Fungrim entry: ed2bf6

$\sigma_{\ell}\!\left(\eta\right) = \frac{\log \Gamma\!\left(1 + \ell + i \eta\right) - \log \Gamma\!\left(1 + \ell - i \eta\right)}{2 i}$
Assumptions:$\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right)$
TeX:
\sigma_{\ell}\!\left(\eta\right) = \frac{\log \Gamma\!\left(1 + \ell + i \eta\right) - \log \Gamma\!\left(1 + \ell - i \eta\right)}{2 i}

\ell \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(1 + \ell + i \eta \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; 1 + \ell - i \eta \notin \{0, -1, \ldots\}\right)
Definitions:
Fungrim symbol Notation Short description
CoulombSigma$\sigma_{\ell}\!\left(\eta\right)$ Coulomb wave function phase shift
LogGamma$\log \Gamma(z)$ Logarithmic gamma function
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("ed2bf6"),
Assumptions(And(Element(ell, CC), Element(eta, CC), And(NotElement(Add(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0)), NotElement(Sub(Add(1, ell), Mul(ConstI, eta)), ZZLessEqual(0))))))