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Appendix C
Effective trapped fraction for Circular Concentric FS

Evaluation of K(ϵ) if the small ϵ approximation

We can rewrite

∫ --1---- ξ0 ---dξ′0---- I (ξ0) = (1+ ϵ) λ2,- 1,0(ξ′) ξ0T 0
(C.1)

with, for ξ0′ > ξ0T ,

( ) ∫ 2π dθ ξ′ λ2,-1,0 ξ′0 = -----′- 0 2π Ψξ0
(C.2)

where

-------------- ′ ∘ ( ′2) ξ = 1- Ψ 1 - ξ0
(C.3)

In the limit ϵ ≪ 1, we have

Ψ ≡---1+-ϵ-- 1 + ϵcosθ
= (1 + ϵ)[ ( 2)] 1 - ϵcosθ + O ϵ
= 1 + ϵ(1- cosθ ) + O( ) ϵ3 (C.4)

so that

ξ′ = ∘ ----------------------------(------) 1 - [1+ ϵ(1 - cosθ)+ O (ϵ2)] 1- ξ′02
= ξ0′∘ --------------(------)--------- (1 - cosθ) 1 - ξ′20 1- ϵ -------ξ′2--------+ O (ϵ2) 0

In fact, the series expansion of the square root must be kept to all orders for the term in ϵ, because the ratio ϵ∕ξ0′2 → 1 for ξ0′→ ξ0T . This gives

{ ∞ [ ( ′2) ]n } ξ′ = ξ′ 1 - ∑ ---(2n---2)!--- ϵ(1--cosθ)--1--ξ0-- + O (ϵ2) 0 22n- 1(n- 1)!n! ξ′20 n=1

where we used

(1 + x)α = 1 + ∑ n=1∞α-(α---1)⋅⋅⋅(α--n-+-1) n!xn (C.5)
(1 + x)1∕2 = 1 + ∑ n=1∞1- 2( ) 1- 2 - 1⋅⋅⋅( ) 1- 2 - n + 1xn- n! (C.6)
= 1 + 1- 2x + ∑ n=2∞ n-1 (--1)---- 2n(1 ) (3) ⋅⋅⋅(2n - 3) n x- n! (C.7)
= 1 + 1- 2x + ∑ n=2∞(--1)n-1(2n---3)! 22(n- 1)(n - 2)!n!xn (C.8)
= 1 + ∑ n=2∞ n-1 (--1)---(2n---2)! 22n-1 (n - 1)!n!xn (C.9)

In addition,

-1 Ψ = -----------1----------- 1 + ϵ(1- cosθ )+ O (ϵ2)
= 1 - ϵ(1- cosθ) + O(ϵ2) (C.10)

We get

λ2,-1,0( ) ξ′0 = ∫ 02π dθ --- 2 π{ [ ] } ∞∑ (2n - 2)! ϵ(1- cosθ) (1- ξ′2) n 1 - ϵ(1- cosθ) - -2n-1---------- ---------′2------0-- n=12 (n- 1)!n! ξ0 (C.11)
= ∫ 02π-dθ 2 π{ ( ′2) 1 - 1ϵ-(1---cosθ)-1-+-ξ0- 2 ξ0′2
∞ [ ( ′2)]n} - ∑ ----(2n---2)!--- ϵ(1--cosθ-)-1--ξ0-- 22n- 1(n- 1)!n! ξ′20 n=2 (C.12)
= 1 -∑ n=1∞λ 2,-1,0(n)( ) ξ′0 + O( ) ϵ2 (C.13)

with

λ2,-1,0(1)( ′) ξ0 = ∫ 02πdθ 2π-ϵ 2- ( ) (1- cosθ) 1 + ξ′02 -------ξ′2-------- 0 (C.14)
= ϵ --′2 2ξ0( ) 1 + ξ′20 (C.15)

and, for n ≥ 2,

λ2,-1,0(n≥2)( ) ξ′0 = ∫ 02πdθ --- 2π (2n- 2)!ϵn -2n-1---------- 2 (n - 1)!n![ ] (1- cosθ )(1- ξ′2) ---------′2-----0-- ξ0n (C.16)
= (2n- 2 )!ϵn -2n-1---------- 2 (n - 1)!n![ ] (1- ξ′2) ----′20-- ξ0n ∫ 02π dθ --- 2 π(1 - cosθ)n (C.17)
= (2n- 2 )!ϵn -2n-1---------- 2 (n - 1)!n![ (1- ξ′2)] ----′20-- ξ0n ∫ 02π dθ --- 2 π[ (θ )] 2 sin2 -- 2n (C.18)
= n -(2n---2)!ϵ---- 2n-1(n - 1)!n![(1 - ξ′2)] ----′2-0- ξ0n ∫ 0πdθ- πsin2nθ (C.19)

