Estimates of the Parabolic Renormalization for 
Local Degree Three

Fei Yang

About this file: 

This file is the Mathematica 12.0 worksheet for checking the numerical estimations in 
the  proof of the Main Theorem in the paper: 
Yang Fei, Parabolic and near–parabolic renormalizations for local degree three.
Last updated: June 29, 2024

§ 2.2  Preliminaries on P

• Proposition 2.2 : the values of μ, (Subscript[cp, P]), (Subscript[cv, P]), ((Subscript[ν, 1])^P) and ((Subscript[ν, 2])^P)

11-4(Sqrt[6])//N
1-(2/3)(Sqrt[6])//N
-(16/(3(8(Sqrt[6])+3)))//N
ν1p=9(Sqrt[6])-(47/2)-(1/2)(Sqrt[996(Sqrt[6])-2439]);
ν2p=9(Sqrt[6])-(47/2)+(1/2)(Sqrt[996(Sqrt[6])-2439]);
ν1p//N
ν2p//N

§ 6  Passing from P to Q

• Lemma 6.1 : the values of cp, cp’, ω, cv, (Subscript[ν, 1]) and (Subscript[ν, 2])

cp=(1/5) (1+4 (Sqrt[6])+2 (Sqrt[2 (9+(Sqrt[6]))]));
cpp=(1/5) (1+4 (Sqrt[6])-2 (Sqrt[2 (9+(Sqrt[6]))]));
ω=((8(Sqrt[6])-3)/25)+((6(Sqrt[6])+4)/25)I;
cv=((3(8(Sqrt[6])+3))/4);
ν1=((2+ν1p)/(-ν1p))+((2(Sqrt[-1-ν1p]))/(-ν1p))I;
ν2=((2+ν2p)/(-ν2p))+((2(Sqrt[-1-ν2p]))/(-ν2p))I;

cp//N
cpp//N
Re[ω]//N
Im[ω]//N
cv//N
Re[ν1]//N
Im[ν1]//N
Re[ν2]//N
Im[ν2]//N

§ 7  Estimates on Q: Part I

• Lemma 7.1 : Figure 9 (left)

a0=-0.06;r0=1.07;ϵ1=0.128;ϵ2=0.007;ϵ3=0.014;

ParametricPlot[{{-0.06+r0 Cos[u],r0 Sin[u]},{Cos[u],Sin[u]},{-1+ϵ1 Cos[u],ϵ1 Sin[u]},{1+ϵ2 Cos[u],ϵ2 Sin[u]},{Re[ω]+ϵ3 Cos[u],Im[ω]+ϵ3 Sin[u]},{Re[ω]+ϵ3 Cos[u],-Im[ω]+ϵ3 Sin[u]}},{u,0,2Pi},PlotRange->All,PlotStyle->{Thick},Axes->False,AspectRatio->Automatic]

• Lemma 7.1 : (7.2*)-(7.5*)

η=3;
((((ϵ1)^6)(((2-ϵ1))^4))/((1+ϵ1)(((2+ϵ1))^8)))
cv (E^(-2 π η))  //N
(((((2-ϵ2))^6)((ϵ2)^4))/((1+ϵ2)(((1+ϵ2))^8)))
(((((2+ϵ3))^6)(((1+ϵ3))^4))/((1-ϵ3)((ϵ3)^4)(((1-ϵ3))^4)))
cv (E^(2 π η)) //N
(((Re[ω]+0.06))^2)+((Im[ω])^2)-(((r0-ϵ3))^2)

• Lemma 7.3: Figure 9 (right)

a4=-0.22+0.69I;
ϵ4=Abs[a4-ω];
a5=0.78+0.21I;
ϵ5=Abs[a5-ω];
a6=ω;
ϵ6=0.41;
ϵ7=0.82;
r1=1.2;

ParametricPlot[{{Cos[u],Sin[u]},{a0+r0 r1 Cos[u],r0 r1 Sin[u]},{Re[a4]+ϵ4 Cos[u],Im[a4]+ϵ4 Sin[u]},{Re[a5]+ϵ5 Cos[u],Im[a5]+ϵ5 Sin[u]},{Re[ω]+ϵ6 Cos[u],Im[ω]+ϵ6 Sin[u]},{-1+ϵ7 Cos[u],ϵ7 Sin[u]},{Re[a4]+ϵ4 Cos[u],-Im[a4]+ϵ4 Sin[u]},{Re[a5]+ϵ5 Cos[u],-Im[a5]+ϵ5 Sin[u]},{Re[ω]+ϵ6 Cos[u],-Im[ω]+ϵ6 Sin[u]}},{u,0,2Pi},PlotRange->All,AspectRatio->Automatic]

• Lemma 7.3 : (7.8*)-(7.11*)

y4p[x_]:=Im[a4]+(Sqrt[((ϵ4)^2)-(((x-Re[a4]))^2)]);
y4m[x_]:=Im[a4]-(Sqrt[((ϵ4)^2)-(((x-Re[a4]))^2)]);
y5p[x_]:=Im[a5]+(Sqrt[((ϵ5)^2)-(((x-Re[a5]))^2)]);
y5m[x_]:=Im[a5]-(Sqrt[((ϵ5)^2)-(((x-Re[a5]))^2)]);
y6p[x_]:=Im[ω]+(Sqrt[((ϵ6)^2)-(((x-Re[ω]))^2)]);
y6m[x_]:=Im[ω]-(Sqrt[((ϵ6)^2)-(((x-Re[ω]))^2)]);
y7p[x_]:=(Sqrt[((ϵ7)^2)-(((x+1))^2)]);
y7m[x_]:=-(Sqrt[((ϵ7)^2)-(((x+1))^2)]);

x60=0.54;x61=Re[ω];x62p=1.07;x62m=1.067;x63=Re[ω]+ϵ6;
(((x60-Re[ω]))^2)+(((y4p[x60]-Im[ω]))^2)-((ϵ6)^2)
(((x62p-Re[ω]))^2)+(((y5p[x62p]-Im[ω]))^2)-((ϵ6)^2)
(((x60-Re[a4]))^2)+(((y6p[x60]-Im[a4]))^2)-((ϵ4)^2)
(((x62m-Re[a5]))^2)+(((y6m[x62m]-Im[a5]))^2)-((ϵ5)^2)

• Lemma 7.3 : (7.12*)-(7.13*)

y4p[x60]-(23/26)(x60+1)

ξ41p[x_]:=((((((x+1))^2)+(((y4p[x]))^2)))^3);
ξ42p[x_]:=((((((x-1))^2)+(((y4p[x]))^2)))^2);
ξ43p[x_]:=((((x^2)+(((y4p[x]))^2)))^(1/2));ξ44p[x_]:=((((((x-Re[ω]))^2)+(((y4p[x]-Im[ω]))^2)))^2);
ξ45p[x_]:=((((((x-Re[ω]))^2)+(((y4p[x]+Im[ω]))^2)))^2);
Ξ4p[x_,xp_]:=((ξ41p[xp]*ξ42p[xp])/(ξ43p[x]*ξ44p[x]*ξ45p[x]));
x601=0.6;
Ξ4p[x60,x601]
Ξ4p[x601,x61]

