- #1
Destroxia
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Homework Statement
Consider the Gaussian Distribution
## p(x) = Ae^{-\lambda(x-a)^{2}} ##,
where ## A ##, ##a##, and ##\lambda## are constants. (Look up any integrals you need.)
(a) Determine ##A##
(I only need help with this (a))
Homework Equations
##\int_{-\infty}^{\infty} p(x)dx = 1##
##\langle x \rangle = \int_{-\infty}^{\infty} xp(x)dx##
##\sigma^{2} \equiv \langle (\Delta x)^{2}) \rangle = \langle x^{2} \rangle - \langle x \rangle^{2} ##
## \int e^{-ax^{2}} = \sqrt{\frac {\pi} {a}}## erf##(x)##
## \int e^{ax^{2}} = -i \sqrt{\frac {\pi} {a}}## erf##(x)## (Where does the -i and sqrt come from?)
The Attempt at a Solution
## p(x) = Ae^{-\lambda(x-a)^{2}} ##
## 1 = A \int_{-\infty}^{\infty} e^{-\lambda(x-a)^{2}}dx##
## u = x-a, du=dx, u: -\infty ## to ## \infty ##
## 1 = A \int_{-\infty}^{\infty} e^{-\lambda u^{2}}dx ##
## 1 = A[\sqrt{\frac {\pi} {\lambda}}## erf##(x) ## ##]_{-\infty}^{\infty}##
(stuck here... how to evaluate this...?)
Okay, so... I have a few questions here...
1) I know that ## \int e^{ax^{2}}## does not evaluate in terms of elementary functions, but an error function... but why does the ##-i## and ##\sqrt{\frac {\pi} {a}}## come into play, being multiplied by the erf(x)?
2) What exactly is a gaussian distribution, and what is the significance of it?
3) How do I evaluate the erf(x) function from ##\infty## to ##\infty##. The book's solution shows it disappears away after they evaluate from ##-\infty## to ##\infty##... ?
4) The answer for A in the book ends up being ## A = \sqrt{\frac {\lambda} {\pi}}##, which I see how they got that, but where did the error function go to in the evaluation? Do we just consider it not there? Shouldn't it be ## A = \sqrt{\frac {\lambda} {\pi}}## erf##(x)##?