From 8c5048dc65f35357088ee1e828197061bb540b41 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 26 Feb 2022 00:19:28 +0100
Subject: Explicit formula for the complex complexity.

---
 stokes.tex | 39 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 39 insertions(+)

diff --git a/stokes.tex b/stokes.tex
index 54b0b64..d4b0d38 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -1273,6 +1273,45 @@ related directly to the magnitude of the imaginary part of $z$, since
 $z^\dagger z=x^Tx+y^Ty=N+2y^Ty$. We therefore define $Y=\frac1N(z^\dagger
 z-N)$, the specific measure of the distance into the complex plane from the
 real sphere. The complexity can then be written
+\begin{equation}
+  \Sigma
+  =
+  \log(p-1)-\frac12\log\left(
+    \frac{1-r^{-2(p-1)}}{1-r^{-2}}
+  \right)
+  -\frac{(\operatorname{Re}\epsilon)^2}{R_+^2}-\frac{(\operatorname{Im}\epsilon)^2}{R_-^2}
+  +I_p(\epsilon/\epsilon_\mathrm{th})
+\end{equation}
+\begin{equation}
+  R_\pm^2=\frac{p-1}2\frac{(r^{p-2}\pm1)\left[
+      r^{2(p-1)}\pm(p-1)r^{p-2}(r^2-1)-1
+  \right]}{
+    1+r^{2(p-2)}\left[p(p-2)(r^2-1)-1\right]
+  }
+\end{equation}
+$I_p(u)=0$ if $|\epsilon|^2<|\epsilon_\mathrm{th}|^2$.
+\begin{eqnarray}
+  I_p(u)
+  &=
+  \left(\frac12+\frac1{r^{p-2}-1}\right)^{-1}(\operatorname{Re}u)^2
+  -
+  \left(\frac12-\frac1{r^{p-2}+1}\right)^{-1}(\operatorname{Im}u)^2\\
+  &\qquad-\log\left(
+    r^{p-2}\left|
+    u+\sqrt{u^2-1}
+    \right|^2
+  \right)+2\operatorname{Re}
+  \left(
+    u\sqrt{u^2-1}
+  \right)
+\end{eqnarray}
+where the branch of the square roots are chosen such that the real part of the
+root has the opposite sign as the real part of $u$, e.g., if
+$\operatorname{Re}u<0$ then $\operatorname{Re}\sqrt{u^2-1}>0$. If the real part
+is zero, then the sign is taken so that the imaginary part of the root has the
+opposite sign of the imaginary part of $u$.
+
+
 
 \subsection{Pure \textit{p}-spin: where are my neighbors?}
 
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