summaryrefslogtreecommitdiffstats
diff options
context:
space:
mode:
authorluccul <luccul@gmail.com>2010-07-06 06:02:26 +0000
committerbnewbold <bnewbold@adelie.robocracy.org>2010-07-06 06:02:26 +0000
commit6597f0b4cc3e8bc179b78e467eb8310ac578e114 (patch)
treec4acd03ee111ca7af551af4191f49e74357aa7bd
parentf73eaf79e845f0548b169a46783c99a106df3fc9 (diff)
downloadafterklein-wiki-6597f0b4cc3e8bc179b78e467eb8310ac578e114.tar.gz
afterklein-wiki-6597f0b4cc3e8bc179b78e467eb8310ac578e114.zip
more formatting
-rw-r--r--ClassJuly5.page3
1 files changed, 2 insertions, 1 deletions
diff --git a/ClassJuly5.page b/ClassJuly5.page
index 19b3d47..fa72182 100644
--- a/ClassJuly5.page
+++ b/ClassJuly5.page
@@ -35,7 +35,7 @@ Rearranging, we get:
$$ \frac{da}{dt}/a = \frac{d^2b}{dx^2}/b $$
-The left hand side only depends on $t$ and the right hand side depends only on $x$. Therefore, neither side depends on either $x$ or $t$. We conclude that both sides are equal to a constant, which for convenience we'll write as $-\lambda^2$.
+The left hand side only depends on $t$ and the right hand side depends only on $x$. Therefore, neither side depends on either $x$ or $t$. We conclude that both sides are equal to a constant, which for convenience we'll write as $-\lambda^2$.
First we solve for $a$. It satisfies the equation
@@ -62,6 +62,7 @@ We would like to assert that any solution takes this form. One way to prove thi
$$ f(x) = \sum_{n = 1}^{\infty} c_n \sin(\frac{\pi n x}{L}) $$
One could then allow the coefficients $c_n$ to vary with $t$ and apply the same method of solution that we used in the case of periodic boundary conditions.
+
In fact, every function of the kind described above does have a Fourier sine expansion. The link above contains a hint of how to do it. First you extend the function $f(x)$ to a certain odd, periodic function $\tilde{f}(x)$ defined on the interval $[-L,L]$. Then you can use convergence of the usual Fourier series for $\tilde{f}(x)$.
## Convergence for not-so-nice Fourier series.