Can Gaussian integrals be done with half integrals?

[cragar]

Is it possible to do Gaussian integrals with half integrals.
we would define then nth derivative of $e^{-x^2}$
and then somehow use that. And this integral is over all space.
any input will be much appreciated.

 

[mathman]

Is it possible to do Gaussian integrals with half integrals.
we would define then nth derivative of $e^{-x^2}$
and then somehow use that. And this integral is over all space.
any input will be much appreciated.

Your question is vague. What do you mean by?

Gaussian integrals with half integrals

 

[cragar]

for example if we had to integrate $e^{ax}$ then nth derivative would be
$a^ne^{ax}$ so the half dervative would be
$a^{.5}e^{ax}$ and the half integral would be
$\frac{e^{ax}}{a^{.5}}$
I was just wondering if we could use this to help us evaluate a Gaussian integral.

 

[lurflurf]

That is called fractional calculus
Half integrals depend on arbitrary constants we might have for the half integral of e^(ax)
e^(ax)/sqrt(a)
or
sqrt(pi/a) e^(a x) erf(sqrt(a x))

I would not be surprising that this could be used, but I am not sure it would be easier or more interesting than other popular methods.

erf(x) function and gamma functions pop out all the time when taking half integrals and your integral is easily expressed in terms of them.

Here is some stuff about all the fun ways to find the integral.
http://en.wikipedia.org/wiki/Gaussian_integral
http://www.york.ac.uk/depts/maths/histstat/normal_history.pdf
http://www.math.uconn.edu/~kconrad/blurbs/analysis/gaussianintegral.pdf