• Lemma 7.3 : (7.14*)-(7.17*)

s61=((Im[a5+ω])/(Re[a5-ω]));
Re[a5]+ϵ5/(Sqrt[1+(s61^2)])
Im[a5]-((Im[ω])/(Re[ω]-1))(Re[a5]-1)

ξ51p[x_]:=((((((x+1))^2)+(((y5p[x]))^2)))^3);
ξ52p[x_]:=((((((x-1))^2)+(((y5p[x]))^2)))^2);
ξ53p[x_]:=((((x^2)+(((y5p[x]))^2)))^(1/2));ξ54p[x_]:=((((((x-Re[ω]))^2)+(((y5p[x]-Im[ω]))^2)))^2);
ξ55p[x_]:=((((((x-Re[ω]))^2)+(((y5p[x]+Im[ω]))^2)))^2);
ξ55pmax=((((Sqrt[((((Re[a5]-Re[ω]))^2)+(((Im[a5]+Im[ω]))^2))])+ϵ5))^4);

x611=0.99;
((ξ51p[x61]*ξ52p[x611])/(ξ53p[x611]*ξ54p[x611]*ξ55pmax))

HatΞ5p[x_,xp_]:=((ξ51p[x]*ξ52p[xp])/(ξ53p[xp]*ξ54p[xp]*ξ55p[x]));
x612=1.05;
x613=1.06;
HatΞ5p[x611,x612]
HatΞ5p[x612,x613]
HatΞ5p[x613,x62p]

• Lemma 7.3 : (7.18*)-(7.23*)

y6p[x_]:=Im[ω]+(Sqrt[((ϵ6)^2)-(((x-Re[ω]))^2)]);
y6m[x_]:=Im[ω]-(Sqrt[((ϵ6)^2)-(((x-Re[ω]))^2)]);

s62=((Im[ω])/(Re[ω]+1));
Re[ω]+ϵ6/(Sqrt[1+(s62^2)])
s63=((Im[ω])/(Re[ω]-1));
Re[ω]-ϵ6/(Sqrt[1+(s63^2)])
s64=((Im[ω])/(Re[ω]));
Re[ω]+ϵ6/(Sqrt[1+(s64^2)])

ξ61p[x_]:=((((((x+1))^2)+(((y6p[x]))^2)))^3);
ξ62p[x_]:=((((((x-1))^2)+(((y6p[x]))^2)))^2);
ξ63p[x_]:=((((x^2)+(((y6p[x]))^2)))^(1/2));
ξ64p[x_]:=((ϵ6)^4);
ξ65p[x_]:=((((((x-Re[ω]))^2)+(((y6p[x]+Im[ω]))^2)))^2);

x621=Re[ω];
x622=0.99;
x623=1.07;
x63=Re[ω]+ϵ6;

((ξ61p[x60]ξ62p[x621])/(ξ63p[x621] ((ϵ6)^4) ξ65p[x621]))

((ξ61p[x621]ξ62p[x622])/((1+ϵ6) ((ϵ6)^4) ξ65p[x621]))

((ξ61p[x622]ξ62p[x623])/(ξ63p[x622] ((ϵ6)^4) ξ65p[x622]))

((ξ61p[x623]ξ62p[x623])/(ξ63p[x622] ((ϵ6)^4) ξ65p[x622]))

((((((((x63+1))^2)+(((Im[ω]))^2)))^3)((((((x63-1))^2)+(((Im[ω]))^2)))^2))/(ξ63p[x623] ((ϵ6)^4) ξ65p[x623]))

ξ61m[x_]:=((((((x+1))^2)+(((y6m[x]))^2)))^3);
ξ62m[x_]:=((((((x-1))^2)+(((y6m[x]))^2)))^2);
ξ63m[x_]:=((((x^2)+(((y6m[x]))^2)))^(1/2));
ξ64m[x_]:=((ϵ6)^4);
ξ65m[x_]:=((((((x-Re[ω]))^2)+(((y6m[x]+Im[ω]))^2)))^2);
Ξ6m[x_,xp_]:=((ξ61m[xp]*ξ62m[xp])/(ξ63m[x]*ξ64m[x]*ξ65m[x]));

x631=1.069;x632=1.072;

Ξ6m[x631,x62m]
Ξ6m[x632,x631]
((ξ61m[x632]ξ62m[x632])/(((((((x63))^2)+(((Im[ω]))^2)))^(1/2))((ϵ6)^4)((((((x63-Re[ω]))^2)+(((2Im[ω]))^2)))^2)))

• Lemma 7.3 : (7.24*)-(7.31*)

x70=-1-ϵ7;x71=-1.095;x72=Re[a4]-ϵ4
x73=-0.77;
(((x71-Re[a4]))^2)+(((y7p[x71]-Im[a4]))^2)-((ϵ4)^2)
(((x71+1))^2)+(((y4p[x71]))^2)-((ϵ7)^2)
y4m[x73]

ξ71p[x_]:=((ϵ7)^6);
ξ72p[x_]:=((((((x-1))^2)+(((y7p[x]))^2)))^2);
ξ73p[x_]:=((((x^2)+(((y7p[x]))^2)))^(1/2));ξ74p[x_]:=((((((x-Re[ω]))^2)+(((y7p[x]-Im[ω]))^2)))^2);
ξ75p[x_]:=((((((x-Re[ω]))^2)+(((y7p[x]+Im[ω]))^2)))^2);
Ξ7p[x_,xp_]:=((ξ71p[xp]*ξ72p[xp])/(ξ73p[x]*ξ74p[x]*ξ75p[x]));

s71=-((Im[ω])/(Re[ω]+1));
-1-ϵ7/(Sqrt[1+(s71^2)])

x701=-1.3;
((ξ71p[x70]*ξ72p[x70])/(ξ73p[x701]*ξ74p[x701]*ξ75p[x70]))
Ξ7p[x701,x70]
Ξ7p[x71,x701]

s72=((Im[a4])/(Re[a4]-1));
Re[a4]-ϵ4/(Sqrt[1+(s72^2)])

((ξ41p[x71]*ξ42p[x71])/(ξ43p[x72]*ξ44p[x71]*ξ45p[x72]))

s73=((Im[a4])/(Re[a4]+1));
Re[a4]-ϵ4/(Sqrt[1+(s73^2)])

s74=((Im[ω-a4])/(Re[ω-a4]));
Re[a4]-ϵ4/(Sqrt[1+(s74^2)])

ξ41m[x_]:=((((((x+1))^2)+(((y4m[x]))^2)))^3);
ξ42m[x_]:=((((((x-1))^2)+(((y4m[x]))^2)))^2);
ξ43m[x_]:=((((x^2)+(((y4m[x]))^2)))^(1/2));ξ44m[x_]:=((((((x-Re[ω]))^2)+(((y4m[x]-Im[ω]))^2)))^2);
ξ45m[x_]:=((((((x-Re[ω]))^2)+(((y4m[x]+Im[ω]))^2)))^2);
Ξ4m[x_,xp_]:=((ξ41m[xp]*ξ42m[xp])/(ξ43m[x]*ξ44m[x]*ξ45m[x]));

x721=-1;
((ξ41m[x72]*ξ42m[x72])/(ξ43m[x721]*ξ44m[x72]*ξ45m[x721]))
Ξ4m[x721,x72]
Ξ4m[x73,x721]
((ξ41m[x73]*ξ42m[x721])/(ξ43m[x73]*ξ44m[x73]*ξ45m[x73]))

