- #1
Nusc
- 752
- 2
Homework Statement
The final answer should be
<br /> \begin{equation}<br /> + g^{2} \int_{-\infty}^{\infty} d \Delta'\; {\cal \rho}(\Delta')\, \frac{1}{s+\gamma+i\Delta'} = + g^{2} \frac{1}{s+\gamma+\gamma_{n}}<br /> \end{equation}<br />
but I get a - in front of g
<br /> \begin{subequations}<br /> \begin{eqnarray}<br /> \int_{-\infty}^{\infty} d \Delta'\; {\cal \rho}(\Delta')\, \frac{1}{s+\gamma+i\Delta'}<br /> &=& <br /> \frac{1}{\pi} \int_{-\infty}^{\infty} d \Delta'\; \frac{\gamma_{n}}{(\Delta'^{2}+\gamma_{n}^{2})}\, \frac{1}{(s+\gamma+i\Delta')}<br /> \\<br /> \nonumber<br /> &=& <br /> \frac{1}{\pi} \int_{-\infty}^{\infty} d \Delta'\; \frac{\gamma_{n}}{(\Delta'+i\gamma_{n})(\Delta'-i\gamma_{n})}\, \frac{1}{(s+\gamma+i\Delta')}<br /> \end{eqnarray}<br /> \end{subequations}<br />
in which case the poles are at $\pm i\gamma_{n}, +i(s+\gamma)$.
<br /> \begin{subequations}<br /> \begin{eqnarray}<br /> \lim \limits_{\Delta' \to i\gamma_{n}} \frac{\gamma_{n}}{(\Delta'+i\gamma_{n})}\, \frac{1}{(s+\gamma+i\Delta')} &=& \frac{1}{2i}\frac{1}{s+\gamma-\gamma_{n}}<br /> \\<br /> \lim \limits_{\Delta' \to -i\gamma_{n}} \frac{\gamma_{n}}{(\Delta'-i\gamma_{n})}\, \frac{1}{(s+\gamma+i\Delta')} &=& \frac{-1}{2i}\frac{1}{s+\gamma+\gamma_{n}}<br /> \\<br /> \lim \limits_{\Delta' \to i(s+\gamma)} \frac{\gamma_{n}}{(\Delta'+i\gamma_{n})(\Delta'-i\gamma_{n})}\frac{1}{(s+\gamma+i\Delta')} &=& \lim \limits_{\Delta' \to i(s+\gamma)} \frac{\gamma_{n}}{(\Delta'+i\gamma_{n})(\Delta'-i\gamma_{n})}\frac{-i}{(-i(s+\gamma)+\Delta')} <br /> \\<br /> \end{eqnarray}<br /> \end{subequations}<br />
It is simpler integrate in down region where we have only one pole $ -i\gamma_{n}$.
While we have two poles in the region above the x:
$\Delta'= i\gamma_n$,
and
$\Delta'= i(s+\gamma)$.
<br /> \begin{subequations}<br /> \begin{eqnarray}<br /> \oint_{\gamma_{R}(t)} f(z) \, dz &=& 2\pi i[\frac{1}{2i}\frac{-1}{s+\gamma+\gamma_{n}} ]<br /> %\frac{-1}{2i}\frac{1}{s+\gamma-\gamma_{n}} + \frac{\gamma_{n}}{(-s-\gamma+\gamma_{n})(s+\gamma+\gamma_{n})}]<br /> \\<br /> %&=&2\pi i[-\frac{(1+i)\gamma_{n}}{(s+\gamma-\gamma_{n})(s+\gamma+\gamma_{n})}]<br /> \end{eqnarray}<br /> \end{subequations}<br />
<br /> \begin{subequations}<br /> \begin{eqnarray}<br /> \int_{-\infty}^{\infty} d \Delta'\; {\cal \rho}(\Delta')\, \frac{1}{s+\gamma+i\Delta'}<br /> &=& <br /> \frac{-\pi}{s+\gamma+\gamma_{n}} <br /> \\<br /> \frac{1}{\pi}\int_{-\infty}^{\infty} d \Delta'\; {\cal \rho}(\Delta')\, \frac{1}{s+\gamma+i\Delta'}<br /> &=& <br /> \frac{-1}{s+\gamma+\gamma_{n}} <br /> \end{eqnarray}<br /> \end{subequations}<br />
I get:
<br /> \begin{equation}<br /> + g^{2} \int_{-\infty}^{\infty} d \Delta'\; {\cal \rho}(\Delta')\, \frac{1}{s+\gamma+i\Delta'} = s + \kappa - g^{2} \frac{1}{s+\gamma+\gamma_{n}}<br /> \end{equation}<br />
