◼ Basic de ﬁ ni � ons Definitions

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if you only want to see the final result just scroll to the very end and only run the subchapter "Final analytic expression for QNEC"

and look at the three input expressions in red *)

Definitions

◼ Basic deﬁni�ons

In[2]:= (*line element in 1805.08782*)

ds2= -(r^2+G1[r]^2)dt^2+dr^2(r^2+G2[r]^2) +r^2 dphi^2;

G1[r_]:=1-8 GN h;

G2[r_]:=1-8 GN h1-1(r^2+1)^2 h-1;

In[5]:= (* transformation to z=1/r *)

rTOz=r→1/z, dr→-1/z^2 dz;

In[6]:= (* h=0 gives global AdS *)

Collectds2/. h→0/. rTOz,dt, dz, dphi

Out[6]= dt² -1- 1

z² + dz²

1+_z¹₂z⁴

+dphi² z²

In[7]:= (* transformation to z=1/r *)

aux1=ds2/. rTOz//Simplify ; f1[z_]:=1+z^21-8 GN h^2;

f2[z_]:=1+z^21-8 GN h1- (1+1/z^2)^1-2 h^2;

replF=ff1→f1, ff2→f2;

(* check if f1 and f2 are defined consistently*)

1/z^2-dt^2 f1[z] +dphi^2+dz^2f2[z] ⩵aux1//Simplify

Out[11]= True

In[12]:= (* standard assumptions to be used in Simplify routines below *)

assume =Assumptions →

x>0, x<1, l>0, l<Pi, h>1/4, zcut>0, zast>0, Nast>0, Cosl>0, Cotl

2²>0;

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In[13]:= (* useful later for simplification *)

repldummy =dummy1 →Hypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2

2,

dummy2 →Hypergeometric2F1 3 2, 2,5

2+2 h,-Tanl 2

2

, dummy3 →Hypergeometric2F1 5

2, 3,7

2+2 h,-Tanl 2² ; subl=l^{-2+4 h}→0, l^{4 h}→0, l^{2+4 h}→0, l^{4+4 h}→0, l^{6+4 h}→0, l^{8+4 h}→0,

1 l

2-8 h

→0, 1 l

-8 h

→0, 1 l

-2-8 h

→0, 1 l

-8-4 h

→0, 1

l

-6-4 h

→0, 1 l

-4-4 h

→0, 1 l

-2-4 h

→0, 1 l

-4 h

→0, 1 l

2-4 h

→0;

◼ Lagrangian and Noether charges

In[15]:= (* the Lagrangian for the area functional *)

Lz_, dz_, dt_:=1/z Sqrt-dt^2 ff1[z] +dz^2ff2[z] +1;

In[16]:= (* Noether charge for spatial homogenity *)

Q1=Lz, dz, dt-dz*DLz, dz, dt, dz-dt*DLz, dz, dt, dt//Simplify

Out[16]= 1

z 1-dt²ff1[z] +_ff2[z]^dz²

In[17]:= (* Q1 being a Noether charge means that it is independent of z

and can be evaluated at z=zast, where dz z=zast=0 per definition*) Q1zast= Q1/. z->zast/. dz->0/. dt->dtzast

replNast=Nast→1/zast/Q1zast;

Out[17]= 1

zast 1-dtzast²ff1[zast]

In[19]:= (* Noether charge Q2 for spatial homogeneity ddydLddt=0 *)

Q2=DLz, dz, dt, dt

Out[19]= - dt ff1[z]

z 1-dt²ff1[z] + ^dz²

ff2[z]

In[20]:= (* after definingΛ:=-Q2/Q1 we can express dt in terms of Λ *)

replDt=SolveΛ== -Q2/Q1, dt[[1]]

repldtzast=dtzast→ dt/. replDt/. z→zast

Out[20]= dt→ Λ

ff1[z]



Out[21]= dtzast→ Λ

ff1[zast]

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In[22]:= (* using Q2 and Λ in the expression for Q1 gives a differential equation for z;

we take the positive branch corresponding to the interval (y=0,z=0) to (y=y*,z=zast) *) replDz=Solve1/ (zast*Nast)⩵Q1/. replDt, dz[[2]]

Out[22]= dz→ z²Λ²-z²ff1[z] +Nast²zast²ff1[z] ff2[z] z ff1[z]



In[23]:= repleps=eps→h*GN;

replGN=GN→epsh;

Calculations

◼ Integral yielding lambda

In[25]:= (* integral for the shifted time : λ=

∫₀^λdt=∫₀^t^*dt+∫_t^λ_*dt⩵2∫₀^zastdz ^Λ ^z ^f1[z]

f1[z] f2[z] z²Λ²-z²f1[z]+Nast²zast²f1[z]

*)

replz=z→zast*x, dz→zast*dx;

dtInt=Simplify2 zast dtdz/. replDz/. replDt/. replF/. replNast/. repldtzast/. replz/. replF, assume ;

int0=SimplifyIntegrateSeriesdtInt/. replGN,{eps, 0, 1}/. eps→0, x, assume ; ExpandNormal SimplifySeries

DSeriesdtInt/. replGN,{eps, 0, 1}, eps/. eps→0,{Λ, 0, 2}, assume ; int1=Integrate-

16(x zast+x^3 zast^3)1+ ¹

x²zast²^{-2 h}Λ 1-x² 1+x²zast²^5/2

, x+

Integrate 48 x³zast³Λ 1-x² 1+x²zast²^5/2

, x; tInt0=Limitint0+eps int1, x→0, assume ; tInt1=Limitint0+eps int1, x→1, assume ;

tInt=Simplify[tInt1-tInt0, assume ]

Out[31]= 32 eps zast³Λ

1+zast²²

+2 ArcTan zastΛ

1+zast² 1+zast²-Λ²

-

8 eps π 1+_zast¹ ₂^{-2 h}zastΛGamma [2+2 h] (1+2 h)1+zast²Gamma ³₂+2 h

In[32]:= (* solving for Λ; we have to pick the positive branch

correspondingt to positive dtdy(zast) for positiveλ *)

solLambda =Solveλ==SimplifySeries[tInt,{Λ, 0, 2}], Assumptions →1+zast^2>0//

Normal ,Λ[[1]];

replLambda =Λ →FullSimplifySeriesΛ/. solLambda ,{eps, 0, 1}, assume //Normal ;

