Volume 9, Issue 2
Tricks of the Trade
In and Out
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We start with some visualizations to get an impression of the complexity of the Rogers-Ramanujan continued fraction. Unfortunately, because it is not a simple continued fraction, we cannot use built-in functions like FromContinuedFraction. So we implement a two-argument version of where the second argument indicates the number of divisions.
Here are the first five levels of the Rogers-Ramanujan continued fraction.
Fortunately, for numerical evaluation, the continued fraction form is very convenient (as opposed to the ratio of two polynomials) and stable, and, even for hundreds of divisions, machine-precision arithmetic is sufficient to get reliable results. (It is important to avoid high-precision arithmetic because we need tens of thousands of values for graphics.) The next input compares a machine- with a high-precision calculation.
The untruncated Rogers-Ramanujan continued fraction can be expressed in closed form using the function DedekindEta. Due to the branch cuts of the logarithm function, this representation does not hold for all .
For any not too near the unit circle, the truncated continued fraction converges quickly. The following graphic shows the first 36 truncations for real . While for the curves converge to one curve, for we see two curves emerge.
These two asymptotic values taken by the truncated continued fraction after divisions for can be expressed through values inside the unit disk :
The next two inputs confirm this.
The following infinite product representation for the Rogers-Ramanujan continued fraction strongly suggests that the -unit circle contains a large set of singularities:
Here is a "togethered" form of a truncated fraction that ignores the factor.
Next, we truncate after 30 divisions and display the zeros of the resulting denominator, a polynomial of degree 240. The roots clearly cluster around the unit circle, and indeed the unit circle is the natural boundary of analyticity for .
On the unit circle, the continued fraction can converge or diverge . The following graphics show the behavior of the truncated continued fractions for various in . To avoid the time-consuming calculation of for different , we implement the function CFConvergents that gives all the convergents at once.
Using the function CFConvergents for arguments outside the unit circle shows that the alternating values (known as a two-period) also occurs for complex -values outside the unit circle.
As a function with a natural boundary at the unit circle, the accumulation of singularities will make a graphic worthwhile. We show contour plots of the real and imaginary parts in the disk .
Because a direct call to ContourPlot yields relatively poor results, we attempt a homogeneous contour spacing and also map the resulting contour plot on a circular domain. The function circularContourPlot does both.
The two yellow and black graphics clearly show the clustering of the contour lines due to the dense set of singularities forming the boundary of analyticity at the unit circle.
We repeat the same calculations for an odd number of divisions to visualize the second value taken asymptotically outside the unit circle. This time we use more psychedelic colors.
We conclude this section with a Riemann surface (in the spirit of previous Trott's Corners). The factor makes a nontrivial Riemann surface with five sheets.
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