• Lemma 7.3 : (7.32*)-(7.41*)

y[x_]:=(Sqrt[(((r0*r1))^2)-(((x-a0))^2)]);

x4=a0+r0*r1;
x1=-1.1;x2=0.54;x3=1.03;

(((x1+1))^2)+(((y[x1]))^2)-((ϵ7)^2)

(((x1-Re[a4]))^2)+(((y[x1]-Im[a4]))^2)-((ϵ4)^2)
(((x2-Re[a4]))^2)+(((y[x2]-Im[a4]))^2)-((ϵ4)^2)

(((x2-Re[ω]))^2)+(((y[x2]-Im[ω]))^2)-((ϵ6)^2)
(((x3-Re[ω]))^2)+(((y[x3]-Im[ω]))^2)-((ϵ6)^2)

(((x3-Re[a5]))^2)+(((y[x3]-Im[a5]))^2)-((ϵ5)^2)
(((x4-Re[a5]))^2)+(((Im[a5]))^2)-((ϵ5)^2)

x1p=-0.5;x2p=0.3;

(((x1p-Re[a4]))^2)+(((Im[a4]))^2)-((ϵ4)^2)
(((x2p-Re[a4]))^2)+(((Im[a4]))^2)-((ϵ4)^2)
(((x2p-Re[a5]))^2)+(((Im[a5]))^2)-((ϵ5)^2)

§ 8  Estimates on Q: Part II

• Lemma 8.1

b0=((2(13+32(Sqrt[6])))/25);
b1=((2029+256(Sqrt[6]))/125);
a11=2 (617+688 (Sqrt[6]));a01=25 (119+16 (Sqrt[6]));
a12=3889250+837000 (Sqrt[6]);a02=2755539+487396 (Sqrt[6]);
a13=31356325+8965425 (Sqrt[6]);a03=66811702+23697378 (Sqrt[6]);
a14=102142212+38104768 (Sqrt[6]);a04=240990025+94826600 (Sqrt[6]);
Q2max[r_]:=((2^4)/(5^5))*((a11*r+a01)/( r (((r-1))^2)))+ ((2^6)/(5^10))*((a12*r+a02)/(((r-1))^4))+ ((2^11)/(5^14))*((a13*r+a03)/(((r-1))^6))+ ((2^12)/(5^16))*((a14*r+a04)/(((r-1))^8));
b0//N
b1//N

• Lemma 8.2 (8.1*) and Lemma 8.3 (8.2*)

Q2max[11]//N
((64(617+688(Sqrt[6])))/3125)//N
LogDQmax[r_]:=(b1/(r^2))+(50/(r^3))+((cp^4)/(2(r^3)(r-cp)));
LogDQmax[5.6]

§ 9  Estimates on φ

• Lemma 9.1: some functions

r0=1.07; a0=-0.06;c00=0.06; c01max=2*r0;
φ1max[r_]:=r0(Sqrt[-Log[1-((((r0/(r-Abs[a0]))))^2)]]); 
LogDφmax[r_]:=-Log[1-((((r0/(r-Abs[a0]))))^2)];

• Lemma 9.2 and the value of h0

Log[11]+((0.06π)/3)
((14(Sqrt[6]))/25)+0.45

• Lemma 9.3: preparation

α1=ArcTan[((Im[a5])/(Re[a5]-a0))]
α2m=0.54;α2p=0.55;
α3=ArcTan[((Im[ω])/(Re[ω]-a0))]
α4=ArcTan[((Im[ω]+ϵ6)/(Re[ω]-a0))]
h0=((14(Sqrt[6]))/25)+0.45;
α5=π-ArcTan[((h0+1)/(a0+1))]
r2[θ_]:=(((E^(π-θ))+1)/((E^(π-θ))-1));

• Lemma 9.3: (9.2*)-(9.7*)

B5[θ_]:=Im[a5]Sin[θ]+(Re[a5]-a0)Cos[θ];
C5[θ_]:=(((Re[a5]-a0))^2)+((Im[a5])^2)-((ϵ5)^2);
r5[θ_]:=(1/(r0 ))(B5[θ]+(Sqrt[((B5[θ])^2)-C5[θ]]));
r5[0]-r2[α1]

(a0+r0 r5[α2p]Cos[α2p]-Re[ω])^2+(r0 r5[α2p]Sin[α2p]-Im[ω])^2-((ϵ6)^2)
r5[α2p]-r2[α2p]

B6[θ_]:=Im[ω]Sin[θ]+(Re[ω]-a0)Cos[θ];
C6[θ_]:=(((Re[ω]-a0))^2)+((Im[ω])^2)-((ϵ6)^2);
r6[θ_]:=(1/(r0 ))(B6[θ]+(Sqrt[((B6[θ])^2)-C6[θ]]));
r6[α4]-r2[α4]
(a0+r0 r6[α2m]Cos[α2m]-Re[a5])^2+(r0 r6[α2m]Sin[α2m]-Im[a5])^2-((ϵ5)^2)
r6[α2m]-r2[α3]

• Lemma 9.3: (9.8*)

r7[θ_]:=((h0-a0)/((Sqrt[2])r0 Sin[θ+(π/4)]));
tt4=UnitVector[24,1];
tt4[[1]]=α4;
For[k=2,k≤ 10,k++,tt4[[k]]=1+0.01*k]
For[k=11,k≤ 15,k++,tt4[[k]]=1.1+0.02*(k-10)]
For[k=16,k≤ 23,k++,tt4[[k]]=1.2+0.05*(k-15)]
tt4[[24]]=α5;

For[k=1,k≤ 23,k++,Print[r7[tt4[[k]]]-r2[tt4[[k+1]]]]]

• Lemma 9.3: (9.9*)

θ51=2.38;θ52=2.6;
r8[θ_]:=((h0+1)/(r0 Sin[θ]));
r8[α5]-r2[θ51]
r8[θ51]-r2[θ52]

§ 10  Lifting Q and φ to X

• Proof of Proposition 3.3: (10.1*)

c00+c01max+φ1max[11]
(cv-11)Sin[(π/6)]//N

§ 11  Estimates on F

• Lemma 11.1: some functions

βmax[r_]:=c01max+(b1/(2r))+Q2max[r]+φ1max[r];
σ1[r_,θ_]:=(((b1 Sin[θ])/(2r))/(b0-c00+((b1 Cos[θ])/(2r))));
σ2[r_,θ_]:=(Sqrt[(((b0-c00))^2)+((((b1/(2r))))^2)+2(b0-c00)((b1/(2r)))Cos[θ]]);
ArgΔFmax[r_,θ_]:=-ArcTan[σ1[r,θ]]+ArcSin[((βmax[r])/(σ2[r,θ]))]; 
ArgΔFmin[r_,θ_]:=-ArcTan[σ1[r,θ]]-ArcSin[((βmax[r])/(σ2[r,θ]))]; 
AbsΔFmax[r_,θ_]:=σ2[r,θ]+βmax[r]; 
AbsΔFmin[r_,θ_]:=σ2[r,θ]-βmax[r]; 

