- #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]