math mystery

Example of order-interchange of limits problem PhL 9.2.14 1. The original problem I encountered this when writing version 2 of lines doc Appendix Q. I have this function and I am interested in u→ 0 for a small ω limit, and also a ωd→ 0 limit for no conductance in a transmission line. See Appendix Q new version.doc. Let's first look at the limit u → 0: Rek = (1/2) [ 2 ( Rdc2C2 ωd2)1/2 - 2RdcC ωd ]1/2 = = (1/2) [ 2 RdcC ωd - 2RdcC ωd ]1/2 = 0 If I then take the limit of this result as ωd→ 0 , the result is of course still 0. Now look at the limit ωd → 0: Rek = (1/2) [ 2 {( Rdc2 + u4Ldc2)(C2t2u4+ u4C2) }1/2 - 2 RdcC t2u4 + 2u4LdcC ]1/2 = (1/2) [ 2 {( Rdc2 + u4Ldc2)Cu2(t2+1) }1/2 - 2 RdcC t2u4 + 2u4 LdcC ]1/2 Now take the limit of this result as u→ 0 Rek = 0 I thought there was going to be a conflict here? Back up a bit. Ignore the overall square root and leading factor Manually simplify this 2C - 2RdcC (ωd+ tu2) + 2u4LdcC Now remove the overall common factor of C f(ωd,u) ≡ 2 - 2Rdc (ωd+ tu2) + 2u4Ldc What can be said about this function of two complex variables ωd and u ? f(ωd=0,u) ≡ 2 - 2Rdc ( tu2) + 2u4Ldc = 2 u2 - 2Rdc ( tu2) + 2u4Ldc = u2 [2 - 2 tRdc + 2u2Ldc ] f(ωd,u=0) ≡ 2 - 2Rdc (ωd) = 0 2. Simpler Case How about this much simpler function f(ωd,u) ≡ I want to expand this in small u with ωd > 0. Thus f(ωd,u) = = = (1/ωd) ≈ (1/ωd) { 1 + (1/2) [2tu2/ωd+ u4(t2+1)/ωd2] +(1/2)(-1/2)(1/2) [2tu2/ωd]2 } ≈ (1/ωd) { 1 + (1/2) [2tu2/ωd+ u4(t2+1)/ωd2] -(1/2) [tu2/ωd]2 } ≈ (1/ωd) { 1 + (1/2) [2tu2/ωd+ u4(t2+1)/ωd2] -(1/2) t2u4/ωd2 } ≈ (1/ωd) { 1 + tu2/ωd+ (1/2)u4(t2+1)/ωd2 - (1/2) t2u4/ωd2 } ≈ (1/ωd) { 1 + tu2/ωd + (1/2)u4(1)/ωd2 + O(u6) } If I then take ωd→ 0, each term blows up! On the other hand, if I first take ωd→ 0 I get f(0,u) = = u2 and this is well-behaved as u→ 0. So what stupidly simple thing is going on here that dumbo fails to comprehend? The series in u must not be a convergent series. Why is that so? 3. Simpler Case Still This is a power series in complex variable y. What is the radius of convergence of this power series? I have a function here of two complex variables and I expand it in z about the point z = 0. 4. Even Simpler Case Still Again, the coefficients all blow up as x → 0. Here is what is going on. Write as f = The small y expansion converges around y = 0 in a disk of radius 1 because that circle hits a branch point, Similarly, the function expanded about z = 0 has a radius of convergence of 1/x: As x→ 0, the radius of convergence → 0 as well. In this limit, the power series expansion in z does not converge! 5. Back to the original problem The two complex variables are z = u and x = ωd . I expand this about u = 0 to get some terms, You can see that the coefficients are blowing up as ωd→ 0. There is surely a branch point for the complicated function Rek shown which is some function of ωd. As ωd → 0, this branch point approaches the origin in u-space, causing the u-expansion radius of convergence to shrink. The expansion becomes less and less convergent and finally diverges. In the limit ωd→ 0, the u expansion is divergent and completely meaningless. ωd ≡ (σd/εd) = 10-15 / 8.8541877 x 10-12 = 1000/8.85 = 113 aHz for PE dielectric fd = ω/2π = 18 Hz. So I guess the question is this: how "small" is such an ωd ? For the Belden cable, ωd = 113 in SI units, maybe that is not so small after all.