From 746e44533d73d61b2a9b63b11a2782fd1016bdc1 Mon Sep 17 00:00:00 2001 From: luccul Date: Sun, 11 Jul 2010 00:47:22 +0000 Subject: checking a formula --- ClassJuly5.page | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/ClassJuly5.page b/ClassJuly5.page index 8a1ac2c..9af2b85 100644 --- a/ClassJuly5.page +++ b/ClassJuly5.page @@ -51,7 +51,7 @@ $$ a(t) = e^{-\lambda^2 t} $$ Next we solve for $b$. It satisfies the equation -$$ \frac{d^2b}{dx^2} = -\lambda b $$ +$$ \frac{d^2b}{dx^2} = -\lambda^2 b $$ However we have to be a bit more careful in picking our solutions because $b$ is supposed to satisfy the boundary conditions $$ b(0) = b(L) = 0$$ To satisfy $b(0) = 0$, we must take $b$ to be (a constant multiple of) a sine function: @@ -73,3 +73,6 @@ In fact, every function of the kind described above does have a Fourier sine exp How do we know that the Fourier series of a square wave or sawtooth function converges? +The answer to this question depends greatly on the type of convergence desired. Aside from the convergence we already proved, the next easiest type of convergence is $L^2$ or root-mean-square convergence. The formal statement is that + +$$ \lim_{N \to \infty} \sqrt{\int_0^{2\pi} | \sum_{n = - N}^N c_n e^{in\theta} - f(\theta) |^2} = 0 $$ \ No newline at end of file -- cgit v1.2.3