$\Delta y = e^{sy}$ or: How I Learned to Stop Worrying and Love the $\Gamma$-function

James David Nixon

$\Delta y = e^{sy}$ or: How I Learned to Stop Worrying and Love the $\Gamma$-function

Complex Variables 2019-10-14 v1

Authors: James David Nixon

Abstract

For a nice holomorphic function $f(s, z)$ in two variables, a respective holomorphic Gamma function $\Gamma = \Gamma_f$ is constructed, such that $f(s, \Gamma(s)) = \Gamma(s + 1)$ . Along the way, we fall through a rabbit hole of infinite compositions, First Order Difference Equations, and absurd functional equations... This paper is orchestrated around an investigation into the unconventional equation $\Delta y = y(s+1)-y(s) = e^{sy(s)}$ and its solutions in the complex plane.

Keywords

holomorphic gamma function first order difference equations infinite compositions functional equations complex plane

Cite

@article{arxiv.1910.05111,
  title  = {$\Delta y = e^{sy}$ or: How I Learned to Stop Worrying and Love the $\Gamma$-function},
  author = {James David Nixon},
  journal= {arXiv preprint arXiv:1910.05111},
  year   = {2019}
}

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$\Delta y = e^{sy}$ or: How I Learned to Stop Worrying and Love the $\Gamma$-function

Abstract

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