I've been making some different graphs of \(\beta\) and I got a good one to share.
Here is \(\beta_{1,1}(s)\) for \(-5 \le \Re(s) \le 10\) and \(-7.5 \le \Im(s) \le 7.5\):
And here's \(\beta_{1+i,1+i}(s)\), it looks super cool. The overflows, again, are mapped to zero. We see a really cool fractal pattern in this one. The domain of \(s\) is the same:
This is \(e^{1+i} = b\) with multiplier \(\lambda = 1+i\). And here's \(\lambda =1\) and \(b = e^{1/2}\):
All of these will be committed towards the asymptotic thesis of the beta function. Which is that the beta method approaches at least an asymptotic expansion at each point, as opposed to a Taylor series. This is compatible with everything I have been saying, and additionally compatible with Sheldon's work. This paper will entirely focus on ASYMPTOTIC behaviour. Which looks like tetration; but if you try and make it tetration, expect a good amount of errors.
Here is \(\beta_{1,1}(s)\) for \(-5 \le \Re(s) \le 10\) and \(-7.5 \le \Im(s) \le 7.5\):
And here's \(\beta_{1+i,1+i}(s)\), it looks super cool. The overflows, again, are mapped to zero. We see a really cool fractal pattern in this one. The domain of \(s\) is the same:
This is \(e^{1+i} = b\) with multiplier \(\lambda = 1+i\). And here's \(\lambda =1\) and \(b = e^{1/2}\):
All of these will be committed towards the asymptotic thesis of the beta function. Which is that the beta method approaches at least an asymptotic expansion at each point, as opposed to a Taylor series. This is compatible with everything I have been saying, and additionally compatible with Sheldon's work. This paper will entirely focus on ASYMPTOTIC behaviour. Which looks like tetration; but if you try and make it tetration, expect a good amount of errors.