LogDQmax[r_]:=(b1/(r^2))+(50/(r^3))+((cp^4)/(2(r^3)(r-cp)));(* LogDQmax is defined in Lemma 8.3(b) *)
LogDFmax[r_]:=LogDQmax[r] +LogDφmax[r];

b0-c00-(b1/(2*6.1))
βmax[6.1]

• Lemma 11.2: (11.4*)-(11.9*)

ϕ1max[r_]:=4 r0 (Sqrt[-Log[1-(((((4r0)/r)))^2)]]);
c00+c01max+ϕ1max[cv-2.35]
(2.35/(Exp[-LogDφmax[cv-2.35-3.5]]))
2.4*Exp[LogDQmax[cv-2.4]]
Q[ζ_]:=(((((-1+ζ))^4) (((1+ζ))^6))/(ζ (((1+ ((6-16(Sqrt[6]))/25)ζ+(ζ^2)))^4)));
2.75+Q[cv]-25.5
(25.5-22)Sin[((7π)/20)]
(b1/20)+Q2max[20]//N

§ 12  Repelling Fatou coordinate (Subscript[Φ, rep]) on X

• (12.1*), (12.2*)

((ϕ1max[125]+(b1/122)+Q2max[122])/5)
LogDFmax[5*25]

§ 13  Attracting Fatou coordinate (Subscript[Φ, attr])

• Lemma 13.1: (13.1*) - (13.5*)

θ1=((3π)/20);θ2=(π/4);
u1θ1=8.5;u2θ1=6.1;u3=22 Cos[θ1];u4=17.3;
u1θ2=9;u2θ2=6.6;

u0θ1=u1θ1/Cos[θ1]
u0θ2=u1θ2/Cos[θ2]//N

u2θ1+c00  Cos[θ1]+c01max+φ1max[u2θ1]
u2θ2+c00  Cos[θ2]+c01max+φ1max[u2θ2]

u4+c00 Cos[θ1]+c01max+φ1max[u4]
u3 //N

ArgΔFmax[u2θ1,θ1]
ArgΔFmax[u2θ1,-θ1]
-ArgΔFmin[u2θ1,θ1]
-ArgΔFmin[u2θ1,-θ1]

ArgΔFmax[u2θ2,θ2]
ArgΔFmax[u2θ2,-θ2]
-ArgΔFmin[u2θ2,θ2]
-ArgΔFmin[u2θ2,-θ2]
(π/2)-θ2 //N

• Lemma 13.2: (13.8*) - (13.12*)

r4=0.34;
u5=u3-u1θ1;
Abs[b0-c00+((b1 (E^(-θ1 I)))/(2 u4))-((2 u5 (r4^2)(E^(θ1 I)))/(1-(r4^2)))]+βmax[u4]-((2 u5 r4)/(1-(r4^2)))
-ArgΔFmin[u4,θ1]+(1/2)LogDFmax[u4]-(1/2)Log[1-(r4^2)]
(π/5)//N
-ArgΔFmax[u4,θ1]-(1/2)LogDFmax[u4]+(1/2)Log[1-(r4^2)]
-((3π)/20) //N
((Exp[(1/2)LogDFmax[u4]])/(AbsΔFmin[u4,θ1](Sqrt[1-(r4^2)])))
((Sqrt[1-(r4^2)])/(AbsΔFmax[u4,θ1]Exp[(1/2)LogDFmax[u4]]))

• Lemmas 13.3 and 13.4: (13.14*) - (13.18*)

Tan[1.245]
((Sqrt[1+(8^2)])/0.083)
u6=10.7;
(b1/u6)+Q2max[u6]
(22-b0)Cos[θ1]-u6
LogDQmax[u6]
LogDφmax[u6]

§ 14  Locating domains

• Proof of Lemma 14.1: (14.2*)

t0=6.5(Sqrt[2])-cp;
Varθ1[t_]:=3ArcTan[(t/(cp-cpp))];
Varθ2[t_]:=ArcTan[((t-Im[ν1])/(cp-Re[ν1]))];Varθ3[t_]:=ArcTan[((t+Im[ν1])/(cp-Re[ν1]))];
Varθ4[t_]:=ArcTan[((t-Im[ν2])/(cp-Re[ν2]))];Varθ5[t_]:=ArcTan[((t+Im[ν2])/(cp-Re[ν2]))];
Varθ6[t_]:=ArcTan[(t/cp)];
Varθ7[t_]:=4ArcTan[((t-Im[ω])/(cp-Re[ω]))];Varθ8[t_]:=4ArcTan[((t+Im[ω])/(cp-Re[ω]))];
Varθ[tp_,tpp_]:=Varθ1[tp]+Varθ2[tp]+Varθ3[tp]+Varθ4[tp]+Varθ5[tp]-Varθ6[tpp]-Varθ7[tpp]-Varθ8[tpp];

tt0=UnitVector[12,1];
For[k=1,k≤ 8,k++,tt0[[k]]=0.5*k]
For[k=9,k≤ 10,k++,tt0[[k]]=0.8+0.4*k]
For[k=11,k≤12,k++,tt0[[k]]=2.8+0.2*k]

-((3π)/4)//N
Varθ[0,tt0[[1]]]
For[k=1,k≤ 11,k++,Print[Varθ[tt0[[k]],tt0[[k+1]]]]]

Print[UpperBound]

(π/2)//N
Varθ[tt0[[1]],0]
For[k=1,k≤ 11,k++,Print[Varθ[tt0[[k+1]],tt0[[k]]]]]

• Proof of Lemma 14.1: (14.4*) - (14.6*)

ArcTan[(t0/cp)]
0.25π
ArcTan[((t0-Im[ω])/(cp-Re[ω]))]
ArcTan[((t0+Im[ω])/(cp-Re[ω]))]

• Proof of Lemma 14.1: (14.7*)

Q3max[r_]:=((2^4)/(5^5))*(a01/( r (((r-1))^2)))+ ((2^6)/(5^10))*((a12*r+a02)/(((r-1))^4))+ ((2^11)/(5^14))*((a13*r+a03)/(((r-1))^6))+ ((2^12)/(5^16))*((a14*r+a04)/(((r-1))^8));
(6.5+b0)Cos[(π/4)]+((2^4)/(5^5))(a11/(((6.5-1))^2))Cos[(π/4)]+Q3max[6.5]
cv Cos[(π/4)]//N

• Proof of Lemma 14.1: (14.9*), (14.10*)

cp+c00-c01max-φ1max[cp]
6.5+c00 Cos[(π/4)]-c01max-φ1max[6.5]

• Proof of Lemma 14.1: (14.12*)

y2[x_]:=h0-x;
θ21[x_]:=6ArcTan[((y2[x])/(x+1))];θ22[x_]:=4ArcTan[((y2[x])/(x-1))];θ23[x_]:=ArcTan[((y2[x])/x)];
θ24[x_]:=4ArcTan[((y2[x]-Im[ω])/(x-Re[ω]))];θ25[x_]:=4ArcTan[((y2[x]+Im[ω])/(x-Re[ω]))];
Θ2[x_,xp_]:=θ21[xp]+θ22[xp]-θ23[x]-θ24[x]-θ25[x];

xx1=UnitVector[16,1];
For[k=1,k≤ 16,k++,xx1[[k]]=0.04*k];(* Evalution *)

