4 + 32 -2 + 3 r2 +
12 r2 -2 + 3 r2 - -4 + 3 r2 + 8 -2 + 3 r2 * 2 - -2 + 3 r2 ^2F
0
A-Edge_CLD_Expression.nb 49
ü simplification
HFactor@ratnlcontr@rDDL ê. : 2 - 5 r2 + 3 r4 Ø rdcndCC, -4 + 3 r2 + 8 -2 + 3 r2 Ø rdcndDD,
56 - 102 r2 + 9 r4 + 32 -2 + 3 r2 + 12 r2 -2 + 3 r2 Ø rdcndEE,
I-2 + 3 r2M -4 + 3 r2 + 8 -2 + 3 r2 Ø rdcndBB * rdcndDD,
I-2 + 3 r2M 56 - 102 r2 + 9 r4 + 32 -2 + 3 r2 + 12 r2 -2 + 3 r2 Ø rdcndBB * rdcndEE> ê.
: -1 + r2 Ø rdcndAA, -2 + 3 r2 Ø rdcndBB, rdcndCC Ø rdcndAA * rdcndBB>
rdcndcomp@r_D := I2592 r3 H-2 + rdcndBBLM *1
2592 r3 H-2 + rdcndBBL
J96 2 + 16 6 p - 288 2 r2 - 96 6 p r2 + 576 r3 + 144 p r3 - 108 2 r4 + 108 p r4 -
36 6 p r4 - 120 2 r2 rdcndAA - 48 2 rdcndBB - 8 6 p rdcndBB + 144 2 r2 rdcndBB +
48 6 p r2 rdcndBB - 288 r3 rdcndBB - 72 p r3 rdcndBB + 54 2 r4 rdcndBB -
54 p r4 rdcndBB + 18 6 p r4 rdcndBB + 60 2 r2 rdcndAA rdcndBB + 42 r2 rdcndDD -
9 r4 rdcndDD - 12 r2 rdcndBB rdcndDD - 18 r2 rdcndEE - 3 r2 rdcndBB rdcndEEN;
SimplifyACoefficientListACoefficientListA
r^4 * ISimplifyAHSimplify@Hrdcndcomp@rDL ê. 8rdcndEE Ø rdcndDD * H2 - rdcndBBL<DL ê.
9rdcndBB2 Ø I-2 + 3 r2M=E ë I2592 r3 H-2 + rdcndBBLMM, rdcndAAE, rEE
SimplifyBSimplifyAHSimplify@Hrdcndcomp@rDL ê. 8rdcndEE Ø rdcndDD * H2 - rdcndBBL<DL ê.
9rdcndBB2 Ø I-2 + 3 r2M=E ë I2592 r3 H-2 + rdcndBBLM -
-6 + 3 p
162 2 * r^3+
3 + 3 p
27 2 * r+
1
36H-4 - pL +
r
144J3 2 + J-3 + 6 N pN +
5 * rdcndAA
108 2 * rF
0
rdcndSimpl@r_D :=
-6 + 3 p
162 2 * r^3+
3 + 3 p
27 2 * r+
1
36H-4 - pL +
r
144J3 2 + J-3 + 6 N pN +
5 * -1 + r2
108 2 * r;
ü checks
Limit@rdcndSimpl@rD - ratnlcontr@rD, r Ø 1, Direction Ø -1D
LimitBrdcndSimpl@rD - ratnlcontr@rD, r Ø 2 , Direction Ø 1F
PlotBrdcndSimpl@rD - ratnlcontr@rD, :r, 1, 2 >F
NormE * HrdcndSimpl@rD + trigcontrSimpl@rDL
50 A-Edge_CLD_Expression.nb
GEAAdd@r_D :=
-1
p V * Sin@aD*
1
36H-4 - pL -
6 + 3 p
162 2 r3+3 + 3 p
27 2 r+
p r
48+
1
144J3 2 + J-3 + 6 N pN r +
5 -1 + r2
108 2 r+
1
18ArcSinB
2 2 r -1 + r2
-2 + 3 r2F -
I-4 + 24 r2 + 9 r4M ArcSinB 1+3 -1+r2
2 -2+3 r2F
108 6 r3;
ü checks
FullSimplify@Limit@GEAAdd@rD - NormE * FaEcldDDold@rD, r Ø 1, Direction Ø -1DD
FullSimplifyBLimitBGEAAdd@rD - NormE * FaEcldDDold@rD, r Ø 2 , Direction Ø 1FF
a = ArcCos@-1 ê 3D; V = 2 í 3; PlotBGEAAdd@rD - NormE * FaEcldDDold@rD, :r, 1, 2 >F
plaa = PlotBGEAAaa@rD, :r, 0, 2 ê 3 >, PlotRange Ø ::0, 2 >, 8-0.005, 0.05<>F;
plbb = PlotBGEAAbb@rD, :r, 2 ê 3 , 3 í 2>, PlotStyle Ø 8Red<F;
plcc = PlotBGEAAcc@rD, :r, 3 í 2, 1>, PlotStyle Ø 8Blue<F;
pldd = PlotBGEAAdd@rD, :r, 1, 2 >, PlotStyle Ø 8Red<F;
Show@plaa, plbb, plcc, plddD
Clear@aD; Clear@VD;
ü the FB case
ü identities
H* 1 < r < 2 *L
:-4 + 4 r2 + 9 r4 + 12 r2 -1 + r2 Ø 3 * r^2 + 2 * -1 + r2 * 2 * -1 + r2 + 3 * r^2 ,
2 -1 + -1 + r2 + r2 5 + 3 -1 + r2 Ø 2 * -1 + r2 + 3 * r^2 * 1 + -1 + r2 ,
,
,
A-Edge_CLD_Expression.nb 51
-8 + 9 r2 + 6 -1 + r2 Ø 1 + 3 * -1 + r2 ,
-4 + r4 - 4 r2 -1 + -1 + r2 Ø r^2 - 2 * -1 + r2 ,
-8 + 12 -1 + r2 - 12 r2 1 + -1 + r2 + 9 r4 2 + -1 + r2 Ø
-4 + 3 r2 + 6 -1 + r2 2 + 3 r2 -1 + r2 ,
ArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F Ø ArcSinB
2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F,
arctanAA Ø -arctanBB + ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F,
ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F Ø 2 * ArcSinB
2 1 - -1 + r2
-2 + 3 r2 3 r2 + 2 -1 + r2
F +
ArcSinB2 2 r -4 + 3 r2 + 6 -1 + r2 2 + 3 r2 -1 + r2
I-2 + 3 r2M 3 r2 + 2 -1 + r22
F ,
ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F Ø
-2 * ArcSinB2 1 - -1 + r2
-2 + 3 r2 3 r2 + 2 -1 + r2
F +p
2+ ArcSinB
2 - r2
-2 + 3 r2F >,
52 A-Edge_CLD_Expression.nb
:ArcSinB3 -1 + -1 + r2
2 -2 + 3 r2F Ø ArcSinB
3 -1 + r2
-2 + 3 r2F - p ê 3 ,
ArcSinB-1 + -1 + r2
2 rF Ø ArcSinB
-1 + r2
rF - p ê 4 ,
ArcSinB3 -1 + r2
-2 + 3 r2F Ø ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F -
p
6>;
TogetherB HFbEcldDold@rDL ê. :ArcTanB4 + 3 r - 6 r2 - 3 r -1 + r2
2 1 + 3 -1 + r2F Ø
ArcSinB--4 - 3 r + 6 r2 + 3 r -1 + r2
3 -2 + 3 r2 -1 + 2 r + -1 + r2F, ArcTanB
4 - 6 r2 + 3 r -1 + -1 + r2
2 1 + 3 -1 + r2F Ø
ArcSinB-4 + 3 r + 6 r2 - 3 r -1 + r2
3 -2 + 3 r2 -1 - 2 r + -1 + r2F> ê.
