I have been trying to find a generalized expression for the multiple
derivatives of a Gaussian. Gaussian functions of course have many
application in physics and math, and high order derivatives of this
Gaussian typically are involved in generating Green functions.

I posted this "puzzle" at the link below along with some hints.

http://jayryablon.files.wordpress.co.../02/puzzle.pdf

It seems to me that there should be a way to do this, and that there
should be a general technique for this sort of thing, besides guesswork
and pattern matching, by using the series expansion.

Can anyone solve this, obtaining a generalized expression for any order
of derivative?

Thanks,

Jay
____________________________
Jay R. Yablon
Email:
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm

Jay R. Yablon wrote:

[...]

Can anyone solve this, obtaining a generalized expression for any order
of derivative?

Yeah, exactly as you've done. Slap the term with an arbitrary derivative,
and carry through the math. The cyclical nature of exp() promises a return
of the function, which you can already see in the x^2j component.

Substitute i = 2j - d, pull the x^d terms out of the summation if you can,
and hope there's a double factorial identity that will help.


Thanks,

Jay
____________________________
Jay R. Yablon
Email:
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm

On Feb 7, 10:41 pm, "Jay R. Yablon" wrote:
I have been trying to find a generalized expression for the multiple
derivatives of a Gaussian. Gaussian functions of course have many
application in physics and math, and high order derivatives of this
Gaussian typically are involved in generating Green functions.

I posted this "puzzle" at the link below along with some hints.

http://jayryablon.files.wordpress.co.../02/puzzle.pdf

It seems to me that there should be a way to do this, and that there
should be a general technique for this sort of thing, besides guesswork
and pattern matching, by using the series expansion.

Can anyone solve this, obtaining a generalized expression for any order
of derivative?

Thanks,

Jay
____________________________
Jay R. Yablon
Email:
co-moderator: sci.physics.foundations
Weblog:http://jayryablon.wordpress.com/
Web Site:http://home.roadrunner.com/~jry/FermionMass.htm



Yoour puzzle has no solution. Let's make the problem a little simpler,
assume A/2=1, so, you are being asked to find the n-th derivative of
exp(x^2)

df/dx=2x*exp(x^2)

Let's assume that:

(d^n)f/df^n=P(x)*exp(x^2) where P(x) is a plynimial in x

Then, the derivative of order n+1 is:

d^(n+1)f/df^(n+1)=(2xP(x)+P'(x))*exp(x^2)

There is no real correlation between the n-th and the n-th+1
derivatives.

Jay R. Yablon schrieb:
I have been trying to find a generalized expression for the multiple
derivatives of a Gaussian. Gaussian functions of course have many
application in physics and math, and high order derivatives of this
Gaussian typically are involved in generating Green functions.

I posted this "puzzle" at the link below along with some hints.

http://jayryablon.files.wordpress.co.../02/puzzle.pdf

It seems to me that there should be a way to do this, and that there
should be a general technique for this sort of thing, besides guesswork
and pattern matching, by using the series expansion.

Can anyone solve this, obtaining a generalized expression for any order
of derivative?

Thanks,

Jay
____________________________
Jay R. Yablon
Email:
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm


http://en.wikipedia.org/wiki/Hermite_polynomials

Ciao

Karl

"karl" wrote in message
...
Jay R. Yablon schrieb:
I have been trying to find a generalized expression for the multiple
derivatives of a Gaussian. Gaussian functions of course have many
application in physics and math, and high order derivatives of this
Gaussian typically are involved in generating Green functions.

I posted this "puzzle" at the link below along with some hints.

http://jayryablon.files.wordpress.co.../02/puzzle.pdf

It seems to me that there should be a way to do this, and that there
should be a general technique for this sort of thing, besides
guesswork
and pattern matching, by using the series expansion.

Can anyone solve this, obtaining a generalized expression for any
order
of derivative?

Thanks,

Jay
____________________________
Jay R. Yablon
Email:
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm


http://en.wikipedia.org/wiki/Hermite_polynomials

Ciao

Karl

Thank you Karl, that was most helpful. Jay

On Mon, 08 Feb 2010 09:14:42 +0100, karl
wrote:



http://en.wikipedia.org/wiki/Hermite_polynomials

Ciao

Karl


[Hammond]
I'll be a sonofagun!

So THATS where the famous Hermite Polynomials come from!

Leave it to Jay to discover a new derivation?

Thanks Karl for keeping your eyes open!

Hi Jay et al.

On Feb 8, 11:29 am, "Jay R. Yablon" wrote:
Thanks to Hans Devries and Karl, I have the answer. It is the Gaussian
Hermite series shown in the file below. Jay

http://jayryablon.files.wordpress.co...2/puzzle-2.pdf

Refering the final equation.

1) We should expect exp(1/2 A x^2) to be dimensionless
and thus a scalar that can be divided out on each side,
that sends us to a general field.

2) A thought is to convert the factorials to Gamma's to
make a continuous function. The hard Sigma really cannot
exist in a single particle system, because there is no way
to define "x".
"x" needs relativity.

3) After (1) a (d/dx)^n operator floats, that may become
more physical using (d'dx^u)^n to enable spacetime.
(u=1,2,3,4).

4) At this point the Sigma operator obtains control due
to the relativity of a 2 particle system.

5) The Hermites are all field, what you want is a couple
inter-action, that way you can get spin and system energy.

I (we) dabble a bit in couples,
http://physics.trak4.com/GR_Charge_Couple.pdf

6) We should expect two interacting particle fields to
output radiation and have gravitational and electrical
interaction.

(That #(6) sounds darn sexy on 2nd read :-).

That was fun.
Regards
Ken S. Tucker

On Feb 9, 12:13 am, "Ken S. Tucker" wrote:
Hi Jay et al.

On Feb 8, 11:29 am, "Jay R. Yablon" wrote:

Thanks to Hans Devries and Karl, I have the answer. It is the Gaussian
Hermite series shown in the file below. Jay

http://jayryablon.files.wordpress.co...2/puzzle-2.pdf

Refering the final equation.

1) We should expect exp(1/2 A x^2) to be dimensionless
and thus a scalar that can be divided out on each side,
that sends us to a general field.

2) A thought is to convert the factorials to Gamma's to
make a continuous function. The hard Sigma really cannot
exist in a single particle system, because there is no way
to define "x".
"x" needs relativity.

3) After (1) a (d/dx)^n operator floats, that may become
more physical using (d'dx^u)^n to enable spacetime.
(u=1,2,3,4).

4) At this point the Sigma operator obtains control due
to the relativity of a 2 particle system.

5) The Hermites are all field, what you want is a couple
inter-action, that way you can get spin and system energy.

I (we) dabble a bit in couples,http://physics.trak4.com/GR_Charge_Couple.pdf

6) We should expect two interacting particle fields to
output radiation and have gravitational and electrical
interaction.

(That #(6) sounds darn sexy on 2nd read :-).

That was fun.
Regards
Ken S. Tucker

On Feb 9, 5:47 pm, eric gisse wrote:

Ken S. Tucker wrote:

[...]

Thanks for the timely demonstration ...

If there was a Nobel Prize for stupidity, Gisse would be front runner.

I think Jay has a chance at ascendancy, two is the hardest number,
it requires a leap to couple, and the relativity involved in that,
furthermore think Jay may create an infinite set of solutions to
the coupled EFE's, but gets fixed to a quantized relativity.
Ken

Ken S. Tucker wrote:
[...]

Thanks for the timely and *repeated* demonstration of the Dunning-Kruger
effect, Ken.