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


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


-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 ,

,


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


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


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;


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


) = 4p V


) = 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;


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


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


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


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


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


1 - 3 -3 + 4 r2

4 -2 + 3 r2F , :r, 3 í 2, 1>F


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


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


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


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


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


1 - 3 -3 + 4 r2

4 -2 + 3 r2F >F ê.

:ArcSinBr + 2 -3 + 4 r2


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


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


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


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


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


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 +


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


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 +


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


GeccNotSimpl@r_D :=1


-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


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


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) }


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


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


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


ü

1 < r < 2ü the FA case


: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,


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


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,


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>


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


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


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


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


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


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 -


-1 + 2 r + -1 + r2F,

> ,



-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 -


-1 + 2 r + -1 + r2F,


-1 - 2 r + -1 + r2F> , r Ø 1, Direction Ø -1FF

FullSimplifyBLimitBArcSinB2 2 r 1 + -1 + r2

3 r2 + 2 -1 + r2F -


-1 + 2 r + -1 + r2F,


-1 - 2 r + -1 + r2F> , r Ø 2 , Direction Ø 1FF


PlotBArcSinB2 2 r 2 -1 + -1 + r2 + r2 5 + 3 -1 + r2

-4 + 9 r4 + 4 r2 1 + 3 -1 + r2F -


-1 + 2 r + -1 + r2F,


-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 -


-1 + 2 r + -1 + r2F,


-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 -


-1 + 2 r + -1 + r2F,

> , , FF



-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 -


-1 + 2 r + -1 + r2F,


-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


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

+

-


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,


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


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


PlotB HarcsinCC - arcsinEE + arctanAA + arctanBBL ê.


-2 + 3 r2 1 + 2 r - -1 + r2F,


-2 + 3 r2 -1 + 2 r + -1 + r2F, arctanAA Ø

ArcSinB2 -1 + r + -1 + r2


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.


-2 + 3 r2 1 + 2 r - -1 + r2F,


-2 + 3 r2 -1 + 2 r + -1 + r2F, arctanAA Ø

ArcSinB2 -1 + r + -1 + r2


2 -1 - r + -1 + r2

-1 - 2 r + -1 + r2F> -


-2 + 3 r2F, r Ø 1, Direction Ø -1FF


FullSimplifyBLimitB HarcsinCC - arcsinEE + arctanAA + arctanBBL ê.


-2 + 3 r2 1 + 2 r - -1 + r2F,


-2 + 3 r2 -1 + 2 r + -1 + r2F, arctanAA Ø

ArcSinB2 -1 + r + -1 + r2


2 -1 - r + -1 + r2

-1 - 2 r + -1 + r2F> -


-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.


-2 + 3 r2 1 + 2 r - -1 + r2F,


-2 + 3 r2 -1 + 2 r + -1 + r2F,

,



-1 + 2 r + -1 + r2F,


-1 - 2 r + -1 + r2F> - p, :r, 1, 2 >F

LimitB HarcsinCC + arcsinEE - arctanAA + arctanBBL ê.


-2 + 3 r2 1 + 2 r - -1 + r2F,


-2 + 3 r2 -1 + 2 r + -1 + r2F,


-1 + 2 r + -1 + r2F,


-1 - 2 r + -1 + r2F> - p, r Ø 1, Direction Ø -1F

LimitB HarcsinCC + arcsinEE - arctanAA + arctanBBL ê.


-2 + 3 r2 1 + 2 r - -1 + r2F,

,



-2 + 3 r2 -1 + 2 r + -1 + r2F,


-1 + 2 r + -1 + r2F,


-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


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,


-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


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,


-2 + 3 r2 -1 + 2 r + -1 + r2F> ì p >, :r, 1, 2 >F


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,


-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


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,


-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,


-2 + 3 r2 -1 + 2 r + -1 + r2F> +

H-arctanAA + arctanBBL ê. :arctanAA Ø ArcSinB2 -1 + r + -1 + r2

-1 + 2 r + -1 + r2F,


-1 - 2 r + -1 + r2F>


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>


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,


-2 + 3 r2 -1 + 2 r + -1 + r2F> , :r, 1, 2 >F

FullSimplifyB


FullSimplifyB

LimitBtrigcontrSimpl@rD - Htrigcontr@rDL ê. :arctanAA Ø ArcTanB2 -1 + r + -1 + r2

