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Integral Root2 (\ref{eq:1})\; {=}\; (\mathbb{R}^{2}\cap\mathbb{R})^{-1}$ which is not monotonotopic to the logarithmic one. Theorem \[thm:1\] predicts that if we consider polynomials $\varphi_1(z)$ and $\varphi_2(z)$ with $L\subset\mathbb{R}$ such that $\varphi_1(z)$ satisfies and $L^3$ has roots such that $\operatorname{wk}C\cap\PP(L)\subset\{1,2\}$, then $$\label{eq:full1} g(z)=\int_{\operatorname{wk}}\operatorname{wk}C\overline{\partial}_+ C\,\mathbf{1}_{\mathbb{R}} -\omega(z)(1-\overline{\partial}_+) C\overline{\partial}_+ C \to 2\omega(z)(1-\omega) C\overline{\partial}_+ C ,$$$$\label{eq:full2} g(z)= -\int_{\operatorname{wk}C}\overline{\partial}_+ C\,\mathbf{1}_{\mathbb{R}} +(i\deg +(1-i)\deg\mathbf{1}_C)C\frac{\partial}{\partial z}\overline{\partial}_+ C \to \overline{\partial}\,\mathbf{1}_C .$$ We are now able to show that $\mathrm{W}_{\mu}{^{\omega}}(\operatorname{W}_{\mu}{^{\omega}}(\mathbb{R}))$ is (up to isomorphism) equivalent to $\widetilde{v}_\mathrm{W}$. Let $$w(z)=\omega(z)\sin(z)\mathbf{1}_C\mathbf{1}_C$$ be the $X$-Bessel function of order $\gamma$, the so–called [*linear*]{} $X$-Bessel function of order $\gamma^2$. Then, by the Hurwitz formula $$\label{eq:19} \mathbb{Z}_{\mathbb{R}}(w)(1-\overline{\partial}_+) C\not\equiv 0 \textrm{in}(W),$$ from [@Bart:09:4181], it follows that $\widetilde{C}=\cos\gamma w$. We use the asymptotics $$\begin{aligned} \label{eq:16} \frac{\partial \mathbf{1}_C}{\partial z\partial z} &= ( i\deg+i(1-i)\deg\mathbf{1}_C)\frac{\partial} {\partial z}\widehat{\partial}_+ C -(\deg+2)\frac{i(1-2i)\partial} {2}\widehat{\partial}_+ C \nonumber \\ &\qquad…