From 2a57d9283f23e211ce8e00e32869ba54b0296a08 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 7 Jun 2022 18:23:19 +0200
Subject: Readded expression for D(mu).

---
 frsb_kac_new.tex | 6 +++++-
 1 file changed, 5 insertions(+), 1 deletion(-)

(limited to 'frsb_kac_new.tex')

diff --git a/frsb_kac_new.tex b/frsb_kac_new.tex
index 552ee9b..e1c5768 100644
--- a/frsb_kac_new.tex
+++ b/frsb_kac_new.tex
@@ -295,7 +295,11 @@ Introducing the parameters:
   \begin{aligned}
     \mathcal D(\mu)
     &=\frac1N\overline{\log|\det(\partial\partial H(s_a)+\mu I)|}
-    =\int d\lambda\,\rho(\lambda+\mu)\log|\lambda| 
+    =\int d\lambda\,\rho(\lambda+\mu)\log|\lambda| \\
+    &=\operatorname{Re}\left\{
+    \frac12\left(1+\frac\mu{2f''(1)}\left(\mu\pm\sqrt{\mu^2-4f''(1)}\right)\right)
+    -\log\left(\frac1{2f''(1)}\left(\mu\pm\sqrt{\mu^2-4f''(1)}\right)\right)
+  \right\}
   \end{aligned}
 \end{equation}
 
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