We can transform, for n ≥ 2,

Y n = ∫ 0πdθ- πsin2nθ
= ∫ 0πdθ- πsinθ sin2n-1θ
= (2n - 1)∫ 0πdθ --- πcos2θ sin2(n- 1)θ
= (2n - 1)[∫ ∫ ] πd-θsin2(n-1)θ - π dθsin2θ 0 π 0 π
= (2n - 1)[Y n-1 - Y n]

so that

(2n - 1) Y n = --------Yn- 1 2n

with

1 Y 1 = -- 2

so that

Y n = (2n---1) 2n(2(n---1)--1)- 2(n - 1)⋅⋅⋅(6---1) 6(4---1) 41- 2
= (2n)! ----------------2 [2n(2n - 2)⋅⋅⋅2]
= --(2n)!- 22n(n!)2

and we find

λ2,-1,0(n≥2)( ) ξ′0 = (2n )!(2n - 2)!ϵn -3n--1-----------3 2 (n- 1)!(n!)[ ] (1- ξ′2) ---′20-- ξ0n (C.20)
= [(2n)!]2 ϵn ---------------4 (2n - 1)23n (n!)[ ] (1- ξ′2) ----′20-- ξ0n (C.21)

We obtain

I(ξ ) 0 = ---1--- (1+ ϵ)∫ ξ0Tξ0 ′ ---dξ0---- λ2,-1,0(ξ′0)
= ---1--- (1+ ϵ)∫ ξ0Tξ0 dξ0′[ ∑∞ [∑∞ ( )]m ( )] 1+ λ(n2),-1,0 ξ′0 + O ϵ2 m=1 n=1
= ∫ ξ0Tξ0 dξ0′[ ∑∞ [ ∞∑ ]m ] 1- ϵ + λ(n) (ξ′) + O (ϵ2) m=1 n=1 2,-1,0 0
= ξ0 -√ -- 2ϵ1∕2 + ∑ m=1∞I(m )(ξ0) + O(ϵ)

with

[ ]m ∫ ξ0 ′ ∑∞ (n) ( ′) I(m)(ξ0) = dξ0 λ2,-1,0 ξ0 ξ0T n=1

where we used

-1--- 1- x = 1 + ∑ m=1∞xm (C.22)
ξ0T = ∘ -2ϵ-- ----- 1+ ϵ = √ -- 2ϵ[ ϵ ( )] 1 - --+ O ϵ2 2 (C.23)
-1- ξ0T = √1-- 2ϵ[ ] 1+ ϵ+ O (ϵ2) 2 (C.24)

We have

[ [( )]n ]m ∫ ξ0 ′ ϵ ( ′2) ∑∞ [(2n )!]2ϵn 1 - ξ′20 I(m )(ξ0) = dξ0 2ξ′2- 1+ ξ0 + ---------3n----4- ---ξ′2--- ξ0T 0 n=2 (2n- 1) 2 (n!) 0

Clearly, the only term of order ϵ-1∕2 comes from the ξ0′-2 contribution integrated and taken on ξ0T . We need to keep only

I(m)(ξ0) = ∫ ξ0Tξ0 dξ0′[ ∞ [ ]n] ∑ ---[(2n)!]2ϵn---- -1- (2n- 1 )23n(n!)4 ξ′02 n=1m + O(ϵ)
= ∫ ξ0Tξ0 dξ0′[ [ ] ] ∑∞ n 1--n ϵ Cn ξ′2 n=1 0m + O(ϵ)
= ∫ ξ0Tξ0 dξ0′∑ i1=1∞∑ i2=1∞⋅⋅⋅∑ im=1∞C i1Ci2⋅⋅⋅Cimϵi1+i2+⋅⋅⋅+im [ ] 1-- ξ′20i1+i2+⋅⋅⋅+im + O(ϵ)
= ∑ i1=1∞∑ i2=1∞⋅⋅⋅∑ im=1∞C i1Ci2⋅⋅⋅Cimϵi1+i2+⋅⋅⋅+im ∫ ξ0Tξ0 dξ0′[ 1 ] -′2- ξ0i1+i2+⋅⋅⋅+im + O(ϵ)
= ∑ i1=1∞∑ i2=1∞⋅⋅⋅∑ im=1∞Ci1Ci2 ⋅⋅⋅Cim ϵi1+i2+⋅⋅⋅+im ------------------------ 2(i1 + i2 + ⋅⋅⋅+ im )- 1[ ] - 1 -′2(i1+i2+-⋅⋅⋅+im)-1 ξ0ξ0Tξ0 + O(ϵ)
= √ -- 2ϵ1∕2 ∑ i1=1∞∑ i2=1∞⋅⋅⋅∑ im=1∞-----------Ci1Ci2 ⋅⋅⋅Cim----------- 2i1+i2+⋅⋅⋅+im [2(i1 + i2 + ⋅⋅⋅+ im) - 1] + O(ϵ)