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In[34]:= CollectFullSimplify2 zast

FullSimplifySeriesΛ/. solLambda ,{eps, 0, 1}, assume //Normal   λ, eps

Out[34]= 1+zast²+eps -16 zast²+

4 π 1+_zast¹ ₂^{-2 h}1+zast²Gamma [1+2 h] Gamma ³₂+2 h

◼ Integral yielding l+lamda

In[35]:= (* integral for the shifted length l+λ=

∫₀^(l+λ)dy=∫₀^y^*dy+∫_y^l+λ_* dy=∫₀^z^*dz+dydz++∫_z⁰_*dz-dydz-=

2∫₀^z^*dz+dydz+=2∫₀^zastdz ^z ^f1[z]

f2[z] z²Λ²-z²f1[z]+Nast²zast²f1[z]

; *)

repldyInt=FullSimplifySolvedz/. replDz ⩵2 dzdy/. replz, dy, zast>0//

Flatten;

dyInt=dy/. repldyInt/. dx→1/. replNast/. repldtzast/. replF;

integrand=CollectFullSimplifyNormal Simplify

SeriesSeriesSimplifyNormal SeriesdyInt/. replGN,{eps, 0, 1}, Assumptions → x>0, x<1, zast>0, h>1/4/.

replLambda ,{eps, 0, 1},{λ, 0, 2},

Assumptions → x>0, x<1, zast>0, h>1/4,{eps,λ}

Out[37]= 2 x zast - 1

-1+x² 1+x²zast²

+ x1+zast² λ²

4 1-x² zast1+x²zast²^3/2 +

eps 1

4 zastx 64 x²zast⁴ 1-x² 1+x²zast²^5/2

+ 64 x⁴zast⁶ 1-x² 1+x²zast²^5/2

-

641+_x₂_zast¹ ₂^{-2 h}zast² 1-x² 1+x²zast²

+

1

4 zastxλ² - 16 zast²

1-x² 1+x²zast²^5/2

+ 24 x²zast² 1-x² 1+x²zast²^5/2

+ 8 x²zast⁴

1-x² 1+x²zast²^5/2

- 8 1+ 1 zast²

2 h 1+ 1

zast² 1+ 1 x²zast²

-2 h

1+zast²  1-x² 1+x²zast²^3/2 + 8 π x² 1+ 1

x²zast²

2 hzast⁴ 1+zast² 1+x²zast² x²zast⁴

1-2 h

Gamma [1+2 h]  1-x² 1+x²zast²^5/2Gamma 3 2+2 h

(5)

In[38]:= I2=SimplifyIntegrateintegrand/. eps→0,{x, 0, 1}, Assumptions →{x>0, x<1, zast>0};

I3=eps IntegrateSimplify

x ^{64 x}²^zast⁴

1-x² 1+x²zast²^52

+ ^{64 x}⁴^zast⁶

1-x² 1+x²zast²^52

4 zast ,

{x, 0, 1}, Assumptions →zast>0;

I4=eps Integrate

x -^641+

1

x²zast²^{-2 h}zast² 1-x² 1+x²zast²

4 zast ,{x, 0, 1}, Assumptions → zast>0, h>1/4; I5=epsλ²Integrate

1

4 zastx - 16 zast²

1-x² 1+x²zast²^5/2

+ 24 x²zast² 1-x² 1+x²zast²^5/2

+ 8 x²zast⁴ 1-x² 1+x²zast²^5/2

, {x, 0, 1}, Assumptions →zast>0;

I6=epsλ²Integrate 1

4 zastx - 8 1+ 1 zast²

2 h 1+ 1

zast² 1+ 1 x²zast²

-2 h

1+zast²  1-x² 1+x²zast²^3/2 +

8 π x² 1+ 1 x²zast²

2 hzast⁴ 1+zast² 1+x²zast² x²zast⁴

1-2 h

Gamma 1+2 h  1-x² 1+x²zast²^5/2Gamma 3

2+2 h , {x, 0, 1}, Assumptions → zast>0, h>1/4;

In[43]:= (* total result to linear order in eps and quadratic inλ *)

yInt=CollectI2+I3+I4+I5+I6//FullSimplify ,{eps,λ}

Out[43]= λ²

4 zast+2 ArcTan[zast] + eps -16 zast

1+zast²+16 ArcTan[zast] +

π zast^{-1+4 h}1+zast²^{-2 h}λ²Gamma [1+2 h] Gamma ³₂+2 h

- 8 π zast^{1+4 h}1+zast²^{-2 h}Gamma [1+2 h]

Hypergeometric2F1 1 2, 1,3

2+2 h,-zast²  Gamma 3 2+2 h

◼ Ge�ng zast

In[44]:= (* master equation determining z_* to all orders in lambda *)

master =Collect

SimplifyNormal SeriesSeriesl+λ-yInt/. zast→z00+z01 eps+z10λ+z11λeps+ z20λ^2+z21λ^2 eps,{eps, 0, 1},{λ, 0, 2},{eps,λ};

(6)

In[45]:= replz00=FlattenSimplifySolve[(master /. λ →0/. eps→0)⩵0, z00], assume [[1]]; replz01=SimplifyFlatten

SolveSimplifymaster /. λ →0/. replz00 eps, assume  ⩵0, z01[[1]];

replz10=FlattenSolve[(D[master ,λ] /.λ →0/. eps→0)⩵0, z10][[1]]; replz11=

FlattenFullSimplifySolveSimplifyD[master ,λ] /.λ →0/. replz10 eps2

1+z00²² ⩵0, z11[[1]];

replz20=SimplifyFlattenSolveD[master ,{λ, 2}] /. eps→0/. replz10/. replz11 ⩵0, z20[[1]];

replz21=SimplifyFlattenSolveSimplifyD[master ,{λ, 2}] /. replz20/. replz10/. replz11/. replz01 eps2 ⩵0, z21[[1]];

In[51]:= z=CollectSimplify

z00+z01 eps+z10λ+z11λeps+z20λ^2+z21λ^2 eps/. replz20/. replz21/.

