Very cool, Leo! Very cool!
I think for the most part you are correct; and I believe you answered your own question. We just have to consider \(z+\Gamma(z)\) with the fixed point at \(-\infty\); and additionally ignore the singularities; by treating the upper and lower half planes separately. Then it becomes a game of Schroder fixed point theory; just a special case analysis. For \(\Im(z) > 0\) we are assured that \(z=\infty\) is a fixed point; as \(\Im(z) \to \infty\) this function diverges, and as \(\Re(z) \to \pm \infty\) this function diverges. Then since \(\Im(z) > 0\) is a simply connected domain, we can apply Schroder. Especially because \(\infty\) is in the interior of \(\Im(z) > 0\). Choose a linear fractional transformation such that \(h:\Im(z) > 0 \to |z|<1\) such that \(h(\infty) = 0\); and now we are doing schroder iterations about \(f(z) = h(z+\Gamma(z))\); which should be doable about \(f\approx 0\); or for large z arguments for \(z+\Gamma(z)\).
Great job!
James
I think for the most part you are correct; and I believe you answered your own question. We just have to consider \(z+\Gamma(z)\) with the fixed point at \(-\infty\); and additionally ignore the singularities; by treating the upper and lower half planes separately. Then it becomes a game of Schroder fixed point theory; just a special case analysis. For \(\Im(z) > 0\) we are assured that \(z=\infty\) is a fixed point; as \(\Im(z) \to \infty\) this function diverges, and as \(\Re(z) \to \pm \infty\) this function diverges. Then since \(\Im(z) > 0\) is a simply connected domain, we can apply Schroder. Especially because \(\infty\) is in the interior of \(\Im(z) > 0\). Choose a linear fractional transformation such that \(h:\Im(z) > 0 \to |z|<1\) such that \(h(\infty) = 0\); and now we are doing schroder iterations about \(f(z) = h(z+\Gamma(z))\); which should be doable about \(f\approx 0\); or for large z arguments for \(z+\Gamma(z)\).
Great job!
James