1 + -1 + r2F,



1 - -1 + r2

1 + -1 + r2F,



-1 + -1 + r2

2 rF,


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,


-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,



1 - -1 + r2

1 + -1 + r2F,



-1 + -1 + r2

2 rF,

,




2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2


2 -2 + -2 + 3 r2

3 rF,


-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,



1 - -1 + r2

1 + -1 + r2F,



-1 + -1 + r2

2 rF,



2 + r - 2 r2 + 2 -1 + r2 + 3 r -1 + r2


2 -2 + -2 + 3 r2

3 rF,


-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 ........


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


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


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 ,

,

,


-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 >,


:ArcSinB3 -1 + -1 + r2


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


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>


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


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


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


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


-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


FactorB HTogether@GEAAdd@rD + GEBBdd@rDDL ê.

:ArcSinB3 -1 + -1 + r2


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


3 -1 + r2

-2 + 3 r2F -

144 6 r2 ArcSinB1 + 3 -1 + r2

2 -2 + 3 r2F + 216 r3 *


-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


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


TogetherB HGEddNotSimpl@rDL ê.

:ArcSinB2 - r2


2 2 r -1 + r2


4 + 4 r2 - 7 r4

I2 - 3 r2M2F ,

ArcSinB3 -1 + r2


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 2 r -1 + r2

-2 + 3 r2F ì p,

ArcSinB3 -1 + r2


1 + 3 -1 + r2

2 -2 + 3 r2F ì p,


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

†


†

Simplification of ArcSinB2 - r2


2 2 r -1 + r2

-2 + 3 r2F

:ArcSinB2 - r2


2 2 r -1 + r2


4 + 4 r2 - 7 r4

I2 - 3 r2M2F >

PlotB: ArcSinB2 - r2


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 2 r -1 + r2


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 2 r -1 + r2

-2 + 3 r2F FF,

Assumptions Ø :1 < r < 2 >F

LimitB ArcSinB2 - r2


2 2 r -1 + r2


4 + 4 r2 - 7 r4

I2 - 3 r2M2F ,

r Ø 1, Direction Ø -1F

LimitB ArcSinB2 - r2


2 2 r -1 + r2


4 + 4 r2 - 7 r4

I2 - 3 r2M2F ,

r Ø 2 , Direction Ø 1F

4 + 4 r2 - 7 r4

I2 - 3 r2M2

0

0


† Simplification of ArcSinB3 -1 + r2


1 + 3 -1 + r2

2 -2 + 3 r2F

: ArcSinB3 -1 + r2


1 + 3 -1 + r2

2 -2 + 3 r2F -

p

6>

PlotB:

ArcSinB3 -1 + r2


1 + 3 -1 + r2

2 -2 + 3 r2F ì p, 1 ê 2, -1 ê 2,

ArcSinB3 -1 + r2


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


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


1 + 3 -1 + r2

2 -2 + 3 r2F -

p

6, r Ø 1, Direction Ø -1F

LimitB ArcSinB3 -1 + r2


1 + 3 -1 + r2

2 -2 + 3 r2F -

p

6, r Ø 2 , Direction Ø 1F


† Simplification of


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


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


3 -1 + r2

-2 + 3 r2F - p ê 3 ,

:ArcSinB-1 + -1 + r2

2 rF Ø ArcSinB

-1 + r2

rF - p ê 4 >;


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


3 -1 + r2

-2 + 3 r2F - p ê 3>;

PlotBArcSinB3 -1 + -1 + r2


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:

>,


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


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 +

- +


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 ;


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 -

-


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 +

- -


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 +


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

-


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 +

-

-


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 +


-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


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


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 :=


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 -

+ +


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

+ +


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


: -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 +

Â * 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


-LogAIH-1 + rL2 H-8 + 3 rL H-1 + tLME + LogB-24 r -2 + 3 r2 H-1 + tLF +

Â * 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>


: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 +

Â * 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


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 >


Date post:	12-Mar-2016
Category:	Documents
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A edge cld expression

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