replz10/. replz11/. replz00/. replz01,{eps,λ}; replzast=zast→z

replzast2=zast0→SeriesCoefficientzast/. replzast,{eps, 0, 0}, zast1→SeriesCoefficientzast/. replzast,{eps, 0, 1};

Out[52]= zast→ 3 Csc[l] 4(1+Cos[l])

+λ Csc[l]² 2(1+Cos[l])

-Cos[2 l]Csc[l]² 2(1+Cos[l])

+λ² Csc[l] 4(1+Cos[l])

-3 Csc[l]²Sin[2 l] 8(1+Cos[l])

- Csc[l]²Sin[3 l]

4(1+Cos[l])

+eps 6 Csc[l] 1+Cos[l]

-4 l Csc[l]² 1+Cos[l]

+4 l Cos[2 l]Csc[l]² 1+Cos[l]

+

4 π Cscl

2^{-1-4 h}Gamma [1+2 h]Hypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2² Secl

2³  Gamma 3

2+2 h-2 Csc[l]²Sin[3 l] 1+Cos[l]

+

λ -8 l Csc[l] 1+Cos[l]

+ 32 π Cscl

2^{-5-4 h}Csc[l]³Gamma [1+2 h]Hypergeometric2F1  1

2, 1,3

2+2 h,-Tanl

2²Secl

2 Gamma 3 2+2 h+ 4 l Csc[l]²Sin[2 l]

1+Cos[l] +

4 π Gamma [1+2 h]Sec^l

2²^{1-2 h}Tan^l

2^{4 h} Gamma ¹₂+2 h

+

λ² - 2 Csc[l] 1+Cos[l]

+10 l Cot[l]Csc[l] 1+Cos[l]

-11 l Csc[l]² 1+Cos[l]

-3 l Cos[2 l]Csc[l]² 1+Cos[l]

+ 48 π Cscl

2^{-7-4 h}Csc[l]⁴Gamma [1+2 h]Hypergeometric2F1  1

2, 1,3

2+2 h,-Tanl

2²Secl

2 Gamma 3 2+2 h+ π Cscl

2^{1-4 h}Gamma [1+2 h]Hypergeometric2F1 1 2, 1,3

2+2 h, -Tanl

2²Secl

2³  2 Gamma 3

2+2 h +3 Csc[l]²Sin[2 l] 1+Cos[l]

+

(7)

-1+16 h² π Cscl

2Gamma [1+2 h]Hypergeometric2F1 1 2, 1,3

2+2 h, -Tanl

2²Secl

2 Secl 2²

-2 hTanl

2^{4 h}  2 Gamma 3

2+2 h + 4 π Gamma [1+2 h] Secl

2^{2 1}

-2 hTanl

2^{1+4 h}  Gamma 1 2+2 h+

(1+4 h) π Gamma [1+2 h]Hypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2² Secl

2^{2 1}

-2 hTanl

2^{1+4 h}  2 Gamma 3

2+2 h + π Gamma [1+2 h]Hypergeometric2F1 1

2, 1,3

2+2 h,-Tanl 2² Secl

2^{2 1}

-2 hTanl

2^{3+4 h}  Gamma 3 2+2 h- 1

2(3+4 h) (5+4 h)Gamma ³

2+2 h

π Gamma [1+2 h] Secl

2 Secl 2²

-2 hTanl 2^{4 h} 15 Cscl

2+32 h Cscl

2+16 h²Cscl

2+ (5+4 h) (7+8 h+ (-3+8 h)Cos[l]) Hypergeometric2F1 3

2, 2,5

2+2 h,-Tanl

2²Secl

2⁵Tanl 2- 24 Hypergeometric2F1 5

2, 3,7

2+2 h,-Tanl

2²Secl

2⁵Tanl 2³

In[54]:= (* small l expansion of z_* *)

Collect

Normal ExpandSeriesz,l, 0, 3/. l^{4 h}→0/. 1 l

-4 h

→0/. l^{1+4 h}→0/. l^{3+4 h}→0, {eps,λ}

Out[54]= l 2+l³

24+ 1 2+l²

8 λ+ - 1 4 l+ l

12+53 l³

1440 λ²+eps -2 l³

3 -2 l²λ+ -4 l 3 -53 l³

45 λ²

In[55]:= CollectFullSimplify[z/. eps→0],λ/.1+Cosl →2Cosl2^2/. Sinl →2 Sinl2Cosl2

Out[55]= 1

2λSecl 2²+1

8λ²(-3 Cot[l] +Csc[l])Secl

2²+Tanl 2

In[56]:= (* this is the exact expression for z_* to leading order in eps *)

LOla=Tanl2+λ  2 Cosl2^2+λ^2/4Tanl2 Cosl2^2-1Sinl

Out[56]= 1

2λSecl

2²+Tanl 2+1

4λ² -Csc[l] +Secl

2²Tanl 2

In[57]:= Simplify[% - %%]⩵0

Out[57]= True

(8)

◼ Area integral

General deﬁni�ons

In[58]:= (* area integrand*)

R[x]:= x²Λ²+ (Nast^2-x^2)ff1[x zast];

dA=2 Nast Sqrtff1[x zast]  x R[x]Sqrtff2[x zast]; dA==

SimplifyLz, dz, dtdy/. replDt/. replDz/. repldyInt/. replz/. dx→1, assume 

Out[60]= True

Divergent part

In[61]:= (* the integral diverges for x→0 *)

Simplify

SeriesSeriesCoefficientdA/. replF/. replGN,{eps, 0, 0},{x, 0, 0}, assume 

Out[61]= 2

x+O[x]¹

In[62]:= (* next we split up the integral∫_zcut/zast¹ Lⅆx=

∫₀¹Lⅆx-∫₀^zcut/zastLⅆx and subtract 0=∫₀¹2/xⅆx-∫₀^zcut/zast2/x ⅆx+2 Log^zcut

zast *) AdivLO=Integrate[2/x, x];

Adiv1LO=AdivLO/. x→1;

AdivcutLO=AdivLO/. x→zcut/zast;

AcutLO=Adiv1LO-AdivcutLO

Out[65]= -2 Logzcut zast

Convergent part, LO

In[66]:= dALOpert=SimplifyNormal 

SeriesNormal SeriesNormal SeriesSeriesdA-2/x/. replNast/. repldtzast/. replF/. replGN/. replLambda ,{λ, 0, 2},{eps, 0, 0}/.

