Let . has a root in (domain colored below) whenever there exists such that or

Do you see the double root at or so? That’s because at that point . There’s a whole necklace of double roots on the right side of the picture.

Let . has a root in (domain colored below) whenever there exists such that or

Do you see the double root at or so? That’s because at that point . There’s a whole necklace of double roots on the right side of the picture.

The Rogers-Ramanujan continued fraction is given by

and looks like this on the unit disk:

The ordinary sine and cosine are averages of exponentials. Here, I play around with replacing the arithmetic average with the arithmetic geometric mean

Let

And since the $\mathrm{agm}(x,y)$ may be represented in terms of a complete elliptic integral of the first kind $K(m)$:

Therefore

Therefore:

Domain colored, with mpmath, it sort of looks like this: