- #1
Nusc
- 760
- 2
Homework Statement
The final answer should be
[tex]
\begin{equation}
+ g^{2} \int_{-\infty}^{\infty} d \Delta'\; {\cal \rho}(\Delta')\, \frac{1}{s+\gamma+i\Delta'} = + g^{2} \frac{1}{s+\gamma+\gamma_{n}}
\end{equation}
[/tex]
but I get a - in front of g
[tex]
\begin{subequations}
\begin{eqnarray}
\int_{-\infty}^{\infty} d \Delta'\; {\cal \rho}(\Delta')\, \frac{1}{s+\gamma+i\Delta'}
&=&
\frac{1}{\pi} \int_{-\infty}^{\infty} d \Delta'\; \frac{\gamma_{n}}{(\Delta'^{2}+\gamma_{n}^{2})}\, \frac{1}{(s+\gamma+i\Delta')}
\\
\nonumber
&=&
\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')}
\end{eqnarray}
\end{subequations}
[/tex]
in which case the poles are at $\pm i\gamma_{n}, +i(s+\gamma)$.
[tex]
\begin{subequations}
\begin{eqnarray}
\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}}
\\
\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}}
\\
\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')}
\\
\end{eqnarray}
\end{subequations}
[/tex]
It is simpler integrate in down region where we have only one pole [tex] $ -i\gamma_{n}$.[/tex]
While we have two poles in the region above the x:
[tex]$\Delta'= i\gamma_n$,[/tex]
and
[tex] $\Delta'= i(s+\gamma)$.[/tex]
[tex]
\begin{subequations}
\begin{eqnarray}
\oint_{\gamma_{R}(t)} f(z) \, dz &=& 2\pi i[\frac{1}{2i}\frac{-1}{s+\gamma+\gamma_{n}} ]
%\frac{-1}{2i}\frac{1}{s+\gamma-\gamma_{n}} + \frac{\gamma_{n}}{(-s-\gamma+\gamma_{n})(s+\gamma+\gamma_{n})}]
\\
%&=&2\pi i[-\frac{(1+i)\gamma_{n}}{(s+\gamma-\gamma_{n})(s+\gamma+\gamma_{n})}]
\end{eqnarray}
\end{subequations}
[/tex]
[tex]
\begin{subequations}
\begin{eqnarray}
\int_{-\infty}^{\infty} d \Delta'\; {\cal \rho}(\Delta')\, \frac{1}{s+\gamma+i\Delta'}
&=&
\frac{-\pi}{s+\gamma+\gamma_{n}}
\\
\frac{1}{\pi}\int_{-\infty}^{\infty} d \Delta'\; {\cal \rho}(\Delta')\, \frac{1}{s+\gamma+i\Delta'}
&=&
\frac{-1}{s+\gamma+\gamma_{n}}
\end{eqnarray}
\end{subequations}
[/tex]
I get:
[tex]
\begin{equation}
+ 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}}
\end{equation}
[/tex]