:ArcSinB-4 - 3 r + 6 r2 + 3 r -1 + r2
3 -2 + 3 r2 -1 + 2 r + -1 + r2F Ø
- ArcSinB-4 + 3 r + 6 r2 - 3 r -1 + r2
3 -2 + 3 r2 -1 - 2 r + -1 + r2F +
2 * ArcSinB2 * 1 - -1 + r2
-2 + 3 r2 * 3 r2 + 2 -1 + r2
F >F ê.
:ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F Ø arctanAA, ArcTanB
2 1 + r - -1 + r2
1 + -1 + r2F Ø arctanBB>
A-Edge_CLD_Expression.nb 53
TogetherB
1
1296 r3-24 2 + 144 r3 - 72 arctanAA r3 - 72 arctanBB r3 + 36 p r3 - 27 2 r4 + 24 2 -1 + r2 -
90 2 r2 -1 + r2 + 108 2 r4 ArcSinB-1 + -1 + r2
2 rF -
8 6 ArcSinB3 -1 + -1 + r2
2 -2 + 3 r2F - 54 6 r4 ArcSinB
3 -1 + -1 + r2
2 -2 + 3 r2F -
144 r3 ArcSinB2 1 - -1 + r2
-2 + 3 r2 3 r2 + 2 -1 + r2
F ê.
:arctanAA Ø -arctanBB + ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F>F
TogetherB
1
1296 r3-24 2 + 144 r3 + 36 p r3 - 27 2 r4 + 24 2 -1 + r2 - 90 2 r2 -1 + r2 + 108
2 r4 ArcSinB-1 + -1 + r2
2 rF - 8 6 ArcSinB
3 -1 + -1 + r2
2 -2 + 3 r2F -
54 6 r4 ArcSinB3 -1 + -1 + r2
2 -2 + 3 r2F - 72 r3 ArcSinB
2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F -
144 r3 ArcSinB2 1 - -1 + r2
-2 + 3 r2 3 r2 + 2 -1 + r2
F ê. :ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F Ø
-2 * ArcSinB2 1 - -1 + r2
-2 + 3 r2 3 r2 + 2 -1 + r2
F +p
2+ ArcSinB
2 - r2
-2 + 3 r2F >F
54 A-Edge_CLD_Expression.nb
TogetherB1
1296 r3-24 2 + 144 r3 - 27 2 r4 + 24 2 -1 + r2 -
90 2 r2 -1 + r2 - 72 r3 ArcSinB2 - r2
-2 + 3 r2F + 108 2 r4 ArcSinB
-1 + -1 + r2
2 rF -
8 6 ArcSinB3 -1 + -1 + r2
2 -2 + 3 r2F - 54 6 r4 ArcSinB
3 -1 + -1 + r2
2 -2 + 3 r2F ê.
:ArcSinB3 -1 + -1 + r2
2 -2 + 3 r2F Ø ArcSinB
3 -1 + r2
-2 + 3 r2F - p ê 3,
ArcSinB-1 + -1 + r2
2 rF Ø ArcSinB
-1 + r2
rF - p ê 4 >F
FbEcldDSimpl@r_D :=
1
3888 r3-72 2 + 8 6 p + 432 r3 - 81 2 r4 - 81 2 p r4 + 54 6 p r4 + 72 2 -1 + r2 -
270 2 r2 -1 + r2 + 324 2 r4 ArcSinB-1 + r2
rF - 216 r3 ArcSinB
2 - r2
-2 + 3 r2F -
24 6 ArcSinB3 -1 + r2
-2 + 3 r2F - 162 6 r4 ArcSinB
3 -1 + r2
-2 + 3 r2F ;
Limit@FbEcldDSimpl@rD - FbEcldDold@rD, r Ø 1, Direction Ø -1D
0
A-Edge_CLD_Expression.nb 55
H* - p
18+
1
36Â LogB1- Â
2F-
1
36Â LogB1+ Â
2F+
1
36Â LogB1- 5 Â
2F-
1
36Â LogB1+ 5 Â
2F+
1
36Â LogB1-2 Â 2 F-
1
36Â LogB1+2 Â 2 F
is equal to zero" *LFullSimplifyB-p
18+
1
36Â * LogB
3
2F - Â * ArcSinB
1
2ì
3
2F -
1
36Â * LogB
3
2F + Â * ArcSinB
1
2ì
3
2F +
1
36Â * LogB3
3
2F + Â * ArcSinB-
5
2ì 3
3
2F -
1
36Â * LogB3
3
2F + Â * ArcSinB
5
2ì 3
3
2F +
1
36Â * JLog@3D + Â * ArcSinB-2 * 2 í 3FN -
1
36Â * JLog@3D + Â * ArcSinB2 * 2 í 3FNF
LimitBFbEcldDSimpl@rD - FbEcldDold@rD, r Ø 2 , Direction Ø 1F
0
PlotBFbEcldDSimpl@rD - FbEcldDold@rD, :r, 1, 2 >, PlotPoints Ø 500F
NormE * FbEcldDSimpl@rD
GEBBdd@r_D := -1
3888 p r3 V * Sin@aD
-72 2 + 8 6 p + 432 r3 - 81 2 r4 - 81 2 p r4 + 54 6 p r4 + 72 2 -1 + r2 -
270 2 r2 -1 + r2 + 324 2 r4 ArcSinB-1 + r2
rF - 216 r3 ArcSinB
2 - r2
-2 + 3 r2F -
24 6 ArcSinB3 -1 + r2
-2 + 3 r2F - 162 6 r4 ArcSinB
3 -1 + r2
-2 + 3 r2F ;
Factor@Together@GEAAdd@rD + GEBBdd@rDDD
56 A-Edge_CLD_Expression.nb
FactorB HTogether@GEAAdd@rD + GEBBdd@rDDL ê.