-((5π)/4.0)

θ21[xx1[[1]]]+(θ22[xx1[[1]]]+4π)-(π/2)-(θ24[0]+4π)-(θ25[0]+4π)(* Θ2[0,xx1[[1]]]-4π+π *)
For[k=1,k≤ 15,k++,Print[Θ2[xx1[[k]],xx1[[k+1]]]-4π]]
θ21[Re[ω]]+θ22[Re[ω]]-θ23[xx1[[16]]]-θ24[xx1[[16]]]-θ25[xx1[[16]]]-4π(* Θ2[xx1[[16]],Re[ω]]-4π+π *)

Text[UpperBound]

θ21[0]+θ22[0]-θ23[xx1[[1]]]-θ24[xx1[[1]]]-θ25[xx1[[1]]]-4π(* Θ2[xx1[[1]],0]-4π+π *)
For[k=1,k≤ 15,k++,Print[Θ2[xx1[[k+1]],xx1[[k]]]-4π]]
θ21[0.64]+(θ22[0.64]+4π)-θ23[Re[ω]]-2π-2π
-((3π)/4.0)

• Proof of Lemma 14.1: (14.13*)

Θ2[-1,-0.975]-5π
Θ2[-0.975,-0.95]-5π
Θ2[-0.95,-0.925]-5π
Θ2[-0.925,-0.9]-5π

6*(π/2)+θ22[-1]-θ23[-0.975]-θ24[-0.975]-θ25[-0.975]-5π
Θ2[-0.95,-0.975]-5π
Θ2[-0.925,-0.95]-5π
Θ2[-0.9,-0.925]-5π

• Proof of Lemma 14.1: (14.14*)

ξ21[x_]:=((((((x+1))^2)+((y2[x])^2)))^3);ξ22[x_]:=((((((x-1))^2)+((y2[x])^2)))^2);ξ23[x_]:=((((x^2)+((y2[x])^2)))^(1/2));
ξ24[x_]:=((((((x-Re[ω]))^2)+(((y2[x]-Im[ω]))^2)))^2);ξ25[x_]:=((((((x-Re[ω]))^2)+(((y2[x]+Im[ω]))^2)))^2);
Ξ2[x_,xp_]:=((ξ21[xp]ξ22[xp])/(ξ23[x]ξ24[x]ξ25[x]));
h2[x_,xp_]:=Ξ2[xp,x] Cos[Θ2[xp,x]-5π];
v2[x_,xp_]:=Ξ2[xp,x] Sin[Θ2[xp,x]-5π]-(h2[x,xp]-4.2(Sqrt[2]));

Ξ2[-0.975,-1]Cos[6*(π/2)+θ22[-1]-θ23[-0.975]-θ24[-0.975]-θ25[-0.975]-5π](*h2[-1,-0.975]*)
h2[-0.975,-0.95]
h2[-0.95,-0.925]
h2[-0.925,-0.9]

-Ξ2[-0.975,-1]-(1.7-4.2(Sqrt[2]))
-Ξ2[-0.95,-0.975]-(1.7-4.2(Sqrt[2]))
-Ξ2[-0.925,-0.95]-(1.7-4.2(Sqrt[2]))
-Ξ2[-0.9,-0.925]-(1.7-4.2(Sqrt[2]))

• Proof of Lemma 14.1: (14.15*)

xx2=UnitVector[22,1];
For[k=4,k≤ 22,k++,xx2[[k]]=-1.1+0.05*k];(* Evalution *)

For[k=4,k≤ 21,k++,Print[Θ2[xx2[[k]],xx2[[k+1]]]-5π]]

Text[UpperBound]
For[k=4,k≤ 20,k++,Print[Θ2[xx2[[k+1]],xx2[[k]]]-5π]]
θ21kmax=θ21[-0.05]+(θ22[-0.05]+4π)-(π/2)-(θ24[0]+4π)-(θ25[0]+4π)

• Proof of Lemma 14.1: (14.16*)

For[k=4,k≤ 20,k++,Print[h2[xx2[[k]],xx2[[k+1]]]]]
Ξ2[0,-0.05] Cos[θ21kmax]

Print[(SuperPlus[v2])]
For[k=4,k≤ 20,k++,Print[v2[xx2[[k]],xx2[[k+1]]]]]
Ξ2[0,-0.05] Sin[θ21kmax]-(Ξ2[0,-0.05] Cos[θ21kmax]-4.2(Sqrt[2]))

• Proof of Lemma 14.1: (14.17*)

y3[x_]:=h0+1;
ξ31[x_]:=((((((x+1))^2)+((y3[x])^2)))^3);ξ32[x_]:=((((((x-1))^2)+((y3[x])^2)))^2);ξ33[x_]:=((((x^2)+((y3[x])^2)))^(1/2));
ξ34[x_]:=((((((x-Re[ω]))^2)+(((y3[x]-Im[ω]))^2)))^2);ξ35[x_]:=((((((x-Re[ω]))^2)+(((y3[x]+Im[ω]))^2)))^2);
θ31[x_]:=6ArcTan[((y3[x])/(x+1))];θ32[x_]:=4ArcTan[((y3[x])/(x-1))];θ33[x_]:=ArcTan[((y3[x])/x)];
θ34[x_]:=4ArcTan[((y3[x]-Im[ω])/(x-Re[ω]))];θ35[x_]:=4ArcTan[((y3[x]+Im[ω])/(x-Re[ω]))];
Ξ3[x_,xp_]:=((ξ31[xp]ξ32[xp])/(ξ33[x]ξ34[x]ξ35[x]));
Θ3[x_,xp_]:=θ31[xp]+θ32[xp]-θ33[x]-θ34[x]-θ35[x]+π;

xx3=UnitVector[40,1];
For[k=1,k≤ 20,k++,xx3[[k]]=-1-0.04*k];     
For[k=21,k≤ 29,k++,xx3[[k]]=2.2-0.2*k];     
For[k=30,k≤ 40,k++,xx3[[k]]=8-0.4*k];    

θ31kmin=6*(π/2)+(θ32[-1]+4π)-(θ33[-1.04]+π)-(θ34[-1.04]+4π)-(θ35[-1.04]+4π)
For[k=1,k≤7,k++,Print[Θ3[xx3[[k+1]],xx3[[k]]]]]

Print[UpperBound]
θ31kmax=Θ3[-1,-1.04]
For[k=1,k≤ 7,k++,Print[Θ3[xx3[[k]],xx3[[k+1]]]]]

• Proof of Lemma 14.1: (14.18*)

4.2(Sqrt[2])-1.7
Ξ3[-1,-1.04]
For[k=1,k≤ 7,k++,Print[Ξ3[xx3[[k]],xx3[[k+1]]]]]

• Proof of Lemma 14.1: (14.19*), (14.20*)

Ξ3[-1,-1.04]Cos[Θ3[-1,-1.04]]
For[k=1,k≤ 7,k++,Print[Ξ3[xx3[[k]],xx3[[k+1]]]Cos[Θ3[xx3[[k]],xx3[[k+1]]]]]]

For[k=8,k≤ 28,k++,Print[Ξ3[xx3[[k]],xx3[[k+1]]]]]

• Proof of Lemma 14.1: (14.21*)

