Statistical models of elasticity in main chain and smectic liquid ...

14 CHAPTER 2. HAIRPIN CHAIN ELASTOMERSof configurations is to express the delta function in terms of its Fourier representation∫ ∞∫Ω (n) dk N∫ yn∫ y2± = 2 dy n dy n−1 ... dy 1−∞ 2π 0 0 0× e −ik[y 1−(y 2 −y 1 )...+(−1) n (N−y n)∓z] . (2.11)These integrals can be decoupled using the following property of the Laplacetransform of a convolution [35]L −1 {f 1 (q)f 2 (q)} =∫ τ0F 1 (τ −σ)F 2 (σ)dσ, (2.12)where f i (q) is the Laplace transform of the function F i (y), denoted by f i (q) =L{F i (y)}. The Laplace transforms required areThe result of using Eq. (2.12) is{L e iky} = f + (q) = 1 (2.13)q −ik{L e −iky} = f − (q) = 1q +ik . (2.14)Ω (n)± = 2 ∫ ∞−∞dk2π e±ikz L −1 {W(q)}, (2.15)where the variable conjugate to q in the Laplace transform is N and W isgiven by{f n 2W(q) = + (q)f n 2 +1− (q) even nf n+1 n+12 2+ (q)f− (q) odd n. (2.16)The inverse Laplace transform can be carried out by using the Bromwichinversion formulaF(N) = 1 ∫ λ+i∞e Nq f(q). (2.17)2πiλ−i∞Here the integration limits are chosen so that all the poles reside to the left ofthe contour of integration. The inverse Laplace transform are thus given bythe residues of the function⎧⎪⎨W(q)e Nq =⎪⎩(1q−ik(1q−ik)n2 ( 1)n+12q+ik(1q+ik)n2 +1 e Nq even n)n+12e Nqodd n(2.18)The most direct way to calculate the number of configurations is to evaluatethe residues of this expression. An alternative method by induction is given