09/29/2021, 03:15 PM
Just out of curiosity. Has anyone solved the complex continuous iterations of the function \( f(z)=z+\Gamma(z) \)?
This will be very challenging because \( \Gamma(z) \) has no zeros, and the zero in the directed complex infinity is non-Botcher-constructable.
My thought is using a function to map the fixed point at \( \pm{i}\infty \) to 0 with a specific function, may be the inverse of \( g(z)=\sqrt{\frac{\pi}{\pm{i}z\sinh(\pm{i}\pi{z})}} \).
This will be very challenging because \( \Gamma(z) \) has no zeros, and the zero in the directed complex infinity is non-Botcher-constructable.
My thought is using a function to map the fixed point at \( \pm{i}\infty \) to 0 with a specific function, may be the inverse of \( g(z)=\sqrt{\frac{\pi}{\pm{i}z\sinh(\pm{i}\pi{z})}} \).