For[k=29,k≤ 39,k++,Print[Θ3[xx3[[k+1]],xx3[[k]]]]]
Print[UpperBound]
For[k=29,k≤ 39,k++,Print[Θ3[xx3[[k]],xx3[[k+1]]]]]

• Proof of Lemma 14.1: (14.22*)

h3[x_,xp_]:=Ξ3[x,xp] Cos[Θ3[xp,x]];
v3[x_,xp_]:=Ξ3[x,xp] Sin[Θ3[xp,x]]-(4.2(Sqrt[2])-h3[x,xp]);

For[k=29,k≤ 39,k++,Print[h3[xx3[[k]],xx3[[k+1]]]]]

Print[(SuperPlus[v3])]
For[k=29,k≤ 39,k++,Print[v3[xx3[[k]],xx3[[k+1]]]]]

• Proof of Lemma 14.1: (14.23*)

2*Q2max[8]-8-1.6+b0

§ Appendix A.  The position of D(a4,ϵ4) and D(a5,ϵ5)

• Proof of Lemma 7.2(a): (A.1*) - (A.5*)

x42m=((2 Re[a4] Im[a4] Im[ω]+((((Re[a4]))^2) -(((Im[a4]))^2) )Re[ω])/((((Re[a4]))^2)+(((Im[a4]))^2)))
y4m[x42m]
x40m=-0.22-ϵ4
x41m=-1;
s41=-(((1+Re[a4])Im[a4]+ϵ4(Sqrt[(((1+Re[a4]))^2)+(((Im[a4]))^2)-((ϵ4)^2)]))/(((ϵ4)^2)-(((1+Re[a4]))^2)))
x41tilde=((Re[a4]+Im[a4] s41-(s41^2))/(1+(s41^2)))
s42=-(((1-Re[a4])Im[a4]+ϵ4(Sqrt[(((1-Re[a4]))^2)+(((Im[a4]))^2)-((ϵ4)^2)]))/((((1-Re[a4]))^2)-((ϵ4)^2)));
x42tilde=((Re[a4]+Im[a4] s42+(s42^2))/(1+(s42^2)))

• Proof of Lemma 7.2(a): (A.6*) - (A.11*)

θ41m[x_]:=6ArcTan[((y4m[x])/(x+1))];θ42m[x_]:=4ArcTan[((y4m[x])/(x-1))];θ43m[x_]:=ArcTan[((y4m[x])/x)];
θ44m[x_]:=4ArcTan[((y4m[x]-Im[ω])/(x-Re[ω]))];θ45m[x_]:=4ArcTan[((y4m[x]+Im[ω])/(x-Re[ω]))];
Θ4m[x_,xp_]:=θ41m[xp]+θ42m[xp]-θ43m[x]-θ44m[x]-θ45m[x]+π;

x401m=-1.096;
Θ4m[x40m,x401m]+2π
Θ4m[x401m,x40m]+2π

6ArcTan[s41]+θ42m[-1]-θ43m[x401m]-θ44m[x401m]-θ45m[x401m]+π+2π
6(-(π/2))+θ42m[x401m]-θ43m[-1]-θ44m[-1]-θ45m[-1]+π+2π

y41m[x_]:=s41(x+1);
x411m=-1.04;
Θ41m[x_,xp_]:=6ArcTan[s41]+4ArcTan[((y41m[xp])/(xp-1))]-ArcTan[((y41m[x])/x)]-4ArcTan[((y41m[x]-Im[ω])/(x-Re[ω]))]-4ArcTan[((y41m[x]+Im[ω])/(x-Re[ω]))]+3π;
Θ41m[x41tilde,x411m]
Θ41m[x411m,x41tilde]
Θ41m[x411m,-1]
Θ41m[-1,x411m]

• Proof of Lemma 7.2(a): (A.12*)

x40p=-0.22-ϵ4;
x41p=-1;
x42p=0;
x43p=Re[ω];

θ41p[x_]:=6ArcTan[((y4p[x])/(x+1))];θ42p[x_]:=4ArcTan[((y4p[x])/(x-1))];θ43p[x_]:=ArcTan[((y4p[x])/x)];
θ44p[x_]:=4ArcTan[((y4p[x]-Im[ω])/(x-Re[ω]))];θ45p[x_]:=4ArcTan[((y4p[x]+Im[ω])/(x-Re[ω]))];
Θ4p[x_,xp_]:=θ41p[xp]+θ42p[xp]-θ43p[x]-θ44p[x]-θ45p[x];

xx40=UnitVector[14,1];
For[k=1,k≤ 6,k++,xx40[[k]]=-1.11+0.005*k];     
For[k=7,k≤ 14,k++,xx40[[k]]=-1.14+0.01*k]; 

Θ4p[xx40[[1]],x40p]+π+2π
For[k=1,k≤13,k++,Print[Θ4p[xx40[[k+1]],xx40[[k]]]+π+2π]]

Print[LowerBound]
Θ4p[x40p,xx40[[1]]]+π+2π
For[k=1,k≤12,k++,Print[Θ4p[xx40[[k]],xx40[[k+1]]]+π+2π]]
-3π+θ42p[xx40[[14]]]-θ43p[xx40[[13]]]-θ44p[xx40[[13]]]-θ45p[xx40[[13]]]+π+2π (*Θ4p[xx40[[13]],xx40[[14]]]+2π*)

• Proof of Lemma 7.2(a): (A.13*)

xx41=UnitVector[50,1];
For[k=1,k≤ 20,k++,xx41[[k]]=-1+0.01*k];     
For[k=21,k≤ 40,k++,xx41[[k]]=-1.2+0.02*k]; 
For[k=41,k≤ 50,k++,xx41[[k]]=-2+0.04*k]; 

3π+θ42p[-1]-θ43p[xx41[[1]]]-θ44p[xx41[[1]]]-θ45p[xx41[[1]]]-5π+2π
For[k=1,k≤48,k++,Print[Θ4p[xx41[[k+1]],xx41[[k]]]-5π+2π]]
θ41p[-0.04]+θ42p[-0.04]+(π/2)-θ44p[0]-θ45p[0]-5π+2π

Print[LowerBound]

Θ4p[-1.0,-0.99]-5π+2π
For[k=1,k≤49,k++,Print[Θ4p[xx41[[k]],xx41[[k+1]]]-5π+2π]]

• Proof of Lemma 7.2(a): (A.14*)

xx42=UnitVector[26,1];
For[k=1,k≤ 11,k++,xx42[[k]]=0.05*k];     
xx42[[12]]=0.575;
xx42[[13]]=x42tilde;

Θ4p[xx42[[1]],0]-4π+2π
For[k=1,k≤12,k++,Print[Θ4p[xx42[[k+1]],xx42[[k]]]-4π+2π]]

Print[LowerBound]

θ41p[xx42[[1]]]+θ42p[xx42[[1]]]-(π/2)-θ44p[0]-θ45p[0]-4π+2π
For[k=1,k≤12,k++,Print[Θ4p[xx42[[k]],xx42[[k+1]]]-4π+2π]]

• Proof of Lemma 7.2(a): (A.15*) - (A.17*)

For[k=14,k≤20,k++,xx42[[k]]=0.625+0.005*(k-14)]; 
For[k=21,k≤ 25,k++,xx42[[k]]=0.659+0.001*(k-21)]; 
xx42[[26]]=0.6635;

