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\begin{document}
\begin{eqnarray*}
\alpha  & = & v_{s}\epsilon \\
K & = & K_{c}+K_{s}\\
r & = & \frac{K_{c}}{\left(n\pi \alpha \right)^{2}}\left(\frac{v_{s}}{v_{c}}\right)^{2}\\
z & = & \epsilon (\omega +i\delta )
\end{eqnarray*}



\subsection{stationary string}

\begin{eqnarray*}
M_{xx}(z) & = & A\epsilon ^{2}D_{\xi }e^{-\pi i(K-1)}\Gamma \left(1-K\right)z^{K-2}
\end{eqnarray*}
\begin{eqnarray*}
A & = & \frac{1}{L_{y}}\left(\frac{2ev_{F}}{\pi ^{2}\alpha }\right)^{2}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}
\end{eqnarray*}



\subsection{rigid string}

for $\omega \ll \omega _{0}$\begin{eqnarray*}
M_{xx}(z) & = & A\epsilon ^{2}D_{\xi }e^{-\pi i(K-1)}\Gamma \left(1-K\right)z^{k-2}
\end{eqnarray*}
for $\omega \gg \omega _{0}$\begin{eqnarray*}
M_{xx}(z) & = & \frac{A\epsilon ^{2}D_{\xi }}{\sqrt{\epsilon \omega _{0}}}e^{-\pi i(K-1/2)}\Gamma \left(\frac{1}{2}-K\right)z^{K-3/2}
\end{eqnarray*}
\begin{eqnarray*}
A & = & \frac{1}{L_{y}}\left(\frac{2ev_{F}}{\pi ^{2}\alpha }\right)^{2}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}\left(\frac{m\omega _{0}}{2\pi }\right)^{\frac{1}{2}}\\
B & = & \frac{2\pi }{L_{y}}\left(\frac{en}{\pi \alpha \rho }\right)^{2}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}\left(\frac{m\omega _{0}}{2\pi }\right)^{\frac{3}{2}}
\end{eqnarray*}



\subsection{gaussian string}

\begin{eqnarray*}
M_{xx}(z) & = & \epsilon ^{2}D_{\xi 0}e^{\left(2-K_{c}\right)l}e^{-\pi i(K-1)}\Gamma \left(1-K\right)z^{K-2}\\
 &  & \left[\frac{A}{\sqrt{2}\lambda \left[l-\ln \frac{z}{\eta }-\pi i\right]^{1/2}}+\frac{\pi Bz^{2}}{\sqrt{2}\lambda ^{3}K\left(K+1\right)\left[l-\ln \frac{z}{\eta }-\pi i\right]^{3/2}}\left(1+\frac{rz^{2}}{\left(K+3\right)\left(K+2\right)}\right)\right]
\end{eqnarray*}
\begin{eqnarray*}
A & = & \frac{1}{L_{y}}\left(\frac{2ev_{F}}{\pi ^{2}\alpha }\right)^{2}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}\left(1-\frac{\lambda ^{2}}{\pi \alpha ^{2}}\right)\\
B & = & \frac{1}{2\pi L_{y}}\left(\frac{en\lambda }{\pi \alpha ^{2}\rho }\right)^{2}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}\eta ^{-2}
\end{eqnarray*}



\subsection{floppy string $n=2$}

\begin{eqnarray*}
M_{xx}(z) & = & \epsilon ^{2}D_{\xi }e^{-\pi i(K-3/4)}\Gamma \left(\frac{3}{4}-K\right)z^{K-7/4}\left[A_{2}+\frac{B_{2}z}{\left(K+\frac{1}{4}\right)}\left(1+\frac{rz^{2}}{\left(K+\frac{9}{4}\right)\left(K+\frac{5}{4}\right)}\right)\right]
\end{eqnarray*}
\begin{eqnarray*}
A_{2} & = & \frac{1}{\lambda L_{y}}\left(\frac{2ev_{F}}{\pi ^{2}\alpha }\right)^{2}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}\left[1-\frac{1}{2\sqrt{\pi s}}\frac{\lambda }{\alpha }\right]\left(\frac{1}{2\sqrt{\pi s}}\frac{\lambda }{\alpha }\right)^{\frac{1}{2}}\\
B_{2} & = & \frac{\sqrt{\pi }}{2\lambda ^{3}L_{y}}\left(\frac{en}{\pi \alpha \rho }\right)^{2}\left(\frac{v_{s}}{v_{c}}\right)^{K_{c}}\left(\frac{1}{2\sqrt{\pi s}}\frac{\lambda }{\alpha }\right)^{\frac{3}{2}}\frac{\lambda }{\sqrt{s}\alpha }
\end{eqnarray*}

\end{document}