:ArcSinB3 -1 + -1 + r2
2 -2 + 3 r2F Ø ArcSinB
3 -1 + r2
-2 + 3 r2F - p ê 3 ,
ArcSinB-1 + -1 + r2
2 rF Ø ArcSinB
-1 + r2
rF - p ê 4 > F
GEddNotSimpl@r_D := -1
3888 p r3 V-144 2 - 4 6 p + 216 2 r2 + 72 6 p r2 -
108 p r3 - 81 2 p r4 + 81 6 p r4 + 72 2 -1 + r2 - 180 2 r2 -1 + r2 +
24 6 * ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F - ArcSinB
3 -1 + r2
-2 + 3 r2F -
144 6 r2 ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F + 216 r3 *
ArcSinB2 2 r -1 + r2
-2 + 3 r2F - ArcSinB
2 - r2
-2 + 3 r2F + 324 2 r4 ArcSinB
-1 + r2
rF -
54 6 r4 3 * ArcSinB3 -1 + r2
-2 + 3 r2F + ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F Csc@aD;
TogetherB HTogether@GEAAdd@rD + GEBBdd@rDDL ê.
:ArcSinB3 -1 + -1 + r2
2 -2 + 3 r2F Ø ArcSinB
3 -1 + r2
-2 + 3 r2F - p ê 3 ,
ArcSinB-1 + -1 + r2
2 rF Ø ArcSinB
-1 + r2
rF - p ê 4 > - GEddNotSimpl@rDF
0
A-Edge_CLD_Expression.nb 57
TogetherB HGEddNotSimpl@rDL ê.
:ArcSinB2 - r2
-2 + 3 r2F Ø ArcSinB
2 2 r -1 + r2
-2 + 3 r2F + ArcSinB
4 + 4 r2 - 7 r4
I2 - 3 r2M2F ,
ArcSinB3 -1 + r2
-2 + 3 r2F Ø ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F -
p
6> F
GEdd@r_D := -1
432 p r3 V * Sin@aD
-16 2 + 24 2 r2 + 8 6 p r2 - 12 p r3 - 9 2 p r4 + 12 6 p r4 + 8 2 -1 + r2 -
20 2 r2 -1 + r2 + 36 2 r4 ArcSinB-1 + r2
rF - 24 r3 ArcSinB
4 + 4 r2 - 7 r4
I2 - 3 r2M2F -
16 6 r2 ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F - 24 6 r4 ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F ;
ü checks
FullSimplify@HLimit@GEdd@rD, r Ø 1, Direction Ø -1D - Limit@GeccNotSimpl@rD, r Ø 1, Direction Ø 1DL ê.8a Ø ArcCos@-1 ê 3D<D
LimitBGEdd@rD, r Ø 2 , Direction Ø 1F
PlotB: ArcSinB2 - r2
-2 + 3 r2F - ArcSinB
2 2 r -1 + r2
-2 + 3 r2F ì p,
ArcSinB3 -1 + r2
-2 + 3 r2F - ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F ì p,
ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F ì p,
+3 * ArcSinB3 -1 + r2
-2 + 3 r2F ì p - 1 ê 2, 1 ê 2, -1 ê 2>, :r, 1, 2 >F
†
58 A-Edge_CLD_Expression.nb
†
Simplification of ArcSinB2 - r2
-2 + 3 r2F - ArcSinB
2 2 r -1 + r2
-2 + 3 r2F
:ArcSinB2 - r2
-2 + 3 r2F Ø ArcSinB
2 2 r -1 + r2
-2 + 3 r2F + ArcSinB
4 + 4 r2 - 7 r4
I2 - 3 r2M2F >
PlotB: ArcSinB2 - r2
-2 + 3 r2F - ArcSinB
2 2 r -1 + r2
-2 + 3 r2F ì p,
ArcSinB2 - r2
-2 + 3 r2F ì p, ArcSinB
2 2 r -1 + r2
-2 + 3 r2F ì p, 1 ê 2, -1 ê 2,
ArcSinB2 - r2
-2 + 3 r2F - ArcSinB
2 2 r -1 + r2
-2 + 3 r2F + ArcSinB
4 + 4 r2 - 7 r4
I2 - 3 r2M2F >, :r, 1, 2 >,
PlotStyle Ø 88Blue<, 8Green<, 8Cyan<, 8Purple<, 8Magenta<, 8Red, [email protected]<<F
FullSimplifyBTrigToExpBSinB ArcSinB2 - r2
-2 + 3 r2F - ArcSinB
2 2 r -1 + r2
-2 + 3 r2F FF,
Assumptions Ø :1 < r < 2 >F
LimitB ArcSinB2 - r2
-2 + 3 r2F - ArcSinB
2 2 r -1 + r2
-2 + 3 r2F + ArcSinB
4 + 4 r2 - 7 r4
I2 - 3 r2M2F ,
r Ø 1, Direction Ø -1F
LimitB ArcSinB2 - r2
-2 + 3 r2F - ArcSinB
2 2 r -1 + r2
-2 + 3 r2F + ArcSinB
4 + 4 r2 - 7 r4
I2 - 3 r2M2F ,
r Ø 2 , Direction Ø 1F
4 + 4 r2 - 7 r4
I2 - 3 r2M2
0
0
A-Edge_CLD_Expression.nb 59
† Simplification of ArcSinB3 -1 + r2
-2 + 3 r2F - ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F
: ArcSinB3 -1 + r2
-2 + 3 r2F Ø ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F -
p
6>
PlotB:
ArcSinB3 -1 + r2
-2 + 3 r2F - ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F ì p, 1 ê 2, -1 ê 2,
ArcSinB3 -1 + r2
-2 + 3 r2F - ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F -
p
6>, :r, 1, 2 >,
PlotStyle Ø 88Blue<, 8Purple<, 8Magenta<, 8Red, [email protected]<<F
SimplifyB FullSimplifyBTrigExpandBSinBArcSinB3 -1 + r2
-2 + 3 r2F - ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2FFF,
Assumptions Ø :1 < r < 2 >F ê.