Θ4phat[x_,xp_]:=θ41p[xp]+θ42p[x]-θ43p[x]-θ44p[x]-θ45p[x]-4π+2π;

For[k=13,k≤25,k++,Print[Θ4phat[xx42[[k+1]],xx42[[k]]]]]
Print[LowerBound]
For[k=13,k≤25,k++,Print[Θ4phat[xx42[[k]],xx42[[k+1]]]]]

Print[x2627]
θ41p[0.6635]+(θ42p[Re[ω]]+4π)-θ43p[Re[ω]]-4(ArcTan[((Im[ω]-0.69)/(Re[ω]+0.22))]+(π/2))-4*(π/2)+2π
Θ4phat[0.6635,Re[ω]]

• Proof of Lemma 14.1(c) and Lemma 7.2(b): (A.18*)

ω=((8(Sqrt[6])-3)/25)+((6(Sqrt[6])+4)/25)I;
a5=0.78+0.21I;
ϵ5=Abs[a5-ω];
x51m=Re[a5]+(Sqrt[1+(((Re[a5]))^2)-2 (Re[a5] Re[ω]+ Im[a5] Im[ω])])
x52m=0.78+ϵ5

• Proof of Lemma 14.1(c) and Lemma 7.2(b): (A.19*)

y5p[x_]:=Im[a5]+(Sqrt[((ϵ5)^2)-(((x-Re[a5]))^2)]);
θ51p[x_]:=6ArcTan[((y5p[x])/(x+1))];θ52p[x_]:=4ArcTan[((y5p[x])/(x-1))];
θ53p[x_]:=ArcTan[((y5p[x])/x)];θ54p[x_]:=4ArcTan[((y5p[x]-Im[ω])/(x-Re[ω]))];
θ55p[x_]:=4ArcTan[((y5p[x]+Im[ω])/(x-Re[ω]))];
Θ5p[x_,xp_]:=θ51p[xp]+θ52p[xp]-θ53p[x]-θ54p[x]-θ55p[x];

((3π)/4)//N

xx50=UnitVector[12,1];
For[k=1,k≤ 11,k++,xx50[[k]]=0.66+0.03*k];    (* Evalution *)
xx50[[12]]=1;

For[k=1,k≤10,k++,Print[Θ5p[xx50[[k]],xx50[[k+1]]]+4π]]
θ51p[1]+4*(-(π/2))-θ53p[xx50[[11]]]-θ54p[xx50[[11]]]-θ55p[xx50[[11]]]+4π

Print[UpperBound]

For[k=1,k≤11,k++,Print[Θ5p[xx50[[k+1]],xx50[[k]]]+4π]]

• Proof of Lemma 14.1(c) and Lemma 7.2(b): (A.20*) - (A.21*)

Θ5p[xx50[[1]],Re[ω]]+4π
θ51p[xx50[[1]]]+θ52p[xx50[[1]]]-θ53p[Re[ω]]-4ArcTan[((0.78-Re[ω])/(Im[ω]-0.21))]-4*(π/2)+4π
((3π)/4)//N

• Proof of Lemma 14.1(c) and Lemma 7.2(b): (A.22*)

xx51p=UnitVector[23,1];
xx51p[[1]]=1.04;
xx51p[[2]]=1.08;

Θ5p[1,xx51p[[1]]]
Θ5p[xx51p[[1]],xx51p[[2]]]

θ51p[1]+4*(π/2)-θ53p[xx51p[[1]]]-θ54p[xx51p[[1]]]-θ55p[xx51p[[1]]]
Θ5p[xx51p[[2]],xx51p[[1]]]

• Proof of Lemma 14.1(c) and Lemma 7.2(b): The real parts of Tilde{x51,x52,x53}

s51=((Im[a5])/(Re[a5]+1));
xTilde51=Re[a5]+ϵ5/(Sqrt[1+(s51^2)])
s52=((Im[a5])/(Re[a5]));
xTilde52=Re[a5]+ϵ5/(Sqrt[1+(s52^2)])
s53=((Im[a5+ω])/(Re[a5-ω]));
xTilde53=Re[a5]+ϵ5/(Sqrt[1+(s53^2)])

• Proof of Lemma 14.1(c) and Lemma 7.2(b):  (A.23*) - (A.24*) for 2≤ k≤ 17

ξ51p[x_]:=((((((x+1))^2)+(((y5p[x]))^2)))^3);ξ52p[x_]:=((((((x-1))^2)+(((y5p[x]))^2)))^2);
ξ53p[x_]:=((((x^2)+(((y5p[x]))^2)))^(1/2));ξ54p[x_]:=((((((x-Re[ω]))^2)+(((y5p[x]-Im[ω]))^2)))^2);
ξ55p[x_]:=((((((x-Re[ω]))^2)+(((y5p[x]+Im[ω]))^2)))^2);
TildeΞ51p[x_,xp_]:=((ξ51p[xp]ξ52p[x])/(ξ53p[x]ξ54p[x]ξ55p[xp]));
h5p1[x_,xp_]:=TildeΞ51p[x,xp] Cos[Θ5p[x,xp]];
v5p1[x_,xp_]:=TildeΞ51p[x,xp] Sin[Θ5p[x,xp]]-(4.2(Sqrt[2])-h5p1[x,xp]);

For[k=3,k≤ 10,k++,xx51p[[k]]=1.1+0.02*(k-3)];    (* Evalution *)
For[k=11,k≤ 16,k++,xx51p[[k]]=1.25+0.01*(k-11)];    (* Evalution *)
xx51p[[17]]=1.305;
xx51p[[18]]=xTilde52;

For[k=2,k≤ 17,k++,Print[Θ5p[xx51p[[k]],xx51p[[k+1]]]]]

Print[UpperBound]
For[k=2,k≤ 17,k++,Print[Θ5p[xx51p[[k+1]],xx51p[[k]]]]]

Print[MaxRealPart]
For[k=2,k≤ 17,k++,Print[h5p1[xx51p[[k]],xx51p[[k+1]]]]]

Print[(SuperPlus[(Subscript[v, 5])])]
For[k=2,k≤17,k++,Print[v5p1[xx51p[[k]],xx51p[[k+1]]]]]

• Proof of Lemma 14.1(c) and Lemma 7.2(b):  (A.23*) - (A.24*) for 18≤ k≤ 20

xx51p[[19]]=1.317;
xx51p[[20]]=1.323;
xx51p[[21]]=xTilde51;
TildeΞ52p[x_,xp_]:=((ξ51p[xp]ξ52p[x])/(ξ53p[xp]ξ54p[x]ξ55p[xp]));
h5p2[x_,xp_]:=TildeΞ52p[x,xp] Cos[Θ5p[x,xp]];
v5p2[x_,xp_]:=TildeΞ52p[x,xp] Sin[Θ5p[x,xp]]-(4.2(Sqrt[2])-h5p2[x,xp]);

For[k=18,k≤ 20,k++,Print[Θ5p[xx51p[[k]],xx51p[[k+1]]]]]

Print[UpperBound]
For[k=18,k≤ 20,k++,Print[Θ5p[xx51p[[k+1]],xx51p[[k]]]]]

