diff options
Diffstat (limited to 'source/know/concept/matsubara-sum')
| -rw-r--r-- | source/know/concept/matsubara-sum/index.md | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/source/know/concept/matsubara-sum/index.md b/source/know/concept/matsubara-sum/index.md index 0e04455..3347476 100644 --- a/source/know/concept/matsubara-sum/index.md +++ b/source/know/concept/matsubara-sum/index.md @@ -23,7 +23,7 @@ $$\begin{aligned} $$g(z)$$ is a *meromorphic* function on the complex frequency plane, i.e. it is [holomorphic](/know/concept/holomorphic-function/) except for a known set of simple poles, -and $$\tau \in [-\hbar \beta, \hbar \beta]$$ is a real parameter. +and $$\tau \in \:]\!-\!\hbar \beta, \hbar \beta[$$ is a real parameter. The Matsubara frequencies $$i \omega_n$$ are defined as follows for bosons (subscript $$B$$) or fermions (subscript $$F$$): @@ -77,9 +77,9 @@ $$\begin{aligned} h(z) \equiv \begin{cases} - n_{B,F}(z) & \mathrm{if}\; \tau \ge 0 + n_{B,F}(z) & \mathrm{if}\; 0 \le \tau < \hbar \beta \\ - -n_{B,F}(-z) & \mathrm{if}\; \tau \le 0 + -n_{B,F}(-z) & \mathrm{if}\; \!-\!\hbar \beta < \tau \le 0 \end{cases} \end{aligned}$$ @@ -107,7 +107,7 @@ $$\begin{aligned} &= \lim_{z \to i \omega_n}\!\bigg( \frac{z - i \omega_n}{e^{\hbar \beta z} + 1} \bigg) = \lim_{\eta \to 0}\!\bigg( \frac{i \omega_n + \eta - i \omega_n}{e^{i \hbar \beta \omega_n} e^{\hbar \beta \eta} + 1} \bigg) \\ - &= \lim_{\eta \to 0}\!\bigg( \frac{\eta}{e^{\hbar \beta \eta} + 1} \bigg) + &= \lim_{\eta \to 0}\!\bigg( \frac{\eta}{-e^{\hbar \beta \eta} + 1} \bigg) = \lim_{\eta \to 0}\!\bigg( \frac{\eta}{- 1 - \hbar \beta \eta + 1} \bigg) = - \frac{1}{\hbar \beta} \end{aligned}$$ |