replzast,{eps, 0, 0},{λ, 0, 2}, assume 

Out[66]= -2 x+

2 _1-x₄_+(-1+x^1+Cos[l]₂

)²Cos[l]

x -

xλ²2-x²+-1+x²Cos[l] ^1+Cos[l]

1-x⁴+(-1+x²)²Cos[l]

1+x²+Cos[l] -x²Cos[l]²

+

2 xλ ^1+Cos[l]

1-x⁴+(-1+x²)²Cos[l] Tan^l₂ -1-x²+-1+x²Cos[l]

(9)

In[67]:= (* the near boundary part ∫₀^zcut/zast... of the

integral is of O(zcut^2) and can be neglected *) CollectIntegrateSimplify

SeriesSeriesCoefficientdA-2/x/. replF/. replGN,{eps, 0, 0},{x, 0, 2}, assume ,{x, 0, zcut/zast}, zcut

Out[67]= -zcut²-1+Nast²zast²+Λ² 2 Nast²zast²

In[68]:= (* Mathematica needs some help to manage

the O(eps^0) integral for the regularized area *)

In[69]:= I7=IntegratedALOpert, x;

I7dn=SimplifyNormal Series[I7,{x, 0, 0}], assume ;

I7up=SimplifyNormal Series[I7,{x, 1, 0}], assume /. x→1;

In[72]:= (* Evaluating the limit of AppellF1 requires

advanced educated guessing and hindsight, see below *)

In[73]:= N -I Cotl

2AppellF11,1 2,1

2, 2, 1 x²,-

Cot^l

2²

x²  x^2-

2 Log[x] -2 I ArcTanCotl

2-Log4 Cosl

2² /. x→10^(-5) /. l→Pi10

Out[73]= 4.8761×10^-11+2.66454×10^-15ⅈ

In[74]:= (* graphical checks that these two

expressions are equivalent up to O(x^2) terms *) PlotRe -I Cotl

2AppellF11,1 2,1

2, 2, 1 x²,-

Cot^l

2²

x²  x^2-

2-Log4 Cosl

2² /. l→Pi10,{x, 0, 0.001}

Out[74]=

0.0002 0.0004 0.0006 0.0008 0.0010

1.×10^-7 2.×10^-7 3.×10^-7 4.×10^-7 5.×10^-7

(10)

In[75]:= PlotIm  -I Cotl

2AppellF11,1 2,1

2, 2, 1 x²,-

Cot^l

2²

x²  x^2-

2-Log4 Cosl

2² /. l→Pi10,{x, 0, 0.001}

Out[75]=

0.0002 0.0004 0.0006 0.0008 0.0010

-2.×10^-15 -1.×10^-15 1.×10^-15 2.×10^-15

In[76]:= AregPertLO=FullSimplify

Normal SeriesI7up-I7dn/.

ⅈAppellF11,¹

2,¹

2, 2, ¹

x²,-^Cot

l 2²

x² Cot^l

2

x² →-2 Log[x] + 2 I ArcTanCotl

2+Log4 Cosl

2²/. replzast/. eps→0,{λ, 0, 2}

Out[76]= - λ²

2+2 Cos[l]

+Log[2(1+Cos[l])] -λTanl 2

In[77]:= (* contribution from the log to LO *)

AcutPertLO=CollectFullSimplify

Normal SeriesSimplifySeriesAcutLO/. replzast,{eps, 0, 0},{λ, 0, 2}, λ 

Out[77]= 2λCsc[l] +λ² -3

2Cot[l]Csc[l] -Csc[l]²

2 -2 Log[zcut(Cot[l] +Csc[l])]

In[78]:= (* trivial sanity check that adding and taking limits commutes *)

AcheckLO=FullSimplifyNormal  SeriesAcutLO+I7up-I7dn/. 1

x²ⅈAppellF11,1 2,1

2, 2, 1 x²,-

Cot^l

2²

x² Cotl 2 → -2 Log[x] +2 I ArcTanCotl

2+Log4 Cosl 2²/. replzast/. eps→0,{λ, 0, 2};

FullSimplifyAcheckLO-AcutPertLO-AregPertLO

Out[79]= 0

(11)

HEE, LO

In[80]:= FullSimplifyCotl+Cscl-Cotl2

Out[80]= 0

In[81]:= (* holographic entanglement entropy O(eps^0) *)

HEEcLO=1/ (4 GN) (AcutPertLO+AregPertLO) //Normal ; HEELO=

FullSimplifyFullSimplifyFullSimplifyHEEcLO/.λ →0, Assumptions →{zcut>0}/.

1+Cosl →2Cosl2^2/.Cotl+Cscl →Cotl2/. Logzcut Cotl

2->Log[zcut] +LogCotl 2/. Log1+Cosl-2 LogCotl

2 →Log2Cosl2^2 Cotl2^2/.

Log2-2 Cosl →Log4Sinl2^2 replGbyc= {GN→3/ (2 c)};

SimplifySeriesHEELO,l, 0, 2, Assumptions →l>0/. replGbyc,l>0, zcut>0

Out[82]=

-2 Log[zcut] +Log4 Sin^l₂² 4 GN

Out[84]= 1

3c Log l

zcut-c l²

72 +O[l]³

QNEC, LO

In[85]:= (* QNEC O(eps^0) *)

QNEC=

Simplify1 2 Pi(D[HEEcLO,{λ, 2}] +6/c D[HEEcLO,{λ, 1}]^2) /.λ →0/. replGbyc

Out[85]= - c 12π

In[86]:= (* Daniels definitionof the lightlike deformation : k=(λ/2,λ/2) O(eps^0) *)

HEElahalf=HEEcLO/.λ → λ/2;

QNEClahalf=

1 2 Pi DHEElahalf,{λ, 2}+6/c DHEElahalf,{λ, 1}^2/.λ →0/. replGbyc//

Simplify

Out[87]= - c 48π

In[88]:= (* lightlike projection of the EMT O(eps^0) *)

Tkk=1 2 Pi(-c/24); QNEClahalf⩵Tkk

Out[89]= True

(12)

