Home

To trace the orbit of the lemniscate defined where the real part of tanh(ln(1+x^2)) vanishes, we can use the iteration function:

$$z_{n+1} = z_n - \frac{\operatorname{Re}\left(\tanh\left(\ln\left(1+z_n^2\right)\right)\right)}{\operatorname{Re}\left(\dfrac{d}{dz}\tanh\left(\ln\left(1+z_n^2\right)\right)\right)}$$

where $z_n$ is the $n$th iterate of the orbit. We start with an initial value $z_0$ in the region where the real part of tanh(ln(1+x^2)) vanishes, and we iterate the function until convergence to the desired accuracy.

This iteration function is derived from the Newton-Raphson method, which is a root-finding algorithm that can be used to find the zeros of a function. In this case, we are looking for the zeros of the real part of tanh(ln(1+x^2)), which correspond to the lemniscate curve.

The denominator of the iteration function corresponds to the derivative of the real part of tanh(ln(1+x^2)), which is given by:

$$\operatorname{Re}\left(\dfrac{d}{dz}\tanh\left(\ln\left(1+z^2\right)\right)\right) = \dfrac{4z}{(1+z^2)^2}\operatorname{sech}^2\left(\ln\left(1+z^2\right)\right).$$

Using this iteration function, we can trace the orbit of the lemniscate curve and study its geometry and properties.