I have a Gaussian defined as follows:
$W(\theta) = j * exp(-0.5 * \theta^2 / \sigma^2)$.
I want to set $j$ such that $\frac{1}{360}\int_{-180}^{180} W(\theta)d\theta = 1$.
I'm using two values for $\sigma$: 36$^\circ$ and 72$^\circ$.
When I solve for $j$ setting $\sigma=36^\circ$, I find that
$j = 3.99 = 360/(\sigma\sqrt{2\pi})$.
However, when I solve for $j$ setting $\sigma=72^\circ$, I get
$j = 2.02,$
which is slightly greater than $1.99 = 360 / (\sigma\sqrt{2\pi})$.
Is there a single expression, like $360 / (\sigma\sqrt{2\pi})$, that captures $j$'s value across all $\sigma$?
EDIT:
I've made some progress on this, but I'm still not satisfied. I've got
$j = 360 / (\sigma\sqrt{2\pi}erf(127.279/\sigma))$,
where $erf()$ is the error function. I need a symbolic representation for 127.279.
I've got it!
$j = 360 / (\sigma \sqrt{2 \pi} erf(\frac{180}{\sigma\sqrt{2}}))$.
(Not quite a "symbolic" representation, but I've gotten rid of that pesky -- read, harbinger of imprecision -- decimal point.)