From e150ff1954e842fe95d437a1cb8a9686670e963b Mon Sep 17 00:00:00 2001
From: Erik Strand <erik.strand@cba.mit.edu>
Date: Thu, 21 Feb 2019 02:46:40 -0500
Subject: [PATCH] Add 3.4 (f)

---
 _psets/2.md | 9 ++++++++-
 1 file changed, 8 insertions(+), 1 deletion(-)

diff --git a/_psets/2.md b/_psets/2.md
index 90ad9f8..c2ca630 100644
--- a/_psets/2.md
+++ b/_psets/2.md
@@ -416,9 +416,16 @@ $$
 So our result above for $$p(\tau)$$ follows because $$E = k T \log(\tau / \tau_0)$$, so $$dE / d\tau
 = k T / \tau$$.
 
-
+Now we must show that $$S(f) \propto 1/f$$.
 
 $$
 \begin{align*}
+S(f) &= \int_0^\infty p(\tau) \frac{2 \tau}{1 + (2 \pi f \tau)^2} \mathrm{d} \tau \\
+&= \int_0^\infty \frac{2 p(E) k T}{1 + (2 \pi f \tau)^2} \mathrm{d} \tau \\
+&= \frac{1}{2 \pi f} \int_0^\infty \frac{2 p(E) k T}{1 + \tau^2} \mathrm{d} \tau \\
+&= \frac{p(E) k T}{\pi f} \int_0^\infty \frac{1}{1 + \tau^2} \mathrm{d} \tau \\
+&= \frac{p(E) k T}{\pi f} \int_0^{\pi/2} \frac{sec^2{\theta}}{1 + tan^2(\theta)} \mathrm{d} \theta \\
+&= \frac{p(E) k T}{\pi f} \int_0^{\pi/2} \mathrm{d} \theta \\
+&= \frac{p(E) k T}{2 f}
 \end{align*}
 $$
-- 
GitLab