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From: Jay R. Yablon on 11 Jan 2010 11:08
In the process of exploring path integration I have come across an
integral of the form:

${-oo,+oo} F(x) (dx)^.5 (1)

Is there any body of literature about how one does such integrals?
Please note, this does not appear to be a fractional integral d^.5x
which is done via the Gamma function. Not does this appear to be a
d(x^.5) which can easily be converted to .5 x^-.5 dx and thus give us a
whole-integer power for dx.

This seems more akin to a differential equation (x' = first derivative,
e.g., dx/dt) which contains an x'^.5 rather than an x'^2.

Thanks,

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

From: eric gisse on 11 Jan 2010 12:48
Jay R. Yablon wrote:

> In the process of exploring path integration I have come across an
> integral of the form:
>
> ${-oo,+oo} F(x) (dx)^.5 (1)
>
> Is there any body of literature about how one does such integrals?

Given that you eliminated fractional derivatives and d(sqrt[x]), "not that I
have seen".

> Please note, this does not appear to be a fractional integral d^.5x
> which is done via the Gamma function. Not does this appear to be a
> d(x^.5) which can easily be converted to .5 x^-.5 dx and thus give us a
> whole-integer power for dx.
>
> This seems more akin to a differential equation (x' = first derivative,
> e.g., dx/dt) which contains an x'^.5 rather than an x'^2.

First thought: You can't.

Second thought: Its' a typo.

Third thought: Try a power series expansion of 'sqrt(dx)'. You are assured
of its' smallness.

I'm sticking with those three thoughts in that order.

>
> Thanks,
>
> Jay
> ____________________________
> Jay R. Yablon
> Email: jyablon(a)nycap.rr.com
> co-moderator: sci.physics.foundations
> Weblog: http://jayryablon.wordpress.com/
> Web Site: http://home.roadrunner.com/~jry/FermionMass.htm

From: Robert on 11 Jan 2010 14:57
fractional derivatives can be defined using integral transforms. an
analogy can be drawn between the way gamma functions allow fractional
factorials.

for integration, perhaps try to generalise cauchy's integration
formula?
From: David C. Ullrich on 12 Jan 2010 06:07
On Mon, 11 Jan 2010 11:08:18 -0500, "Jay R. Yablon"
<jyablon(a)nycap.rr.com> wrote:

>In the process of exploring path integration I have come across an
>integral of the form:
>
>${-oo,+oo} F(x) (dx)^.5 (1)

Where did you run across this? The notation makes very little
sense.

>Is there any body of literature about how one does such integrals?
>Please note, this does not appear to be a fractional integral d^.5x
>which is done via the Gamma function. Not does this appear to be a
>d(x^.5) which can easily be converted to .5 x^-.5 dx and thus give us a
>whole-integer power for dx.
>
>This seems more akin to a differential equation (x' = first derivative,
>e.g., dx/dt) which contains an x'^.5 rather than an x'^2.
>
>Thanks,
>
>Jay
>____________________________
>Jay R. Yablon
>Email: jyablon(a)nycap.rr.com
>co-moderator: sci.physics.foundations
>Weblog: http://jayryablon.wordpress.com/
>Web Site: http://home.roadrunner.com/~jry/FermionMass.htm

From: Jay R. Yablon on 12 Jan 2010 09:59
"David C. Ullrich" <ullrich(a)math.okstate.edu> wrote in message
news:ktlok5dlse03abvibjkli3f086rm38si26(a)4ax.com...
> On Mon, 11 Jan 2010 11:08:18 -0500, "Jay R. Yablon"
> <jyablon(a)nycap.rr.com> wrote:
>
>>In the process of exploring path integration I have come across an
>>integral of the form:
>>
>>${-oo,+oo} F(x) (dx)^.5 (1)
>
> Where did you run across this? The notation makes very little
> sense.

Replicating what I posted elsewhere in the thread, I'll show you where.
Look at Zee's derivation posted at
http://jayryablon.files.wordpress.com/2009/12/sakurai-and-zee.pdf.

Look at $Dq just before equation (4). Following some manipulation and
in the limiting case, one can arrive for a given fixed time slice, at
the expression:

(-2pi i)^.5 $ (m/dt)^.5 dq = (-2pi i)^.5 $ p^.5 dq^.5 (1)

if one uses the momentum:

p = m dq/dt (2)

to absorb the m/dt term. One is left in this event, at any fixed time
slice, to deal with the weird integral:

$ p^.5 dq^.5

which has dimensions of the square root of action.

Jay.





>
>>Is there any body of literature about how one does such integrals?
>>Please note, this does not appear to be a fractional integral d^.5x
>>which is done via the Gamma function. Not does this appear to be a
>>d(x^.5) which can easily be converted to .5 x^-.5 dx and thus give us
>>a
>>whole-integer power for dx.
>>
>>This seems more akin to a differential equation (x' = first
>>derivative,
>>e.g., dx/dt) which contains an x'^.5 rather than an x'^2.
>>
>>Thanks,
>>
>>Jay
>>____________________________
>>Jay R. Yablon
>>Email: jyablon(a)nycap.rr.com
>>co-moderator: sci.physics.foundations
>>Weblog: http://jayryablon.wordpress.com/
>>Web Site: http://home.roadrunner.com/~jry/FermionMass.htm
>

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