Print[MaxRealPart]
For[k=18,k≤ 20,k++,Print[h5p2[xx51p[[k]],xx51p[[k+1]]]]]

Print[(SuperPlus[(Subscript[v, 5])])]
For[k=18,k≤ 20,k++,Print[v5p2[xx51p[[k]],xx51p[[k+1]]]]]

• Proof of Lemma 14.1(c) and Lemma 7.2(b):  (A.23*) - (A.24*) for 21≤ k≤ 22

xx51p[[22]]=1.329;
xx51p[[23]]=0.78+ϵ5;
TildeΞ53p[x_,xp_]:=((ξ51p[x]ξ52p[x])/(ξ53p[xp]ξ54p[x]ξ55p[xp]));
h5p3[x_,xp_]:=TildeΞ53p[x,xp] Cos[Θ5p[x,xp]];
v5p3[x_,xp_]:=TildeΞ53p[x,xp] Sin[Θ5p[x,xp]]-(4.2(Sqrt[2])-h5p3[x,xp]);

Θ5p[xx51p[[21]],xx51p[[22]]]
Θ5p[xx51p[[22]],xx51p[[21]]]
h5p3[xx51p[[21]],xx51p[[22]]]
v5p3[xx51p[[21]],xx51p[[22]]]

Θ5p1[x_,xp_]:=6ArcTan[((Im[a5])/(xp+1))]+4ArcTan[((Im[a5])/(xp-1))]-θ53p[x]-θ54p[x]-θ55p[x];
Θ5p2[x_,xp_]:=θ51p[xp]+θ52p[xp]-ArcTan[((Im[a5])/x)]-4ArcTan[((Im[a5]-Im[ω])/(x-Re[ω]))]-4ArcTan[((Im[a5]+Im[ω])/(x-Re[ω]))];
Θ5p1[xx51p[[22]],xx51p[[23]]]
Θ5p2[xx51p[[23]],xx51p[[22]]]

TildeΞ53p3[x_,xp_]:=((ξ51p[x]ξ52p[x])/(((((xp^2)+(((Im[a5]))^2)))^(1/2))ξ54p[x]((((((xp-Re[ω]))^2)+(((Im[a5]+Im[ω]))^2)))^2)));

h5p33[x_,xp_]:=TildeΞ53p3[x,xp] Cos[Θ5p1[x,xp]];
v5p33[x_,xp_]:=TildeΞ53p3[x,xp] Sin[Θ5p1[x,xp]]-(4.2(Sqrt[2])-h5p33[x,xp]);
h5p33[xx51p[[22]],xx51p[[23]]]
v5p33[xx51p[[22]],xx51p[[23]]]

• Proof of Lemma 14.1(c) and Lemma 7.2(b):  (A.25*) - (A.28*)

x51m=Re[a5]+(Sqrt[1+(((Re[a5]))^2)-2 (Re[a5] Re[ω]+ Im[a5] Im[ω])]);
Q[z_]:=(((((-1+z))^4) (((1+z))^6))/(z (((1+ ((6-16(Sqrt[6]))/25)z+(z^2)))^4)));
Q[x51m]
(1/4)*(0.7/(Q'[x51m]))
y5m[x_]:=Im[a5]-(Sqrt[((ϵ5)^2)-(((x-Re[a5]))^2)]);
x511m=1.293;
x511m-x51m
y5m[x511m]
(0.0175/(Sqrt[2]))
s54=((Im[ω])/(Re[x51m-ω]));
(((1.2-Re[a5]))^2)+(((s54(1.2-x51m)-Im[a5]))^2)-((ϵ5)^2)

• Proof of Lemma 14.1(c) and Lemma 7.2(b):  (A.29*)

θ51m[x_]:=6ArcTan[((y5m[x])/(x+1))];θ52m[x_]:=4ArcTan[((y5m[x])/(x-1))];θ53m[x_]:=ArcTan[((y5m[x])/x)];
θ54m[x_]:=4ArcTan[((y5m[x]-Im[ω])/(x-Re[ω]))];θ55m[x_]:=4ArcTan[((y5m[x]+Im[ω])/(x-Re[ω]))];
Θ5m[x_,xp_]:=θ51m[xp]+θ52m[xp]-θ53m[x]-θ54m[x]-θ55m[x];

xx51m=UnitVector[11,1];
xx51m[[1]]=1.293;
xx51m[[2]]=1.297;
xx51m[[3]]=1.30;
xx51m[[4]]=1.305;
xx51m[[5]]=1.31;
xx51m[[6]]=1.315;
xx51m[[7]]=1.32;
xx51m[[8]]=1.325;
xx51m[[9]]=1.327;
xx51m[[10]]=1.329;
xx51m[[11]]=0.78+ϵ5;

For[k=1,k≤ 9,k++,Print[Θ5m[xx51m[[k+1]],xx51m[[k]]]]]
Θ5m1110=θ51m[xx51m[[10]]]+θ52m[xx51m[[10]]]-ArcTan[((Im[a5])/(xx51m[[11]]))]-4ArcTan[((Im[a5]-Im[ω])/(xx51m[[11]]-Re[ω]))]-4ArcTan[((Im[a5]+Im[ω])/(xx51m[[11]]-Re[ω]))]
Print[LowerBound]
For[k=1,k≤ 9,k++,Print[Θ5m[xx51m[[k]],xx51m[[k+1]]]]]
6ArcTan[((Im[a5])/(xx51m[[11]]+1))]+4ArcTan[((Im[a5])/(xx51m[[11]]-1))]-θ53m[xx51m[[10]]]-θ54m[xx51m[[10]]]-θ55m[xx51m[[10]]]

• Proof of Lemma 14.1(c) and Lemma 7.2(b):  (A.30*)

ξ51m[x_]:=((((((x+1))^2)+(((y5m[x]))^2)))^3);ξ52m[x_]:=((((((x-1))^2)+(((y5m[x]))^2)))^2);
ξ53m[x_]:=((((x^2)+(((y5m[x]))^2)))^(1/2));ξ54m[x_]:=((((((x-Re[ω]))^2)+(((y5m[x]-Im[ω]))^2)))^2);
ξ55m[x_]:=((((((x-Re[ω]))^2)+(((y5m[x]+Im[ω]))^2)))^2);
TildeΞ5m[x_,xp_]:=((ξ51m[xp]ξ52m[xp])/(ξ53m[x]ξ54m[xp]ξ55m[x]));

For[k=1,k≤9,k++,Print[TildeΞ5m[xx51m[[k]],xx51m[[k+1]]]]]
TildeΞ5m1011=((((((((xx51m[[11]]+1))^2)+(((Im[a5]))^2)))^3)((((((xx51m[[11]]-1))^2)+(((Im[a5]))^2)))^2))/(ξ53m[xx51m[[10]]]((((((xx51m[[11]]-Re[ω]))^2)+(((Im[a5]-Im[ω]))^2)))^2)ξ55m[xx51m[[10]]]))

• Proof of Lemma 14.1(c) and Lemma 7.2(b):  (A.31*)

For[k=1,k≤9,k++,Print[TildeΞ5m[xx51m[[k]],xx51m[[k+1]]]Cos[Θ5m[xx51m[[k+1]],xx51m[[k]]]]]]
TildeΞ5m1011*Cos[Θ5m1110]