: I-1 + r2M r2 - 2 -1 + r2 Ø -1 + r2 * 1 - -1 + r2 >F
LimitBArcSinB3 -1 + r2
-2 + 3 r2F - ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F -
p
6, r Ø 1, Direction Ø -1F
LimitB ArcSinB3 -1 + r2
-2 + 3 r2F - ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F -
p
6, r Ø 2 , Direction Ø 1F
60 A-Edge_CLD_Expression.nb
† Simplification of
ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F + 3 * ArcSinB
3 -1 + r2
-2 + 3 r2F ã
H* ArcSinB 3 -1+r2
-2+3 r2FØ*L
3 * ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F -
p
6+ ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F ã
4 * ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F -
p
2
PlotB ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F + 3 * ArcSinB
3 -1 + r2
-2 + 3 r2F -
4 * ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F -
p
2, :r, 1, 2 >F
A-Edge_CLD_Expression.nb 61
H* PRROF OF THE IDENTITIES *L
:-4 + 4 r2 + 9 r4 + 12 r2 -1 + r2 Ø 3 * r^2 + 2 * -1 + r2 * 2 * -1 + r2 + 3 * r^2 ,
2 -1 + -1 + r2 + r2 5 + 3 -1 + r2 Ø 2 * -1 + r2 + 3 * r^2 * 1 + -1 + r2 ,
-8 + 9 r2 + 6 -1 + r2 Ø 1 + 3 * -1 + r2 ,
-4 + r4 - 4 r2 -1 + -1 + r2 Ø r^2 - 2 * -1 + r2 ,
-8 + 12 -1 + r2 - 12 r2 1 + -1 + r2 + 9 r4 2 + -1 + r2 Ø
-4 + 3 r2 + 6 -1 + r2 2 + 3 r2 -1 + r2 ,
ArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F Ø ArcSinB
2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F,
arctanAA Ø -arctanBB + ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F,
ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F Ø 2 * ArcSinB
2 1 - -1 + r2
-2 + 3 r2 3 r2 + 2 -1 + r2
F +
ArcSinB2 2 r -4 + 3 r2 + 6 -1 + r2 2 + 3 r2 -1 + r2
I-2 + 3 r2M 3 r2 + 2 -1 + r22
F ,
ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F Ø -2 * ArcSinB
2 1 - -1 + r2
-2 + 3 r2 3 r2 + 2 -1 + r2
F +
p
2+ ArcSinB
2 - r2
-2 + 3 r2F ,
ArcSinB3 -1 + -1 + r2
2 -2 + 3 r2F Ø ArcSinB
3 -1 + r2
-2 + 3 r2F - p ê 3 ,
:ArcSinB-1 + -1 + r2
2 rF Ø ArcSinB
-1 + r2
rF - p ê 4 >;
62 A-Edge_CLD_Expression.nb
ExpandB2 -1 + -1 + r2 + r2 5 + 3 -1 + r2 - 2 * -1 + r2 + 3 * r^2 * 1 + -1 + r2 F
ExpandB-4 + 4 r2 + 9 r4 + 12 r2 -1 + r2 - 3 * r^2 + 2 * -1 + r2 * 2 * -1 + r2 + 3 * r^2 F
ExpandB-8 + 9 r2 + 6 -1 + r2 - 1 + 3 * -1 + r2 ^2F
ExpandB-4 + r4 - 4 r2 -1 + -1 + r2 - r^2 - 2 * -1 + r2 ^2F
ExpandB-8 + 12 -1 + r2 - 12 r2 1 + -1 + r2 +
9 r4 2 + -1 + r2 - -4 + 3 r2 + 6 -1 + r2 2 + 3 r2 -1 + r2 F
:ArcSinB3 -1 + -1 + r2
2 -2 + 3 r2F Ø ArcSinB
3 -1 + r2
-2 + 3 r2F - p ê 3>;
PlotBArcSinB3 -1 + -1 + r2
2 -2 + 3 r2F - ArcSinB
3 -1 + r2
-2 + 3 r2F - p ê 3 , :r, 1, 2 >F
PlotBArcSinB-1 + -1 + r2
2 rF - ArcSinB
-1 + r2
rF - p ê 4 , :r, 1, 2 >F
H* THAT ALLOW TO GET THE FOLLOWING IDENTITY *L
:ArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F Ø ArcSinB
2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F> ;
H* and consequenlty the identity
:arctanAAØ-arctanBB+ArcSinB2 2 r 2 -1+ -1+r2 +r2 5+3 -1+r2
-4+9 r4+4 r2 1+3 -1+r2F>
becomes *L
:arctanAA Ø -arctanBB + ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F>
PlotB:
>,
A-Edge_CLD_Expression.nb 63
PlotB:
ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F - ArcSinB
2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F>,
:r, 1, 2 >, PlotStyle Ø 88Blue<<, PlotPoints Ø 1000F
LimitBArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F -
ArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F, r Ø 1, Direction Ø -1F
LimitBArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F -
ArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F, r Ø 2 , Direction Ø 1F
PlotB: ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F - 2 * ArcSinB
2 1 - -1 + r2
-2 + 3 r2 3 r2 + 2 -1 + r2
F ì p,
ArcSinB2 2 r 1 + -1 + r2
3 r2 + 2 -1 + r2F - ArcSinB
2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F,
-1 ê 2, 1 ê 2>, :r, 1, 2 >,
