On 15 May, 22:51, "Jay R. Yablon" wrote:
In the file linked below, I have written out a particular calculation of
a Gaussian integral, and would like to know if (6) in this file is is
correct. (If link does not work, right click to download and then
open.)

http://jayryablon.files.wordpress.co...alculation-of-...

This is not rocket science nor is it particularly earthshaking. But, I
am having some conceptual trouble thinking about this integral (6) in
which the plane wave coefficient B=0, and would appreciate if someone
can help me straighten out my thinking about this.

It would seem OK up to a point but it is not the way I would tackle
it.

Now e^x = 1+x+x^2/2! + x^3/3! + etc

Hence e^f(x) = 1+f(x)+f(x)^2/2! + f(x)^3/3! + etc

We know that Int (x^n)dx = x^(n+1)/(n+1) Hence if we have a power
series integration is trivial.

Now f(x) = -x^2 + V(x) we have a power series for f(x) and getting a
term by term evaluation is a matter of a term by term multiplication.
This gives your your equations for what you have labelled as C(n).

You can also constuct a series using Taylor's theorem, that is to say
you differentiate e^(f(x)) repeatedly.

You get f'(x)e^(f(x)) differentiating this again (f''(x) +
(f'(x)^2)e^(f(x))

Can you see one thing? The process of Taylor differentiation is
isomorphic with binomial multiplication (we would be surprised if it
wasn't).

You want clarification. I think the isomorphism of binomial
multiplication and Taylor's theorem does clarify things consiserably.

The physical meaning. There isn't a way to do things that isn't a
little bit messy.

- Ian Parker