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

 

[CellCoree]

I can provide a response to your question. Gaussian integrals, also known as normal integrals, are a class of integrals that involve the Gaussian function e^{-x^2}. These integrals can be challenging to solve, and there are various techniques and methods used to evaluate them.

In response to your question, it is possible to solve Gaussian integrals using half integrals. Half integrals are a type of fractional calculus, which is a branch of mathematics that deals with integrals and derivatives of non-integer orders. In fractional calculus, the half integral is defined as the n-th derivative of a function at a fractional order of 1/2.

Using this definition, we can apply it to the Gaussian function e^{-x^2} and solve the integral by taking the half integral of the function. This approach has been used in various studies and has been shown to be effective in solving Gaussian integrals.

However, it is essential to note that this method may not always be the most efficient or practical way to solve Gaussian integrals. Other techniques, such as contour integration, may be more suitable in some cases. It is also essential to have a good understanding of fractional calculus and its applications before attempting to use half integrals in solving Gaussian integrals.

In conclusion, yes, it is possible to solve Gaussian integrals using half integrals, but it may not always be the most efficient or practical approach. I hope this response helps clarify your question.