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Just out of curiosity. Has anyone solved the complex continuous iterations of the function

?

This will be very challenging because

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 to 0 with a specific function, may be the inverse of .

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A wild guess without basis, maybe this function is just a special case of Fox H

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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

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