09/24/2021, 04:37 PM
Having a little tiny break from my tons of homework, I contemplated the very case,
\( f(z)=z+\Gamma(z) \) (Conjugate to more general case \( a\ne0,f_a(z)=z+a\Gamma(z) \))
Since \( \Gamma(z) \) has no zeros, f has no fixed points. But f is not that like the one already constructed: \( z+e^z \) which has a well behaviored fixed point at -infinity.
My first attempt is to map the fixed point at \( \pm{i}\infty \) to 0, using \( T(z)=ln(z)/i,F=T^{-1}fT \). however, F won't have a second derivative at z=0, so I got stuck.
How would someone construct the family of superfunctions of this f? This will be challenging. I'd like to invite everyone to discuss this.
\( f(z)=z+\Gamma(z) \) (Conjugate to more general case \( a\ne0,f_a(z)=z+a\Gamma(z) \))
Since \( \Gamma(z) \) has no zeros, f has no fixed points. But f is not that like the one already constructed: \( z+e^z \) which has a well behaviored fixed point at -infinity.
My first attempt is to map the fixed point at \( \pm{i}\infty \) to 0, using \( T(z)=ln(z)/i,F=T^{-1}fT \). however, F won't have a second derivative at z=0, so I got stuck.
How would someone construct the family of superfunctions of this f? This will be challenging. I'd like to invite everyone to discuss this.