NLO

In[90]:= (* O(eps^1) integral for the regularized area; this is length,

so we split it into O(1) int8, O(λ) int9 and O(λ^2) int10 parts;

thos parts have to be massaged individuallyfor Mathematica to succeed *) preparedANLOpert=

SeriesSeriesdA/. replNast/. repldtzast/. replF/. replGN/. replLambda , {λ, 0, 2},{eps, 0, 1}//Normal ;

dANLOpert=eps SeriesSeriesCoefficientpreparedANLOpert/. zast→(zast0+eps zast1), {eps, 0, 1}/. replzast2,{λ, 0, 2}//Normal ;

(13)

In[92]:= (* let us start with O(1) *)

int8=FunctionExpandSimplifyExpanddummy3 dANLOpert/.λ →0, assume /. 16 π x -1+x² -1-x²-Cosl+x²Cosl

1+Cosl Cscl 2

-4 h

Gamma 1+2 hHypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2² 

-1+x² -1-x²-Cosl+x²Cosl²Gamma 3

2+2 h - 16 π x Cosl

-1+x² -1-x²-Cosl+x²Cosl

1+Cosl

Cscl 2^{-4 h}

Gamma 1+2 hHypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2² 

2+2 h → dummy1 dummy3 /. 1+

Cot^l

2² x²

-2 h

→dummy2 dummy3 ;

int81=IntegrateSimplifyint8/. dummy1 →0/. dummy2 →0/. dummy3 →1, assume , x;

int82=IntegrateSimplifyint8/. dummy1 →0/. dummy2 → 1+ Cot^l

2² x²

-2 h

/. dummy3 →0, assume , x;

int83=IntegrateSimplifyint8/. dummy1 →

16 π x√-1+x² -1-x²-Cosl+x²Cosl  1+CoslCscl 2^{-4 h} Gamma 1+2 hHypergeometric2F1 1

2, 1,3

2+2 h,-Tanl 2² 

2+2 h -

16 π x Cosl√-1+x² -1-x²-Cosl+x²Cosl  1+Cosl

Cscl

2^{-4 h}Gamma 1+2 h Hypergeometric2F1 1

2, 1,3

2+2 h,-Tanl 2² 

2+2 h /. dummy2 →0/. dummy3 →0, assume , x;

(14)

In[96]:= I8up=Simplify

SimplifySimplifyNormal Seriesint81+int82+int83,{x, 1, 0}, assume /. Gamma 2+2 h → 1+2 h2 h Gamma 2 h/.

Hypergeometric2F1 2 h,1

2+2 h,3

2+2 h,-Tanl 2²+ 4 h Hypergeometric2F1 1

2+2 h, 1+2 h,3

2+2 h,-Tanl 2² →

4 h+1 Cosl2^4 h, assume ;

I8dn=SimplifyNormal Seriesint81+int82+int83,{x, 0, 0}/. Cot^l

2² x²

-2 h

→0;

I8=SimplifyI8up-I8dn, assume 

Out[98]= - 8 eps π Cscl

2^{-2-4 h}Gamma [1+2 h] Hypergeometric2F1 1

2, 1,3

2+2 h,-Tanl

2²Secl

2²  Gamma 3

2+2 h - 4 eps(1+4 h) π Gamma [2 h]Sin^l

2^{4 h} Gamma ³

2+2 h

+8 eps l Tanl 2

In[99]:= Simplify

Dint91+int92+int93+int94+int95, x-ExpandDdANLOpert,λ/.λ →0/. l→Pi2/. h→2/. eps→1/10, assume 

Out[99]= - 1

1575-1+x² 1+x²⁶

4 x 1-x⁴ 315π -2+x² 1+x²³-2 243+x² 972-80 Hypergeometric2F1 1

2, 1,11

2 ,-1 - 32 Hypergeometric2F1 1

2, 1,11 2 ,-1+ 4 x⁶ -387+4 Hypergeometric2F11

2, 1,11

2 ,-1 - 6 x⁴ -243+8 Hypergeometric2F11

2, 1,11

2 ,-1 + 2 x⁸ 279+8 Hypergeometric2F11

2, 1,11 2 ,-1

In[100]:= (* now consider O(λ);

to avoid issues when expanding around x=1 we take instead the directed limit from below otherwise the integrals do not evaluate correctly! *) int9=ExpandDdummy8 dANLOpert,λ/.λ →0/.

Hypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2

2

 →dummy4 dummy8 /.

Tanl

2^{4 h}→dummy5 dummy8 /. 1+ 161+Cosl²Sinl²

x²-3+CsclSin3 l²

-1-2 h

→dummy6 dummy8 /.

(15)

1+ 161+Cosl²Sinl² x²-3+CsclSin3 l²

-2 h

→dummy7 dummy8 ;

int91=IntegrateSimplifyint9/. dummy4 →0/. dummy5 →0/. dummy6 →0/.

dummy7 →0/. dummy8 →1, assume , x;

int92=IntegrateSimplifyint9/. dummy4 →Hypergeometric2F1  1

2, 1,3

2+2 h,-Tanl

2²/. dummy5 →0/.

dummy6 →0/. dummy7 →0/. dummy8 →0, assume , x; int93=IntegrateSimplifyint9/. dummy4 →0/. dummy5 →Tanl

2

4 h

/. dummy6 →0/. dummy7 →0/. dummy8 →0,

Assumptions → x>0, x<1, l>0, l<Pi, h>1/4, x; int94=IntegrateSimplifyint9/. dummy4 →0/. dummy5 →0/.

dummy6 → 1+ 161+Cosl²Sinl² x²-3+CsclSin3 l²

-1-2 h

/. dummy7 →0/. dummy8 →0, assume , x;

int95=IntegrateSimplifyint9/. dummy4 →0/. dummy5 →0/. dummy6 →0/. dummy7 → 1+ 161+Cosl²Sinl²

x²-3+CsclSin3 l²

-2 h

/. dummy8 →0, assume , x; I9up1=SimplifyLimitint91, x→1, Direction→1, assume ,

Assumptions → x>0, x<1, l>0, l<Pi, h>1/4; I9up2=SimplifyLimitint92, x→1, Direction→1, assume ,

