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\begin{document}

\title{Residue of Series of Functions}



\author[1]{Benedict Irwin}%
\affil[1]{University of Cambridge}%


\vspace{-1em}



  
  \date{May 17, 2024}


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\section*{Abstract}
We consider the values of residues of functions.
We know the residues of some functions at their poles.
For example
$$
Res(\Gamma(s),-k) = \frac{(-1)^k}{k!}, k\in \mathbb{Z}
$$
but then some functions are hard, for example
$$
Res(e^{\Gamma(s)},-k) = ?
$$
if we consider a series expansion of $e^{\Gamma[s]}$, we get terms
$$
Res\left(\sum_{l=0}^n \frac{\Gamma(s)^l}{l!}, -k\right)
$$
for some such expansions it appears that there is a convergent limit, for large $n$. 
i.e.
$$
\lim_{n \to \infty}Res\left(\sum_{l=0}^n \frac{\Gamma(s)^l}{l!}, s=0\right) \approx 0.80562017...
$$
and
$$
\lim_{n \to \infty}Res\left(\sum_{l=0}^n \frac{\Gamma(s)^l}{l!}, s=-1\right) \approx -1.21469623...
$$
If we call this
$$
Q[e^{\Gamma(s)}](k) = \lim_{n \to \infty}Res\left(\sum_{l=0}^n \frac{\Gamma(s)^l}{l!}, s=-k\right)
$$
then we could define a function
\begin{equation}
f(x) = \sum_{k=0}^\infty Q[e^{\Gamma(s)}](k)x^k \approx 0.806 - 1.215x + 1.02621x^2 - 0.13931294x^3 + 0.04446964782x^4 - ...
\end{equation}
where interestingly, the higher terms are more precise with only a few terms required in the truncated version of Q.

For the function f(x), it seems like the minimum is around $x_0=0.658734$ with a value of $v=0.418402$, with some likely bounds
$0.401076 < v < 0.454371$ and $0.62907 < x_0 < 0.726154$.

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