with

2 C = -----[(2n-)!]----- n (2n - 1)23n (n!)4

and finally we get

√ -- [ ∑∞ ∑∞ ∑∞ ∑∞ ] I (ξ0) = ξ0- 2ϵ1∕2 1- ⋅⋅⋅ -----------Ci1Ci2 ⋅⋅⋅Cim----------- +O (ϵ) m=1 i1=1 i2=1 im=1 2i1+i2+⋅⋅⋅+im [2(i1 + i2 + ⋅⋅⋅+ im) - 1]

∫ -11dξ 0σξ0H(|ξ |- ξ ) 0 0TI(|ξ |) 0
= 2∫ ξ0T1dξ 0ξ02 - ξ 0ϵ1∕2√ -- 2
×[ ∞ ∞ ∞ ∞ ] 1- ∑ ∑ ∑ ⋅⋅⋅∑ -----------Ci1Ci2 ⋅⋅⋅Cim----------- 2i1+i2+⋅⋅⋅+im [2(i1 + i2 + ⋅⋅⋅+ im )- 1] m=1 i1=1i2=1 im=1 + O(ϵ)
= 2- 3 - ϵ1∕2√ -- 2[ ] ∑∞ ∑∞ ∞∑ ∑∞ ------------Ci1Ci2-⋅⋅⋅Cim----------- 1 - ⋅⋅⋅ 2i1+i2+ ⋅⋅⋅+im [2 (i1 + i2 + ⋅⋅⋅+ im)- 1] m=1 i1=1i2=1 im=1 + O(ϵ)

and therefore

|-----------------------------------------------------------------------------------| | √ --[ ∞∑ ∑∞ ∑∞ ∑∞ ] ( ) | K (ϵ) = 3--2 1- ⋅⋅⋅ -----------Ci1Ci2 ⋅⋅⋅Cim----------- + O ϵ1∕2 | | 2 m=1 i1=1i2=1 im=1 2i1+i2+⋅⋅⋅+im [2(i1 + i2 + ⋅⋅⋅+ im )- 1] | -------------------------------------------------------------------------------------
(C.25)

with

|----------------------| | 2 | |Cn = -----[(2n-)!]----4 | ------(2n---1)23n-(n!)--

For example, C1 = 1∕2, C2 = 3∕16, C3 = 5∕32, ⋅⋅⋅

m
i1
i2
correction
1
1
-
1∕4
1
2
-
1∕64
1
3
-
1∕256
1
4
-
25∕16384
1
5
-
17∕22737
2
1
1
1∕48
2
1
2
3∕1280 × 2
2
2
2
9∕28672
2
1
3
5∕7168 × 2
2
2
3
5∕49152 × 2
2
3
3
6∕173015

The sum of all these coefficient gives K = 1.486, which is already much better that 9√ -- 2∕8 = 1.591

PIC


Figure C.1: Bootstrap current coefficient κ as a function of the highest terms M and N kept in the series.

We compute the coefficient K for several values of the highest terms M and N in the series (C.25). The results are shown on Fig. C.1. We first see that K converges for high M and N, towards a value that seems close to 1.46. The best combination seems to be for M = N, as both high M and high N are needed for convergence. Therefore, we take M = N and calculate K as a function of M. Results are shown on Fig. C.2. We can see that K indeed converges, and that the asymptotic limit is about K = 1.46. For M = N = 6, we find K = 1.467.

PIC


Figure C.2: Bootstrap current coefficient κ as a function of the highest terms M and N kept in the series, for M = N

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