old App G REVIEWED

Old App G no longer used as support for Chapter 4 PhL 1.17.14 Appendix G: Chapter 4 Support This Appendix provides technical support for certain claims made in Chapter 4. G.1 Evaluation of the integral (4.1.9) (a) The first claim is that the integral on the left below has the form shown on the right: In(x) = !Syntax Error, Idz' (z'-z)n = sn fn(βs) (G.1) where R = and s = . Here In(x) = I(x,y,z) is just our name for this integral, please do not confuse this with the modified Bessel function In(x). To show the claim, the first step is to define z" ≡ z'-z to get In(x) = !Syntax Error, Idz" (z")n R = (G.2) which shows that I(x) is independent of z so we can think of (G.2) as (G.1) with z = 0. Next define x ≡ z"/s to get dz" (z")n = sn+1 dx xn R = = s so that In(x) = sn !Syntax Error, I dx xn = sn fn(sβ) which then confirms the claimed form in (G.1). (b) Our next task is then to compute this integral fn(α) = !Syntax Error, I dx xn . α ≡ sβ For odd integer n, the integral clearly vanishes since the integrand is then an odd function of x. For even n we replace the integral with double its positive-side value and replace n = 2m with m = 1,2,3 ... to list off the non-vanishing integrals : f2m(α) = 2!Syntax Error, I dx x2m . m = 0,1,2... (G.3) Finally, change to y = which says x2 = y2 - 1 and so xdx = ydy. Then dx x2m = (y/x)dy x2m = y dy x2m-1 = y dy (y2-1)m-1/2 and so f2m(α) = 2 !Syntax Error, Idy (y2-1)m-1/2 e-jαy . (G.4) Finally we have something we can find in the standard tables. GR7 3.387 3 (p 350) claims, with ν = m+1/2 and μ = jα = jsβ. Recall from (1.5.1) that β2 = μεω2 - jωμσ so β2 and therefore β has a small negative imaginary part causing μ = jsβ to have a small positive real part which then allows the integration to converge due to the e-μx factor. Therefore, f2m(α) = (2/) (2/jα)m Γ(m+1/2) Km(jα) (G.5) = (2m+1/) Γ(m+1/2) Km(jα) / (jα)m where Km(z) is a modifed Bessel function. For m = 0 one has Γ(m+1/2) = . For m = 1,2,3 one can write Γ(m+1/2) = (2m-1)!! /2m Spiegel 16.6 (G.6) where for example 5!! = 5*3*1. These can be combined into a single formula if one interprets (-1)!! = 1 as well as (0)!! = 1. Then, f2m(α) = (2m+1/) [(2m-1)!! /2m] Km(jα) / (jα)m = 2 (2m-1)!! Km(jα) / (jα)m (G.7) and then restoring 2m = n we have fn(α) = 2 (n-1)!! Kn/2(jα) / (jα)n/2 for n even fn(α) = 0 . for n odd (G.8) Using then the fact that sn fn(βs) = sn 2 (n-1)!! Kn/2(jβs) / (jβs)n/2 (G.9) we conclude that In(x) = !Syntax Error, Idz' (z'-z)n = sn fn(βs) = (G.10) In particular, I0(x) = !Syntax Error, Idz' = 2 K0(jβs) . (G.11) The modified Bessel function Km(z) is described for example in NIST Chapter 10 p 248 and elsewhere. For small z, we find this useful series expansion for Km(z) in GR7 p 919, which we rewrite as, K0(z) ≈ -ln(z/2) [ 1 + (z/2)2 + O(z4) ] + ψ(1) + (z/2)2 ψ(2) + O(z4) m = 0 Km(z) ≈2m-1(m-1)! z-m - 2m-3 (m-2)! z-m+2 + O( z-m+4) . m = 1,2,3... (G.12) Here ψ(z) ≡ ∂z(lnΓ(z)) is the "digamma function", and ψ(1) and ψ(2) are just certain constants, The leading terms in the