PlotStyle Ø 88Blue<, 8Red, [email protected]<, 8Green<, 8Magenta<<F
64 A-Edge_CLD_Expression.nb
THE EDGE CASE
ü the following expressions have been copied from "octahedron_E_FA_FNL.nb" The names of the functions have been changed passing from FAintgrl[] to FaEcld[]old The functions ..OLD[r] were worked out in "octahedron_A_FA.nb". We added the OLD and changed the prefix as specified above
FaEcldAAold@r_D :=1
36H-4 + 3 p - 4 ArcSec@-3DL +
9 + 2 3 p
216 2* r;
FaEcldBBold@r_D := -p
27 6 * r^3+
1
9 * r
2
3p +
-1
9-25 p
192+11 ArcCotB 2 F
144-
5
288ArcCotB2 2 F +
1
18ArcTanB2 2 F + r *
1
24 2+
p
9 6;
H* result obtained in "octahedron_A_Fa" *L
FaEcldBBOLD@r_D :=1
20 736 r3Jp J-128 6 + 768 6 r2 - 1368 r3 + 3 J63 + 128 6 N r4N -
9 r3 J8 J32 - 22 ArcCscB 3 F + 21 ArcSec@-3DN +
r J-48 2 + 42 ArcCscB 3 F + 21 ArcSec@-3DNNN;
FaEcldCCold@r_D :=J9 + 2 3 pN r
216 2+
p J-4 6 + 24 6 r2 - 72 r3 + 9 6 r4N
648 r3+
1
576J-64 - 11 p + 44 ArcCotB 2 F - 10 ArcCotB2 2 F + 32 ArcTanB2 2 FN +
1
3888 r3-36 -6 + 8 r2 + 180 r2 -6 + 8 r2 +
- +
A-Edge_CLD_Expression.nb 65
6 I-20 + 144 r2 + 45 r4M ArcSinB-1 - 3 -3 + 4 r2
4 -2 + 3 r2F - 432 r3 ArcSinB
r - 2 -3 + 4 r2
3 -2 + 3 r2F +
54 6 r4 ArcSinB3 - -3 + 4 r2
4 rF - 54 6 r4 ArcSinB
3 + -3 + 4 r2
4 rF +
432 r3 ArcSinBr + 2 -3 + 4 r2
3 -2 + 3 r2F + 20 6 ArcSinB
-1 + 3 -3 + 4 r2
4 -2 + 3 r2F -
144 6 r2 ArcSinB-1 + 3 -3 + 4 r2
4 -2 + 3 r2F - 45 6 r4 ArcSinB
-1 + 3 -3 + 4 r2
4 -2 + 3 r2F -
4 6 ArcTanB1 - 3 -3 + 4 r2
3 1 + -3 + 4 r2F + 9 6 r4 ArcTanB
1 - 3 -3 + 4 r2
3 1 + -3 + 4 r2F +
4 6 ArcTanB1 + 3 -3 + 4 r2
3 1 - -3 + 4 r2F - 9 6 r4 ArcTanB
1 + 3 -3 + 4 r2
3 1 - -3 + 4 r2F ;
FaEcldCCOLD@r_D := H* old result octahedron_A_FA.nb" *LJ9 + 2 3 pN r
216 2+
1
576J-64 - 11 p + 44 ArcCotB 2 F - 10 ArcCotB2 2 F + 32 ArcTanB2 2 FN -
1
1296 r3 -3 + 4 r2
2 -36 + 228 r2 + 2 p -9 + 12 r2 - 15 r4 16 + 3 p -9 + 12 r2 + 144 r3 -3 + 4 r2
ArcCotB2 r -6 + 8 r2
6 - 7 r2F - 18 6 r4 -3 + 4 r2 ArcCscB
4 r
3 - -3 + 4 r2F +
1
2-3 + 4 r2
-24 p J 6 - 6 rN r2 + 4 6 p - 6 p r2 + 18 p r4 + 9 r4 ArcCscB4 r
3 + -3 + 4 r2F - 4
I1 - 6 r2 + 18 r4M ArcTanB -1
9 - 12 r2I-5 + 6 r2MF + 81 r4 ArcTanB
-5 + 6 r2
-9 + 12 r2F ;
66 A-Edge_CLD_Expression.nb
FaEcldDDold@r_D :=
1
5184 r3 -2 + -2 + 3 r2192 2 + 32 6 p - 576 2 r2 - 192 6 p r2 + 1152 r3 +
288 p r3 - 216 2 r4 + 216 p r4 - 72 6 p r4 - 240 2 r2 -1 + r2 - 96 2 -2 + 3 r2 -
16 6 p -2 + 3 r2 + 288 2 r2 -2 + 3 r2 + 96 6 p r2 -2 + 3 r2 -
576 r3 -2 + 3 r2 - 144 p r3 -2 + 3 r2 + 108 2 r4 -2 + 3 r2 - 108 p r4 -2 + 3 r2 +
36 6 p r4 -2 + 3 r2 + 120 2 r2 2 - 5 r2 + 3 r4 + 84 r2 -4 + 3 r2 + 8 -2 + 3 r2 -
18 r4 -4 + 3 r2 + 8 -2 + 3 r2 - 24 r2 I-2 + 3 r2M -4 + 3 r2 + 8 -2 + 3 r2 -
36 r2 56 - 102 r2 + 9 r4 + 32 -2 + 3 r2 + 12 r2 -2 + 3 r2 -
6 r2 I-2 + 3 r2M 56 - 102 r2 + 9 r4 + 32 -2 + 3 r2 + 12 r2 -2 + 3 r2 -
162 2 r4 ArcCscB3 r
2 2 - -2 + 3 r2F +
81 2 r4 -2 + 3 r2 ArcCscB3 r
2 2 - -2 + 3 r2F + 162 2 r4 ArcSinB
-1 + -1 + r2
2 rF -
81 2 r4 -2 + 3 r2 ArcSinB-1 + -1 + r2
2 rF - 64 6 ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F +
384 6 r2 ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F + 144 6 r4 ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F +
32 6 -2 + 3 r2 ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F - 192 6 r2 -2 + 3 r2 ArcSinB
1 + 3 -1 + r2
2 -2 + 3 r2F -
-
A-Edge_CLD_Expression.nb 67
72 6 r4 -2 + 3 r2 ArcSinB1 + 3 -1 + r2
2 -2 + 3 r2F -
576 r3 ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F -
216 r4 ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F +
288 r3 -2 + 3 r2 ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F +
108 r4 -2 + 3 r2 ArcSinB2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2
-2 + 3 r2 1 + 2 r - -1 + r2F -
162 2 r4 ArcSinB2 -2 + -2 + 3 r2
3 rF + 81 2 r4 -2 + 3 r2
ArcSinB2 -2 + -2 + 3 r2
3 rF + 576 r3 ArcSinB
r + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F -
216 r4 ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F -
288 r3 -2 + 3 r2 ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F +
108 r4 -2 + 3 r2 ArcSinBr + 2 r2 + 3 r -1 + r2 - 2 1 + -1 + r2
-2 + 3 r2 -1 + 2 r + -1 + r2F +
- -
68 A-Edge_CLD_Expression.nb
162 2 r4 ArcTanB1 - -1 + r2
1 + -1 + r2F - 81 2 r4 -2 + 3 r2 ArcTanB
1 - -1 + r2
1 + -1 + r2F -
576 r3 ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F - 216 r4 ArcTanB
2 1 + r - -1 + r2
1 + -1 + r2F +
288 r3 -2 + 3 r2 ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F +
108 r4 -2 + 3 r2 ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F -
576 r3 ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F + 216 r4 ArcTanB
2 -1 + r + -1 + r2
1 + -1 + r2F +
288 r3 -2 + 3 r2 ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F -
108 r4 -2 + 3 r2 ArcTanB2 -1 + r + -1 + r2
1 + -1 + r2F ;
FaEcldDDOLD@r_D := -1
4 3 r2 + 2 -1 + r2r 1 + -1 + r2
2 r2 - 2 -1 + r2
6 r2 + 4 -1 + r2
ArcCosB
3 r2 + 2 -1 + r2 2 -
2 1+ -1+r2 4 r4-2 -1+r2 +r2 -3+4 -1+r2
3 r2+2 -1+r2
8 r r2 + -1 + r2F +
A-Edge_CLD_Expression.nb 69
144 2 r3 9 r6 1 - -1 + r2 - 8 -1 + r2 1 - -1 + r2 +
r2 -28 1 - -1 + r2 + 8 -1 + r2 1 - -1 + r2 +
r4 28 1 - -1 + r2 + 30 -1 + r2 1 - -1 + r2 ArcCotBr2 - 2 -1 + r2
3 r2 + 2 -1 + r2F +
1
3 r2 + 2 -1 + r2
r -2048 2 - 2048 r - 512 p r + 1024 2 r2 + 144 p r2 + 4096 r3 +
1024 p r3 + 4096 2 r4 - 288 p r4 + 25 600 r5 + 6400 p r5 - 16 896 2 r6 - 1800 p r6 -
27 648 r7 - 6912 p r7 + 24 192 2 r8 + 1944 p r8 - 10 368 r9 - 2592 p r9 + 729 p r10 -
1280 2 -1 + r2 + 8704 2 r2 -1 + r2 + 12 288 r3 -1 + r2 + 3072 p r3 -1 + r2 -
10 880 2 r4 -1 + r2 - 864 p r4 -1 + r2 - 12 288 r5 -1 + r2 - 3072 p r5 -1 + r2 +
17 280 2 r6 -1 + r2 + 864 p r6 -1 + r2 - 27 648 r7 -1 + r2 - 6912 p r7 -1 + r2 +
9072 2 r8 -1 + r2 + 1944 p r8 -1 + r2 + 384 2 1 - -1 + r2 +
1104 2 r2 1 - -1 + r2 - 6600 2 r4 1 - -1 + r2 + 2844 2 r6 1 - -1 + r2 +
2754 2 r8 1 - -1 + r2 + 384 2 -1 + r2 1 - -1 + r2 -
2640 2 r2 -1 + r2 1 - -1 + r2 - 984 2 r4 -1 + r2 1 - -1 + r2 +
5508 2 r6 -1 + r2 1 - -1 + r2 + 486 2 r8 -1 + r2 1 - -1 + r2 - 18 r2
16 + 81 r8 + 216 r6 1 + -1 + r2 - 32 r2 1 + 3 -1 + r2 + 8 r4 -25 + 12 -1 + r2
ArcSinB2 1 + -1 + r2
3 r2 + 2 -1 + r2
F + 128 r 16 + 81 r8 + 216 r6 1 + -1 + r2 -
+ ì
-
70 A-Edge_CLD_Expression.nb
32 r2 1 + 3 -1 + r2 + 8 r4 -25 + 12 -1 + r2 ArcTanB1 + -1 + r2
2 rF ì
1152 r2 3 r2 + 2 -1 + r23ê2
-4 + 9 r4 + 4 r2 1 + 3 -1 + r2 -
1
331 776 r332 p J64 6 - 384 6 r2 + 9 J-171 + 128 6 N r4N -
1
r2 + -1 + r22
3 r2 + 2 -1 + r22
1024 6 p I1 - 6 r2 + 18 r4M r2 + -1 + r22
3 r2 + 2 -1 + r22
-1
3 r2 + 2 -1 + r2312 2
I-64 + 384 r2 + 243 r4M r2 - 2 -1 + r2 1 + -1 + r2 3 r2 + 2 -1 + r22
+
r4 r2 - 2 -1 + r2 3 + 3 -1 + r25
+ 2 I32 + 192 r2 + 243 r4M
r2 - 2 -1 + r2 3 r2 + 2 -1 + r2 2 -1 + -1 + r2 + r2 3 + -1 + r2 -
252 r2 1 + -1 + r24
-4 + 4 r6 + 4 r2 3 + -1 + r2 - r4 11 + 4 -1 + r2 -
8 I-16 + 23 r2M 1 + -1 + r22
3 r2 + 2 -1 + r2
-4 + 4 r6 + 4 r2 3 + -1 + r2 - r4 11 + 4 -1 + r2 +
-
-
A-Edge_CLD_Expression.nb 71
68 3 r3 + 2 r -1 + r22
-4 + 4 r6 + 4 r2 3 + -1 + r2 - r4 11 + 4 -1 + r2 -
98 496 r4 r2 + -1 + r22
ArcSinB2 1+ -1+r2
3 r2+2 -1+r2
F
3 r2 + 2 -1 + r22
- 64 6 ArcTanB
3 -4 + 4 r6 + 4 r2 3 + -1 + r2 - r4 11 + 4 -1 + r2
6 r4 - 2 -1 + r2 + r2 -5 + 6 -1 + r2F +
1
3 r2 + 2 -1 + r2264
32 6 I1 - 6 r2 + 18 r4M r2 + -1 + r22
ArcTanB3 1 + -1 + r2
r2 - 2 -1 + r2
F + 36 r I-18 +
47 r2M r2 + -1 + r22
ArcTanB
r r2-2 -1+r2
4 r4-2 -1+r2 +r2 -3+4 -1+r2
2F - 6 -12 + 288
r8 + 192 r6 1 + 3 -1 + r2 + 4 r2 27 + 5 -1 + r2 - r4 377 + 192 -1 + r2
ArcTanB
3 -4 + 4 r6 + 4 r2 3 + -1 + r2 - r4 11 + 4 -1 + r2
6 r4 - 2 -1 + r2 + r2 -5 + 6 -1 + r2F -
1
32 r2J5 p r3 - 4 6 p r3 - 4 Â Log@2D + 14 Â r2 Log@2D - Â 6 r3 Log@54D +
2 Â I-2 + 7 r2M Log@rD + Â I-2 + 7 r2M LogA-2 + 3 r2E +