Assumptions → x>0, x<1, l>0, l<Pi, h>1/4; I9up=I9up1+I9up2+I9up3+I9up4+I9up5;

I9dn=

SimplifyExpandNormal Seriesint91+int92+int93+int94+int95,{x, 0, 0}/. Cot^l

2² x²

-2 h

→0, assume ; I9=SimplifyI9up-I9dn, assume  λ

(16)

Out[113]= 1

2epsλCscl 2^{-4 h} 1

Gamma [3+2 h]

π Cscl

2Gamma [1+2 h] -32(1+h) (1+h+h Cos[l])Gamma [2+2 h] Gamma ³

2+2 h

- Gamma [3+2 h] (5+4 h) (7+8 h+ (-2+8 h)Cos[l])Gamma 5

2+2 h- 2(13+16 h+4(1+4 h)Cos[l])Gamma 7

2+2 h  (1+Cos[l])Gamma 5

2+2 hGamma 7

2+2 h Secl 2+ 32 Cosl

2³Sinl

2 -2 π Gamma [1+2 h] (1+Cos[l])Gamma 3 2+2 h+ Gamma 1

2+2 hHypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2² + Cscl

2^{2+4 h}Gamma 1

2+2 hGamma 3

2+2 hSin[l]l+Sin[l]  (1+Cos[l])³Gamma 1

2+2 hGamma 3

2+2 h + 8 π Gamma [2+2 h]Tan^l₂

(1+2 h)Gamma ³₂+2 h

In[114]:= (* finally consider O(λ^2); same remarks as above;

evaluation of this part takes a while *)

int10=ExpandDdummy8 dANLOpert,{λ, 2}/.λ →0/. Hypergeometric2F1 3

2, 2,5

2+2 h,-Tanl

2² →dummy1 dummy8 /. Hypergeometric2F1 1

2, 1,3

2+2 h,-Tanl

2, 3,7

2+2 h,-Tanl

2² →dummy3 dummy8 /. 1+ 161+Cosl²Sinl²

x²-3+CsclSin3 l²

-2 h

→dummy4 dummy8 /.

-1-2 h

→dummy5 dummy8 /.

-2-2 h

→dummy6 dummy8 /.

1+161+Cosl²Sinl²

-3+CsclSin3 l²

-2 h

→dummy7 dummy8 ; int101=IntegrateSimplify

int10/. dummy1 →0/. dummy2 →0/. dummy3 →0/. dummy4 →0/. dummy5 → 1+ 161+Cosl²Sinl²

x²-3+CsclSin3 l²

-1-2 h

/. dummy6 →0/.

(17)

dummy7 →0/. dummy8 →0, assume , x;

int102=IntegrateSimplifyint10/. dummy1 →0/. dummy2 →0/. dummy3 →0/. dummy4 → 1+ 161+Cosl²Sinl²

x²-3+CsclSin3 l²

-2 h

/. dummy5 →0/. dummy6 →0/. dummy7 →0/. dummy8 →0, assume , x;

int103=IntegrateSimplifyint10/. dummy1 →0/. dummy2 →0/.

dummy3 →Hypergeometric2F1 5 2, 3,7

2+2 h,-Tanl 2²/. dummy4 →0/. dummy5 →0 /. dummy6 →0/.

dummy7 →0/. dummy8 →0, assume , x; int104=IntegrateSimplify

int10/. dummy1 →0/. dummy2 →Hypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2

2

/.

dummy3 →0/. dummy4 →0/. dummy5 →0/. dummy6 →0/. dummy7 →0/. dummy8 →0, assume , x; int105=IntegrateSimplify

int10/. dummy1 →Hypergeometric2F1 3 2, 2,5

2+2 h,-Tanl

2²/. dummy2 →0/. dummy3 →0/. dummy4 →0/. dummy5 →0/.

dummy6 →0/. dummy7 →0/. dummy8 →0, assume , x;

int106=IntegrateSimplifyint10/. dummy1 →0/. dummy2 →0/. dummy3 →0/.

dummy4 →0/. dummy5 →0 /. dummy6 → 1+ 161+Cosl²Sinl²

x²-3+CsclSin3 l²

-2-2 h

/. dummy7 →0/. dummy8 →0, assume , x;

int107=IntegrateSimplifyint10/. dummy1 →0/. dummy2 →0/. dummy3 →0/. dummy4 →0/. dummy5 →0 /. dummy6 →0/.

dummy7 → 1+161+Cosl²Sinl²

-3+CsclSin3 l²

-2 h

/. dummy8 →0, assume , x; int10rest=int10/. dummy1 →0/. dummy2 →0/. dummy3 →0/. dummy4 →0/.

dummy5 →0 /. dummy6 →0/. dummy7 →0/. Tanl

2^{4 h}→dummy9 dummy8 /. Tanl

2^{1+4 h}→dummy10 dummy8 ; int108=IntegrateSimplifyint10rest/. dummy8 →0/. dummy9 →0/.

dummy10 ->Tanl

2^{1+4 h}, assume , x;

int109=IntegrateSimplifyint10rest/. dummy8 →0/. dummy9 →Tanl 2^{4 h}/. dummy10 →0, assume , x;

int110=IntegrateSimplifyint10rest/. dummy8 →1/. dummy9 →0/. dummy10 →0, assume , x;

I10up=SimplifyLimitint101+int102+int103+int104+int105+int106+

(18)

int107+int108+int109+int110, x→1, Direction→1, assume , Assumptions → x>0, x<1, l>0, l<Pi, h>1/4;

I10dn=SimplifyExpandSimplifyNormal Seriesint101+int102+int103+int104+ int105+int106+int107+int108+int109+int110,{x, 0, 0}, assume /. x Tanl

2

4 h

→0, assume ; I10=SimplifyI10up-I10dn, assume  λ^2/2

Simplify::time :

Time spent on a transformation exceeded 300.` seconds, and the transformation was aborted. Increasing the value of TimeConstraint option may improve the result of simplification .

Simplify::time :

Time spent on a transformation exceeded 300.` seconds, and the transformation was aborted. Increasing the value of TimeConstraint option may improve the result of simplification .

