MinimalSurface

To form a minimal surface from the real part of $g(x) = \tanh(\ln(1+x^2))$, we can use the Enneper-Weierstrass parameterization, which expresses a minimal surface as a complex function $f(z)$ that satisfies the Weierstrass representation formula:

$$f(z) = z - i \int_0^z \frac{\text{Re}(g(\zeta))}{\sqrt{1 + |\partial_\zeta f|^2}} d\zeta,$$

where $z = x + iy$ is a complex variable and $\partial_\zeta f$ denotes the derivative of $f$ with respect to $\zeta$. This formula relates the real part of $g$ to the shape of the minimal surface.

To apply this parameterization to the real part of $g(x)$, we first need to extend it to a holomorphic function of a complex variable $z$. One way to do this is to define $g(z)$ as:

$$g(z) = \tanh(\ln(1+z^2)),$$

where the complex logarithm $\ln$ is taken to be the principal branch. Note that $g(z)$ has a branch cut along the imaginary axis, where the argument of $z^2$ changes by $\pi$. However, since we are only interested in the real part of $g(z)$, this branch cut does not affect the Enneper-Weierstrass parameterization.

We can then compute the complex function $f(z)$ using the formula above. Note that the denominator $\sqrt{1 + |\partial_\zeta f|^2}$ can be simplified in this case to $|\partial_x f|$, since $f$ is a real function and does not depend on $y$. Thus, we have:

$$f(z) = z - i \int_0^z \frac{\text{Re}(g(\zeta))}{|\partial_x f(\zeta)|} d\zeta.$$

To compute this integral, we can use the fact that $f(z)$ is a real function and therefore satisfies the Cauchy-Riemann equations, which relate its partial derivatives with respect to $x$ and $y$. Specifically, we have:

$$\partial_x f(z) = 1 - i \int_0^z \frac{\text{Re}(g(\zeta))}{|\partial_x f(\zeta)|^3} \partial_y f(\zeta) d\zeta,$$

where we have used the fact that $\partial_x f$ is constant along horizontal lines in the $z$-plane. We can then solve for $\partial_y f(z)$ using this equation and substitute into the formula for $f(z)$ to obtain a single integral that depends only on the real part of $g(z)$.

In practice, the computation of the Enneper-Weierstrass parameterization can be challenging, especially for complex functions with singularities or branch cuts. Numerical techniques, such as the Cauchy integral method or the boundary element method, may be needed to accurately compute the shape of the minimal surface in these cases.