Equation (2) applies to angles theta in the range
For larger angles, it is replaced by two other equations.
These are both solved explicitly giving, on [90°, 135°] equation (3) or its simplified form
and on [135°, 180°], a modified secant function:
Mandelbrot and Frame's Figure (9) shows the graph of the critical r, the
solution of equation (2) and these other two equations,
as a function of theta. My version of that graph is shown
at the right (theirs shows nicer detail, being printed on nice
Looking at the graph, we might make the following guesses:
1. We have already observed that the cosine equation makes it easy to prove that r is continuous on [0°, 90°]. A little algebra shows that r is continuous at 90° and 135°, with value 1 / sqrt(2) at both of these angles.
2. & 3. The formula
gives value r = .5 at theta = 180, and the
derivative there is zero.
At the other end (theta = 0) we show (elsewhere) that the r-values are asymptotic to 1 / 2cos(theta) and so r has the same limit one half) and has a zero derivative at r=0, even though the derivative of r has points of discontinuity approaching 0.
4. & 5. We already observed (without using calculus) that r is increasing on [0, 90], and that it is smooth in the intervals [90/n, 90/(n-1)]. The two formulas which apply for larger angles are certainly differentiable.
6. For angles in the range [0, 90], the derivative of r is given by the implicit function theorem:
The key here is to remember that the sums are taken over exactly those successive k-values for which the cosine values are positive. As theta increases past an angle 90/n (n>1), the sums in the top and bottom of this fraction both lose their last terms. At that point the cosine in the lost term is zero so the value of the bottom is unchanged. However the sine value is one, so the value of the top (and hence the fraction) decreases, exhibiting a jump discontinuity.
7. It seems that it would be straightforward to show that the second derivative of r is positive, but I have not yet wrestled this problem into submission.
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