(19)

Out[128]= 1 32epsλ²

1 (1+2 h)Gamma ⁵

2+2 h

π -128(1+2 h)²3+7 h+6 h²+8 h(1+h)Cos[l] +h(1+2 h)Cos[2 l]

Cscl

2^{-4 h}Csc[l]²Gamma [1+2 h] + 1

1+h2¹³²^{-2 h}Cosl

2³(1+Cos[l])^{-3-2 h} Csc[l]^{-4 h} -(1+h) (-2+Cos[l]) 1+Cos[l] (2+2 h+Cos[l] +2 h Cos[l])

Gamma [2+2 h] + 2 (1+2 h)Cosl

2³(-1+Cos[l])Gamma [3+2 h] + 4^-h(1+Cos[l])^{2-2 h}Cscl

2²Secl

2⁶ 2(1+2 h) (5+3 Cos[l])

3+7 h+6 h²+8 h(1+h)Cos[l] +h(1+2 h)Cos[2 l]Gamma [1+2 h] + 24 h(3+Cos[l]) (2+2 h+Cos[l] +2 h Cos[l])

Gamma [2+2 h]Sinl

2² Sin[l]^{4 h} - 8 Cscl

2^-4(1+h)Secl

2² 215+32 h+16 h² π Cscl

2¹⁰Gamma [1+2 h] Gamma 3

2+2 hSin[l]⁶-1

4Gamma 1 2+2 h 815+32 h+16 h² l+2 Cotl

2 Cscl

2^{8+4 h}Gamma 3

2+2 hSin[l]⁵+ π Gamma [1+2 h] 960 Cosl

2²Cos[l]Cotl

2⁴-6415+32 h+16 h²

3+h+12 h²+-5+16 h²Cos[l] +h(-1+4 h)Cos[2 l]

Cotl

2⁴Hypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2²+ 128(5+4 h) (7+8 h+ (-3+8 h)Cos[l])Cotl

2² Hypergeometric2F1 3

2, 2,5

2+2 h,-Tanl 2²- 3072 Hypergeometric2F1 5

2, 3,7

2+2 h,-Tanl 2²+

15 Cscl

2¹⁰Sin[l]⁶+32 h Cscl

2¹⁰Sin[l]⁶+ 16 h²Cscl

2¹⁰Sin[l]⁶+32 h Cos[l]Cscl

2¹⁰Sin[l]⁶+ 16 h²Cos[l]Cscl

2¹⁰Sin[l]⁶-15 Cscl

2¹²Sin[l]⁸- 32 h Cscl

2¹²Sin[l]⁸-16 h²Cscl

2¹²Sin[l]⁸  (3+4 h) (5+4 h) (1+Cos[l])³Gamma 1

2+2 hGamma 3 2+2 h

In[129]:= (* after this tour de force we add all results to get the O(eps) contribution

to the area integral apart from the log-term that we split away *) ApertNLO=I8+I9+I10

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Out[129]= - 8 eps π Cscl

2^{-2-4 h}Gamma [1+2 h] Hypergeometric2F1 1

2, 1,3

2+2 h,-Tanl

2²Secl

2²  Gamma 3

2+2 h - 4 eps(1+4 h) π Gamma [2 h]Sin^l₂^{4 h}

Gamma ³

2+2 h

+ 1

32epsλ² 1

(1+2 h)Gamma ⁵

2+2 h

π -128(1+2 h)²3+7 h+6 h²+8 h(1+h)Cos[l] +h(1+2 h)Cos[2 l]Cscl 2^{-4 h} Csc[l]²Gamma [1+2 h] + 1

1+h2¹³²^{-2 h}Cosl

2³(1+Cos[l])^{-3-2 h}Csc[l]^{-4 h} -(1+h) (-2+Cos[l]) 1+Cos[l] (2+2 h+Cos[l] +2 h Cos[l])Gamma [

2+2 h] + 2 (1+2 h)Cosl

2³(-1+Cos[l])Gamma [3+2 h] + 4^-h(1+Cos[l])^{2-2 h}Cscl

2²Secl

2⁶ 2(1+2 h) (5+3 Cos[l])

3+7 h+6 h²+8 h(1+h)Cos[l] +h(1+2 h)Cos[2 l]Gamma [1+2 h] + 24 h(3+Cos[l]) (2+2 h+Cos[l] +2 h Cos[l])Gamma [2+2 h]Sinl

2² Sin[l]^{4 h} - 8 Cscl

2^-4^(1+h)Secl 2² 215+32 h+16 h² π Cscl

2¹⁰Gamma [1+2 h]Gamma 3

2+2 hSin[l]⁶- 1

4Gamma 1

2+2 h 815+32 h+16 h² l+2 Cotl

2 Cscl 2^{8+4 h} Gamma 3

2+2 hSin[l]⁵+ π Gamma [1+2 h] 960 Cosl

2²Cos[l]Cotl

2⁴-6415+32 h+16 h²

3+h+12 h²+-5+16 h²Cos[l] +h(-1+4 h)Cos[2 l]

Cotl

2⁴Hypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2²+ 128(5+4 h) (7+8 h+ (-3+8 h)Cos[l])Cotl

2² Hypergeometric2F1 3

2, 2,5

2+2 h,-Tanl

2²-3072 Hypergeometric2F1 5

2, 3,7

2+2 h,-Tanl

2²+15 Cscl

2¹⁰Sin[l]⁶+32 h Cscl

2¹⁰Sin[l]⁶+16 h² Cscl

2¹⁰Sin[l]⁶+32 h Cos[l]Cscl

2¹⁰Sin[l]⁶+16 h²Cos[l]Cscl

2¹⁰Sin[l]⁶-15 Cscl

2¹²Sin[l]⁸-32 h Cscl

2¹²Sin[l]⁸-16 h²Cscl

2¹²Sin[l]⁸  (3+4 h) (5+4 h) (1+Cos[l])³Gamma 1

2+2 hGamma 3

2+2 h +8

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eps l Tanl 2+ 1

2 eps λ Csc

l 2^{-4 h}

1

Gamma [3+2 h]

π Cscl

2Gamma [1+2 h] -32(1+h) (1+h+h Cos[l])Gamma [2+2 h]

Gamma ³

2+2 h

-

Gamma [3+2 h] (5+4 h) (7+8 h+ (-2+8 h)Cos[l])Gamma 5 2+2 h- 2(13+16 h+4(1+4 h)Cos[l])Gamma 7