2 Â 6 r3 LogA-6 + 9 r2EN +
72 A-Edge_CLD_Expression.nb
-2 rr2 - 2 -1 + r2
3 r2 + 2 -1 + r2+ 2 r3
r2 - 2 -1 + r2
3 r2 + 2 -1 + r2+
2 r3 r2 - 2 -1 + r2 1 + -1 + r22
3 r2 + 2 -1 + r23ê2
+
2 r3 1 + -1 + r2 -4 + 4 r6 + 4 r2 3 + -1 + r2 - r4 11 + 4 -1 + r2
3 r2 + 2 -1 + r23ê2
+
5 r3 -2 +
8 r2 r2 + -1 + r2
3 r2 + 2 -1 + r2ArcSinB
2 1 + -1 + r2
3 r2 + 2 -1 + r2
F -
2 6 r3 -2 +
8 r2 r2 + -1 + r2
3 r2 + 2 -1 + r2ArcTanB
3 1 + -1 + r2
r2 - 2 -1 + r2
F -
6 r3 2 -
8 r2 r2 + -1 + r2
3 r2 + 2 -1 + r2LogB
8 6 r2 + -1 + r2
3 r2 + 2 -1 + r2F -
2 2 -
8 r2 r2 + -1 + r2
3 r2 + 2 -1 + r2
LogB2 r r -1 +
2 1 + -1 + r22
3 r2 + 2 -1 + r2+ -2 +
8 r2 r2 + -1 + r2
3 r2 + 2 -1 + r2F + 7
r2
A-Edge_CLD_Expression.nb 73
r2 2 -
8 r2 r2 + -1 + r2
3 r2 + 2 -1 + r2
LogB2 r r -1 +
2 1 + -1 + r22
3 r2 + 2 -1 + r2+ -2 +
8 r2 r2 + -1 + r2
3 r2 + 2 -1 + r2F +
6 r3 2 -
8 r2 r2 + -1 + r2
3 r2 + 2 -1 + r2LogB
1
3 r2 + 2 -1 + r24 -6 r4 + 2 -1 + r2 +
r2 5 - 6 -1 + r2 + 3 4 - 4 r6 - 4 r2 3 + -1 + r2 + r4 11 + 4 -1 + r2 F ì
16 r2 -2 +
8 r2 r2 + -1 + r2
3 r2 + 2 -1 + r2;
ü THE Fb Caseü 0 < r < 2ê3
ü the following expressions have been copied by "octahedron_E_FB_FNL.nb" The names of the functions have been changed passing from FBintgrl[A,B,C,or D] to FbEcld[]old The functions ..OLD[r] were worked out in "octahedron_A_FB.nb". We added the OLD and changed the prefix as specified above
H* 0 < r < 2ê3 *L
FbEcldAold@r_D :=1
864J2 p J30 + 2 J-9 + 5 3 N rN + 3 J6 2 r - 72 ArcCotB 2 F -
11 ArcSec@-3D - 2 ArcSecB 3 F + 8 J4 - 4 2 + ArcTanB2 2 FNNN;
H* 2ê3 < r < 3 í 2 *L
74 A-Edge_CLD_Expression.nb
H* 2ê3 < r < 3 í 2 *L
FbEcldBold@r_D := FbEcldAold@rD;
H* 3 í 2 < r < 1 *L
FbEcldCold@r_D :=1
864J2 p J30 + 2 J-9 + 5 3 N rN + 3 J6 2 r - 72 ArcCotB 2 F - 11 ArcSec@-3D -
2 ArcSecB 3 F + 8 J4 - 4 2 + ArcTanB2 2 FNNN +
1
1296 r3-90 6 r4 ArcCscB
2 r
-3 + 4 r2F + 4 6 I2 + 9 r2M ArcSinB
1
2
9 - 12 r2
2 - 3 r2F +
3 -8 -6 + 8 r2 + 30 r2 -6 + 8 r2 - 9 -3 + 4 r2 ArcSinB2
3F -
24 r2 -3 + 4 r2 ArcSinB2
3F + 6 6 r2 I-2 + 3 r2M ArcSinB
-9 + 12 r2
2 -2 + 3 r2F +
9 -3 + 4 r2 ArcTanB 2 F + 24 r2 -3 + 4 r2 ArcTanB 2 F +
24 r3 ArcTanB4 - 6 r2 - 3 r -3 + 4 r2
2F - 24 r3 ArcTanB
4 - 6 r2 + 3 r -3 + 4 r2
2F ;
FbEcldCOLD@r_D :=1
1296 r33 2 J9 - 9 p + 5 3 pN r4 - 24 -6 + 8 r2 +
90 r2 -6 + 8 r2 - 90 6 r4 ArcCscB2 r
-3 + 4 r2F +
2 6 I4 + 27 r4M ArcTanB -9 + 12 r2 F - 36 r3 -p + 2 -2 + 2 2 + ArcTanB2 2 F +
ArcTanB4 + 3 r -2 r + -3 + 4 r2
2F + ArcTanB
-4 + 3 r 2 r + -3 + 4 r2
2F ;
H* 1 < R < 2 *LFbEcldDold@r_D :=
A-Edge_CLD_Expression.nb 75
1
1296 r3108 2 r4 ArcSinB
-1 + -1 + r2
2 rF - 2 6 I4 + 27 r4M ArcSinB
3 -1 + -1 + r2
2 -2 + 3 r2F -
3 8 2 - 48 r3 - 12 p r3 + 9 2 r4 - 8 2 -1 + r2 + 30 2 r2 -1 + r2 +
24 r3 ArcTanB2 1 + r - -1 + r2
1 + -1 + r2F + 24 r3 ArcTanB
2 -1 + r + -1 + r2
1 + -1 + r2F + 24 r3
ArcTanB4 + 3 r - 6 r2 - 3 r -1 + r2
2 1 + 3 -1 + r2F - 24 r3 ArcTanB
4 - 6 r2 + 3 r -1 + -1 + r2
2 1 + 3 -1 + r2F ;
FbEcldDOLD@r_D :=4 + p
36+
1
5184 r3-108 2 - 324 2 r2 + 135 2 r4 + 108 2 r2 -1 + r2 - 192 2 r2 - 2 -1 + r2 +
396 2 r2 r2 - 2 -1 + r2 + 72 2 I-1 + r2M r2 - 2 -1 + r2 -
216 2 I-1 + r2M r2 + 2 -1 + r2 - 180 r2 2 r2 + 4 -1 + r2 +
18 2 I-1 + r2M 9 r4 - 4 r2 5 + 3 -1 + r2 + 4 3 + 4 -1 + r2 -
12 18 r4 - 8 r2 5 + 3 -1 + r2 + 8 3 + 4 -1 + r2 +
81 r2 18 r4 - 8 r2 5 + 3 -1 + r2 + 8 3 + 4 -1 + r2 -
+ +
76 A-Edge_CLD_Expression.nb
36 11 r2 r2 - 2 -1 + r2 + 2 I-1 + r2M r2 - 2 -1 + r2
ArcCosB2 2
9 r2 + 6 -1 + r2
F + 198 r2 r2 - 2 -1 + r2 ArcSec@-3D +
36 I-1 + r2M r2 - 2 -1 + r2 ArcSec@-3D - 432 2 r4 ArcSinBr2 - 2 -1 + r2
2 rF -
396 r2 r2 - 2 -1 + r2 ArcSinBr2 - 2 -1 + r2
3 r2 + 2 -1 + r2F -
72 I-1 + r2M r2 - 2 -1 + r2 ArcSinBr2 - 2 -1 + r2
3 r2 + 2 -1 + r2F + 32 6
ArcTanB 3r2 - 2 -1 + r2
-8 + 9 r2 + 6 -1 + r2F + 216 6 r4 ArcTanB 3
r2 - 2 -1 + r2
-8 + 9 r2 + 6 -1 + r2F +