above small-z expansions are K0(z) ≈ -ln(z/2) m = 0 Km(z) ≈2m-1(m-1)! z-m. m = 1,2,3... (G.13) Notice that K0 is fundamentally different from the other Km in this limit: K0(z) is logarithmically divergent at z = 0, whereas Km(z) diverges as a power z-m for m > 0. G.2 Examination of higher order terms in (4.2.2) Equation (4.2.2) reads, V(z) = !Syntax Error, I q(2m)(z) * { !Syntax Error, Idx' dy' a1(x',y') [ s1m Km (jβs1) - s2m Km (jβs2)] - !Syntax Error, Idx' dy' a2(x',y') [ s1m Km (jβs1) - s2m Km (jβs2)] } (G.14) We wish to show that in the small β limit, all terms with m= 1,2,3... can be neglected. Consider these factors which contain all the β dependence in the above expression (jβ)-m [ s1m Km (jβs1) - s2m Km (jβs2)] . For m = 0 the leading terms for small β give [K0 (jβs1) - K0 (jβs2)] ≈ -ln((jβs1/2) [ 1 + ((jβs1/2)2] + ln((jβs2/2) [ 1 + ((jβs2/2)2 ] = -ln((jβs1/2) [ 1 +O(β2)] + ln((jβs2/2) [ 1 + O(β2) ] = -{ln(β) + ln(js1/2)} [ 1 +O(β2)] + {ln(β) + ln(js2/2)} [ 1 +O(β2)] = ln(s2/s1) + O(β2) . For m > 0, this same factor behaves as [Km (jβs1) - Km (jβs2)] = {2m-1(m-1)! (jβs1)-m - 2m-3 (m-2)! (jβs1)-m+2 + O((jβs1)-m+4)} - {2m-1(m-1)! (jβs2)-m - 2m-3 (m-2)! (jβs2)-m+2 + O((jβs2)-m+4)} = (jβs1)-m {2m-1(m-1)! - 2m-3 (m-2)! (jβs1)2 + O((jβs1)4)} - (jβs2)-m {2m-1(m-1)! - 2m-3 (m-2)! (jβs2)2 + O((jβs2)4)} = (jβs1)-m {2m-1(m-1)! + O(β2) } - (jβs2)-m {2m-1(m-1)! + O(β2) } = (jβ)-m 2m-1(m-1)! { s1-m - s2-m + O(β2) } Thus we compare the factors as they appear in (G.14) [ ln(s2/s1) + O(β2)] m = 0 2m-1(m-1)! (jβ)-m { s1-m - s2-m + O(β2) } = 2m-1(m-1)! { s1-m - s2-m + O(β2) } m = 1,2,3... The m = 0 term is seen to be order β0 = 1 while all other terms are order β V(z) = !Syntax Error, I q(2m)(z) * { !Syntax Error, Idx' dy' a1(x',y') [ s1m Km (jβs1) - s2m Km (jβs2)] - !Syntax Error, Idx' dy' a2(x',y') [ s1m Km (jβs1) - s2m Km (jβs2)] } (G.14) Inserting the above small argument expansion for Km(z) we find (jβ)-m [ s1m Km (jβs1) - s2m Km (jβs2)] = (jβ)-m [ s1m {2m-1(m-1)! (jβs1)-m - 2m-3 (m-2)! (jβs1)-m+2 + O( (jβs1)-m+4)} - s2m {2m-1(m-1)! (jβs2)-m - 2m-3 (m-2)! (jβs2)-m+2 + O( (jβs2)-m+4)} ] Let us insert the small z expansion (G.12) for Km(z) into the square bracketed factors in (G.14). We get: (jβ)-m [ s1m Km (jβs1) - s2m Km (jβs2)] = + 2m-1(m-1)! [ 1 - 1 ] /leading term - 2m-3 (m-2)! (jβ)2 [ s12 - s22 ] / second term, order(β2) + 2m-5 (m-3)! (jβ)4 [ s14 - s24 ] / third term, order(β4) + ... The leading term, which would have been significant, completely cancels. The remaining terms are of order β2 and smaller. For example, the second term has a coefficient which contains β2 and involves finite integrals over the charge density of the following form: !Syntax Error, Idx' dy' a1(x',y') [ s12 - s22 ] Thus, each term in the sum over m in (9) is of order β2 or less. That is, it might happen that the integral above vanishes, so then each term would be of order β4. Since β = 2π/λ is a smallness parameter, we conclude that only the m=0 term is significant. This term was analyzed in the text of Chapter 4.