2+2 h  (1+Cos[l])Gamma 5

2+2 hGamma 7

2+2 h Secl 2+

32 Cosl

2³Sinl

2 -2 π Gamma [1+2 h] (1+Cos[l])Gamma 3 2+2 h+

Gamma 1

2+2 hHypergeometric2F1 1 2, 1,3

2+2 h,-Tanl 2² + Cscl

2^{2+4 h}Gamma 1

2+2 hGamma 3

2+2 hSin[l]l+Sin[l]  (1+Cos[l])³Gamma 1

2+2 hGamma 3

2+2 h + 8 π Gamma [2+2 h]Tan^l₂

(1+2 h)Gamma ³₂+2 h

In[130]:= (* now let us add again the log part and expand

to first order in eps and second order in lambda *)

Acuttotal=SimplifyNormal SeriesSeriesAcutLO/. replzast,{eps, 0, 1},{λ, 0, 2}

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Out[130]= 2λCsc[l] -1

2λ²(1+3 Cos[l])Csc[l]²-2 Logzcut Cotl

2+16 eps(1+Cos[l])Csc[l]² -1

32Cscl

2Secl

2³4 l+4λ+9 lλ²+4 lλ²Cos[l] +-4 l-4λ+3 lλ²Cos[2 l] - 6 Sin[l] -4λ²Sin[l] -4 lλSin[2 l] -6λ²Sin[2 l] +2 Sin[3 l]+ π Cscl

2^-4^(2+h)Gamma [1+2 h] Secl 2²

-2 hSin[l] -415+32 h+16 h² λ

(-1+λCot[l])Cscl

2^{9+4 h}Gamma 3

2+2 hSecl

2³Sin[l]Tanl 2^{4 h}+ Gamma 1

2+2 h -λ²Cscl

2^{5+4 h}Secl

2 15 Cscl

2⁴+32 h Cscl 2⁴+16 h²Cscl

2⁴+ (5+4 h) (7+8 h+ (-3+8 h)Cos[l])Cscl 2² Hypergeometric2F1 3

2, 2,5

2+2 h,-Tanl

2²Secl 2⁶- 24 Hypergeometric2F1 5

2, 3,7

2+2 h,-Tanl

2²Secl 2⁸ Tanl

2^{4 h}+1

415+32 h+16 h²Cscl 2⁹ Hypergeometric2F1 1

2, 1,3

2+2 h,-Tanl

2²Secl 2⁵ 4 Secl

2^{2 2 h}+9λ² Secl

2^{2 2 h}-4λ Secl 2^{2 2 h} Sin[2 l] +2λ²Cscl

2^{4 h}Tanl

2^{4 h}+2 hλ²Cscl 2^{4 h} Tanl

2^{4 h}+24 h²λ²Cscl

2^{4 h}Tanl

2^{4 h}+2λ²Cos[l] 2 Secl

2^{2 2 h}+-3+16 h²Cscl

2^{4 h}Tanl 2^{4 h} + Cos[2 l] -4+3λ² Secl

2^{2 2 h}+ 2 h(-1+4 h)λ²Cscl

2^{4 h}Tanl

2^{4 h}  16(3+4 h) (5+4 h)Gamma 1

2+2 hGamma 3 2+2 h

In[131]:= (* simple check that limits of small eps and small lambda commute *)

Acutcheck=

SimplifyNormal SeriesSeriesAcutLO/. replzast,{λ, 0, 2},{eps, 0, 1}; SimplifyAcutcheck-Acuttotal

Out[132]= 0

Total area

In[133]:= (* area consist of three parts: the regular O(1) piece,

the regular O(eps) piece and the log-contribution,

both to O(1) and O(eps); we treat different orders in λ separately, called area0, area1 and area2 the latter is split further into

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four pieces to reduce the time for Mathematica to Simplify *) arearaw=CollectSimplifyAregPertLO+ApertNLO+Acuttotal, assume ,{eps,λ}; area0=Simplifyarearaw/. Hypergeometric2F1 1

2, 1,3

2+2 h,-Tanl 2

2

 →dummy1 /. Hypergeometric2F1 3

2, 2,5

2+2 h,-Tanl 2

2

 →dummy2 /. Hypergeometric2F1 5

2, 3,7

2+2 h,-Tanl 2

2

 →dummy3 /.

Sqrt1+Cosl →Sqrt[2]Cosl2/.λ →0, assume //FullSimplify ; area1=SimplifyDarearaw/. Hypergeometric2F11

2, 1,3

2+2 h,-Tanl

2² →dummy1 /. Hypergeometric2F1 3

2, 2,5

2+2 h,-Tanl

2² →dummy2 /. Hypergeometric2F1 5

2, 3,7

2+2 h,-Tanl 2

2

 →dummy3 /.

Sqrt1+Cosl →Sqrt[2]Cosl2,λ/.λ →0, assume //FullSimplify ; area21=SimplifyDdummy4 arearaw/. Hypergeometric2F11

2, 1,3

2+2 h,-Tanl 2² → dummy1 dummy4 /. Hypergeometric2F1 3

2, 2,5 2+2 h, -Tanl

2² →dummy2 dummy4 /. Hypergeometric2F1  5

2, 3,7

2+2 h,-Tanl

2² →dummy3 dummy4 /.

Sqrt1+Cosl →Sqrt[2]Cosl2,{λ, 2} 2/. dummy1 →0/. dummy2 →0/. dummy3 →0/. dummy4 →1, assume ;

area22=SimplifyCollectDdummy4 arearaw/. Hypergeometric2F1 1 2, 1,3

2+ 2 h,-Tanl

2, 2,5

2+2 h,-Tanl 2² → dummy2 dummy4 /. Hypergeometric2F1 5

2, 3,7

2+2 h,-Tanl

2² →dummy3 dummy4 /.

Sqrt1+Cosl →Sqrt[2]Cosl2,{λ, 2} 2, dummy4 /. dummy1 →0/. dummy2 →0/. dummy4 →0, assume ;

area23=SimplifyCollectDdummy4 arearaw/. Hypergeometric2F1 1 2, 1,3

2+ 2 h,-Tanl

2, 2,5

2+2 h,-Tanl 2² →