288 r3 ArcTanB
2 -r + r2 - 2 -1 + r2
r2 + 2 -1 + r2
F - 288 r3
+ +
A-Edge_CLD_Expression.nb 77
ArcTanB
2 r + r2 - 2 -1 + r2
r2 + 2 -1 + r2
F + 288 r3 ArcTanB4 - 6 r2 - 3 r r2 - 2 -1 + r2
2 -8 + 9 r2 + 6 -1 + r2
F +
288 r3 ArcTanB-4 + 6 r2 - 3 r r2 - 2 -1 + r2
2 -8 + 9 r2 + 6 -1 + r2
F ;
IDENTITIES FOR THE Fa function
IDENTITIES for the range 2 ê 3 < r < 3 í 2
: LogB2 - 3 r2 - 2 -2 + 3 r2 + 3 r -2 + 3 r2 F Ø
LogB -2 + 3 r2 F + LogB-2 + 3 * r - -2 + 3 r2 F >
: LogB-2 + 3 r2 - 2 -2 + 3 r2 + 3 r -2 + 3 r2 F Ø
LogB -2 + 3 r2 F + LogB-2 + 3 * r + -2 + 3 r2 F >
: LogB-2 + 3 r2 + 2 -2 + 3 r2 + 3 r -2 + 3 r2 F Ø
LogB -2 + 3 r2 F + LogB2 + 3 * r + -2 + 3 r2 F >
: LogB2 - 3 r2 + 2 -2 + 3 r2 + 3 r -2 + 3 r2 F Ø LogB -2 + 3 r2 F + LogB2 + 3 * r - -2 + 3 r2 F >
: LogB-1
2 - 3 r + -2 + 3 r2F Ø -LogB- 2 - 3 r + -2 + 3 r2 F >
: -1 + 3 r2 + 2 -2 + 3 r2 Ø 1 + -2 + 3 r2 >
: -1 + 3 r2 - 2 -2 + 3 r2 Ø 1 - -2 + 3 r2 >
IDENTITIES 3 /2 < r < 1
78 A-Edge_CLD_Expression.nb
: -1 + 3 r2 + 2 -2 + 3 r2 Ø 1 + -2 + 3 r2 >
: -1 + 3 r2 - 2 -2 + 3 r2 Ø 1 - -2 + 3 r2 >
: LogB-2 - -2 + 3 r2 + Â -2 + 6 r2 - 4 -2 + 3 r2 F Ø
Log@3 * rD + Â * p - ArcSinB2 * 1 - -2 + 3 r2
3 rF >
: LogB-2 + -2 + 3 r2 + Â -2 + 6 r2 + 4 -2 + 3 r2 F Ø
Log@3 * rD + Â * p - ArcSinB2 * 1 + -2 + 3 r2
3 rF >
IDENTITIES J 3 í 2 < r < 1 N
: LogB4 Â
3- 2 Â 3 r t + 2 -2 + 4 r t + r2 I1 - 3 t2M F Ø
LogB2 -2 + 3 r2
3F + Â * ArcSinB
2 - 3 r t
-2 + 3 r2F ,
LogB 24 -2 Â + Â r2 H1 + 3 tL + 2 -2 + 4 r t + r2 I1 - 3 t2M +
r -2 Â + 2 Â t + 2 -2 + r2 + 4 r t - 3 r2 t2 ì IH1 + rL2 H8 + 3 rL H1 + tLMF Ø
-LogAIH1 + rL2 H8 + 3 rL H1 + tLME + LogB24 r -2 + 3 r2 H1 + tLF +
 * ArcSinB-2 - 2 r + r2 + 2 r t + 3 r2 t
r -2 + 3 r2 H1 + tL
F ,
LogB- 24 -2 Â - Â r2 H-1 + 3 tL - 2 -2 + 4 r t + r2 I1 - 3 t2M +
r 2 Â + 2 Â t + 2 -2 + r2 + 4 r t - 3 r2 t2 ì IH-1 + rL2 H-8 + 3 rL H-1 + tLMF Ø
A-Edge_CLD_Expression.nb 79
-LogAIH-1 + rL2 H-8 + 3 rL H-1 + tLME + LogB-24 r -2 + 3 r2 H-1 + tLF +
 * ArcSinB-2 + r2 H1 - 3 tL + 2 r H1 + tL
r -2 + 3 r2 H-1 + tL
F >
: r + 4 r2 - 3 r -3 + 4 r2 - 2 1 + -3 + 4 r2 Ø 3 + 4 r - -3 + 4 r2 *r - 2 * -3 + 4 r2
3>
: 4 r2 - 2 1 + -3 + 4 r2 + r -1 + 3 -3 + 4 r2 Ø
-3 + 4 r + -3 + 4 r2 *r + 2 * -3 + 4 r2
3>
: -2 + r + 4 r2 + 2 -3 + 4 r2 + 3 r -3 + 4 r2 Ø 3 + 4 * r + -3 + 4 r2 *r + 2 * -3 + 4 r2
3>
: 2 + r - 4 r2 - 2 -3 + 4 r2 + 3 r -3 + 4 r2 Ø 3 - 4 * r + -3 + 4 r2 *r - 2 * -3 + 4 r2
3>
: -1 + 2 r2 - -3 + 4 r2 Ø1 - -3 + 4 r2
2>
: -1 + 2 r2 + -3 + 4 r2 Ø1 + -3 + 4 r2
2>
ü IDENTITIES 1 < r < 2
: 2 + 3 r2 - 4 -2 + 3 r2 Ø 2 - -2 + 3 r2 >
: -2 + 6 r2 + 4 -2 + 3 r2 Ø 2 + 2 * -2 + 3 r2 ^2>
:-4 + 3 r2 + 8 -2 + 3 r2
-2 + 6 r2 + 4 -2 + 3 r2Ø
-4 + 3 r2 + 8 -2 + 3 r2
2 * 1 + -2 + 3 r2>
80 A-Edge_CLD_Expression.nb
:4 + 6 r2 - 8 -2 + 3 r2
-4 + 3 r2 + 8 -2 + 3 r2Ø
2 * 2 - -2 + 3 r2
-4 + 3 r2 + 8 -2 + 3 r2
>
:
4 + 3 r - 2 -2 + 3 r2 2+3 r2-4 -2+3 r2
-4+3 r2+8 -2+3 r2
-2 + -2 + 3 r2Ø
-4 - 3 r + 2 -2 + 3 r2
-4 + 3 r2 + 8 -2 + 3 r2
>
:
4 - 3 r - 2 -2 + 3 r2 2+3 r2-4 -2+3 r2
-4+3 r2+8 -2+3 r2
-2 + -2 + 3 r2Ø
-4 + 3 r + 2 -2 + 3 r2
-4 + 3 r2 + 8 -2 + 3 r2
>
:LogB4 Â
3- 2 Â 3 r t + 2 -2 + r2 + 4 r t - 3 r2 t2 F Ø
LogB2 -2 + 3 r2
3F + Â * ArcSinB-
-2 + 3 r t
-2 + 3 r2F > ;
:LogB 24 -2 Â - Â r2 H-1 + 3 tL - 2 -2 + r2 + 4 r t - 3 r2 t2 +
r 2 Â + 2 Â t + 2 -2 + r2 + 4 r t - 3 r2 t2 ì IH-1 + rL2 H-8 + 3 rL H-1 + tLMF Ø
LogAH24 L ë IH-1 + rL2 H-8 + 3 rL H-1 + tLME + LogB -r -2 + 3 r2 H-1 + tLF +
 * ArcSinB2 - 2 r - r2 - 2 r t + 3 r2 t
r -2 + 3 r2 H-1 + tL
F > ;
: LogB-1
H1 + rL2 H8 + 3 rL H1 + tL24 Â -2 + r2 H1 + 3 tL - Â 2 -2 + r2 + 4 r t - 3 r2 t2 +
F Ø
A-Edge_CLD_Expression.nb 81
r -2 + 2 t - Â 2 -2 + r2 + 4 r t - 3 r2 t2 F Ø LogB24
H1 + rL2 H8 + 3 rL H1 + tLF +
LogBr -2 + 3 r2 H1 + tLF + Â * p - ArcSinB2 - 2 r H-1 + tL - r2 H1 + 3 tL
r -2 + 3 r2 H1 + tL
F >
82 A-Edge_CLD_Expression.nb