Exercise 18

Vasco,

See if you think this plot represents the problem accurately.

dl.dropboxusercontent.com/u/35393965/nh.ch4.ex18.pdf

Since posting this, a crucial correction has been made to the entities passed to amplitwist in two instances.

Gary

Vasco,

I have no particular concerns. I think the code now reflects the theory accurately, but there is no claim to infallibility.

I think two things made this tricky: (1) distinguishing the z in M(z) from the one in M'(z); and (2) distinguishing a z being passed to a function with a variable named "z" from an origin-centred z. M(z) normally applies to a point (z+q), whether centred at the origin or not. In this case we begin with z's on origin-centred circles recentred to -d/c, so z+q = z-d/c. We can see in the decomposition from (3) p. 124, that a point under M gets recentred to the origin before amplification and twisting, so we can pass (z+q) to M in the program. M'(z) requires a point of emanation centred at the origin and applies to an infinitesimal vector by multiplication. In the program, the amplitwist[z] function has a factor of \(1/(z-q)^2\). The z that is passed to it is the point of emanation (z+q) just as with M. Then z-q in the denominator works out to be (z+(q=(-d/c)) - (-d/c)) = z (origin-centred).  It could be done differently, but it just seemed more convenient at the time. If we actually take the derivative of M, the result includes a factor of \(1/(cz+d)^2 = 1/(c^2(z+d/c)^2)\). If we pass (z+q) = (z-d/c) to the amplitwist[] function, we get \(1/c^2((z-d/c)+d/c)^2) = 1/c^2z^2\) where z is origin-centred and \(|z| = \rho\). I think this is what we want.

Gary

Vasco,

Thank you for the comments. I believe the program is equivalent to what you have proposed but I have revised it to make the variable names consistent with your note and to eliminate some obsolete comments, which I think were adding to the confusion created by the similarity of variable names from the problem and the names of function variables. There is added discussion of your interesting comment on the use of Figure [1], p. 124 to understand the amplitwist in this problem. I benefited by working through that.

Here is how I understand it using the z and w notation. In this problem, where we wish to preserve concentricity, M is applied to z = w + q, where |w| = \(\rho\) and q = -d/c, because M(z) automatically recentres circles to the origin under those conditions, but M' is calculated on an expression in w (A/w in your document), after recentering to the origin. The result is then applied to the \(\epsilon\) vectors, which after amplitwisting, emanate from M(z), so the essence of the effect on the \(\epsilon\) vectors is M(z) + amplitwist(w) \(\epsilon\). I think this could also be written:

      \(M(z) + (|A|/|w|^2)e^{i (-2arg(w)+arg(A))}  \epsilon\), where \(A = (ad-bc)/c^2\)

Gary

Vasco,

Thank you. It was a careless oversight caused by not reading your note carefully enough. I have changed it to z-q.

Gary