Improve knowledge baseHEADmaster

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 @@ $$\begin{aligned}
     &= \lim_{z \to i \omega_n}\!\bigg( \frac{z - i \omega_n}{e^{\hbar \beta z} + 1} \bigg)
     = \lim_{\eta \to 0}\!\bigg( \frac{i \omega_n + \eta - i \omega_n}{e^{i \hbar \beta \omega_n} e^{\hbar \beta \eta} + 1} \bigg)
     \\
-    &= \lim_{\eta \to 0}\!\bigg( \frac{\eta}{e^{\hbar \beta \eta} + 1} \bigg)
+    &= \lim_{\eta \to 0}\!\bigg( \frac{\eta}{-e^{\hbar \beta \eta} + 1} \bigg)
     = \lim_{\eta \to 0}\!\bigg( \frac{\eta}{- 1 - \hbar \beta \eta + 1} \bigg)
     = - \frac{1}{\hbar \beta}
 \end{aligned}$$