02/06/2021, 08:47 PM
(This post was last modified: 02/06/2021, 09:32 PM by sheldonison.)
Thanks James for a superbly written paper, especially the first half of your paper where you carefully prove the convergence of \( \phi(s) \), and that it is entire. Here are a couple of graphs of \( \phi \) at the \( \pi\\i \) line, and in the complex plane. \( \phi(s+1)=\exp(\phi(s)+s) \)
The first graph goes from -2, to +7 at the real axis, and -0.5 to 3.5 in the imaginary.
The pi i contour is also interesting, where it is real valued. It starts out negative near zero as the asymptotic behavior of \( \phi(s)=\approx\exp(s-1) \), but it doesn't grow quite as fast, and you get these weird bump around 6+pi*i. The graph goes from -2 to +12. Eventually, the phi(s) becomes large enough negative that exp(phi(s)) becomes really small compared to exp(s), so the phi(s+1) gets close to zero, but slightly negative. Then phi(s+2) is nearly exp(s+1).
Finally, here is phi(s) at pi*i in the complex plane from +4 .. +13 from pi*I-3 to pi*i+1, showing some of the interesting behavior.
The first graph goes from -2, to +7 at the real axis, and -0.5 to 3.5 in the imaginary.
The pi i contour is also interesting, where it is real valued. It starts out negative near zero as the asymptotic behavior of \( \phi(s)=\approx\exp(s-1) \), but it doesn't grow quite as fast, and you get these weird bump around 6+pi*i. The graph goes from -2 to +12. Eventually, the phi(s) becomes large enough negative that exp(phi(s)) becomes really small compared to exp(s), so the phi(s+1) gets close to zero, but slightly negative. Then phi(s+2) is nearly exp(s+1).
Finally, here is phi(s) at pi*i in the complex plane from +4 .. +13 from pi*I-3 to pi*i+1, showing some of the interesting behavior.
- Sheldon