05/11/2014, 04:29 PM
If \( F \) is holomorphic for \( \Re(z) < b \) and \( |F(z)| < C e^{\alpha |\im(z)| + \rho|\Re(z)|} \) then:
\( \beta(x) = \sum_{n=0}^\infty \frac{(-x)^n}{n!}F(-n) \)
When F has singularities we see we pull on a second balancing function \( \psi \) such that:
\( \beta(x) = \psi(x) + \sum_{n=0}^\infty \frac{(-x)^n}{n!}F(-n) \)
For simple functions like \( F(z) = 1/(z^2 - 1) \) then \( \psi \) is very easy to calculate. For more complicated functions like tetration, which no doubt has essential singularities instead of poles, it becomes much more complicated. But the general result on essential singularities is just applying cauchy's residue formula around all the poles of \( F \) on the function \( \G(z)x^{-z} \)
As in, if \( F(z) = G(z) + \sum_{n=0}^\infty \frac{a_n}{(z-\alpha)^{n+1} \) where \( G(z) \) is analytic in a neighbourhood of \( \alpha \) then:
\( \psi(x) = g(x) + \sum_{n=0}^\infty a_n [\frac{d^n}{dz^n} \G(z)x^{-z} ]_{z=\alpha} \)
where \( g(x) \) carries information about the other poles of \( F \)
I haven't looked into much of how these balancing functions behave. I'm more familiar with just working with entire \( \beta \) and entire \( F \). I wish I could help you more on this but I feel discouraged looking at tetration. I think my iteration method might be restricted to simpler functions that don't behave quite as eraddictly.
And on a different note. I've successfully shown that, if \( \phi \) is holomorphic in the strip \( 0< a-1 < \Re(z) < b \) and satisfies the bounds \( |\phi(z)| < Ce^{\alpha |\Im(z)|} \) for \( 0 \le \alpha < \pi/2 \). Then for \( a < \Re(z) < b \) and \( \Re(s) > 0 \) I can calculate:
\( \bigtriangledown_z^{-s} \phi(z) \)
where it satisfies the composition rule and interpolates the iterated continuum sum at natural values. Pcha!! Holomorphic in z and s. Pcha!
\( \beta(x) = \sum_{n=0}^\infty \frac{(-x)^n}{n!}F(-n) \)
When F has singularities we see we pull on a second balancing function \( \psi \) such that:
\( \beta(x) = \psi(x) + \sum_{n=0}^\infty \frac{(-x)^n}{n!}F(-n) \)
For simple functions like \( F(z) = 1/(z^2 - 1) \) then \( \psi \) is very easy to calculate. For more complicated functions like tetration, which no doubt has essential singularities instead of poles, it becomes much more complicated. But the general result on essential singularities is just applying cauchy's residue formula around all the poles of \( F \) on the function \( \G(z)x^{-z} \)
As in, if \( F(z) = G(z) + \sum_{n=0}^\infty \frac{a_n}{(z-\alpha)^{n+1} \) where \( G(z) \) is analytic in a neighbourhood of \( \alpha \) then:
\( \psi(x) = g(x) + \sum_{n=0}^\infty a_n [\frac{d^n}{dz^n} \G(z)x^{-z} ]_{z=\alpha} \)
where \( g(x) \) carries information about the other poles of \( F \)
I haven't looked into much of how these balancing functions behave. I'm more familiar with just working with entire \( \beta \) and entire \( F \). I wish I could help you more on this but I feel discouraged looking at tetration. I think my iteration method might be restricted to simpler functions that don't behave quite as eraddictly.
And on a different note. I've successfully shown that, if \( \phi \) is holomorphic in the strip \( 0< a-1 < \Re(z) < b \) and satisfies the bounds \( |\phi(z)| < Ce^{\alpha |\Im(z)|} \) for \( 0 \le \alpha < \pi/2 \). Then for \( a < \Re(z) < b \) and \( \Re(s) > 0 \) I can calculate:
\( \bigtriangledown_z^{-s} \phi(z) \)
where it satisfies the composition rule and interpolates the iterated continuum sum at natural values. Pcha!! Holomorphic in z and s. Pcha!
