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 Has anyone solved iterations of z+Γ(z)? Leo.W Junior Fellow Posts: 29 Threads: 3 Joined: Apr 2021 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 $g(z)=\sqrt{\frac{\pi}{\pm{i}z\sinh(\pm{i}\pi{z})}}$. Ember Edison Fellow Posts: 66 Threads: 7 Joined: May 2019 09/29/2021, 07:52 PM A wild guess without basis, maybe this function is just a special case of Fox H tommy1729 Ultimate Fellow Posts: 1,491 Threads: 355 Joined: Feb 2009 10/03/2021, 09:13 PM I see no reasons why iterations of this type should not be possible with the same techniques we used for tetration and related ones. I once started a thread here about the superfunction of exp(z)+z. Notice exp(z)+z also has no FINITE fixpoint just like your z + Γ(z). But exp(z) + z has a fixpoint at negative infinity. On the other hand , exp(z) + z has no poles. And exp(z) + z has a nice asymp to z. So the situation is different. One could also wonder about the generalization z +  Γ(z,v) for various v. In particular v = 1 for obvious reasons. On the other hand Γ(z,1) does have zero's. ---- the infinite composition/functional equation : f(s+1) = t(s) f(s) + Γ( t(s) f(s) ) - where t(s) is like with the gaussian method - should work , not ? Interesting question, thank you. regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,491 Threads: 355 Joined: Feb 2009 10/03/2021, 09:17 PM Again , nice question ! « Next Oldest | Next Newest »

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