| Topic: |
Science > Physics |
| User: |
"Iwonder" |
| Date: |
12 Jan 2004 10:53:52 AM |
| Object: |
Technical math question |
Where can I find a readable account (not some differential form formulation
on complex varieties full of theorems without use) of how to tackle 2D
vector calculus, especially so as to integrate in the complex plane, namely,
integrate on the plane (not on a contour as one usually find it done when it
comes to integration in C) for a function f(z,z*) stuff of the like
(@/@z + @/@z*)f(z,z*)dz dz*
@^2/@z@z* f(z,z*) dz dz*
(2D, or complex, divergence and laplacian)
I am turning everything to real variables,
@/@z -> (1/2)(@/@x -i @/@y), dz dz* -> 2dx dy
but I feel it lacks the power of complex tools.
Thanks for any hint
--
(with @ partial derivative, z* complex conjugate of z, and so on)
.
|
|
| User: "Snubis" |
|
| Title: Re: Technical math question |
13 Jan 2004 04:00:01 PM |
|
|
"Iwonder" <yssual@yahoo.com> wrote in message news:<4002d121$0$7146$626a54ce@news.free.fr>...
Where can I find a readable account (not some differential form formulation
on complex varieties full of theorems without use) of how to tackle 2D
vector calculus, especially so as to integrate in the complex plane, namely,
integrate on the plane (not on a contour as one usually find it done when it
comes to integration in C) for a function f(z,z*) stuff of the like
(@/@z + @/@z*)f(z,z*)dz dz*
@^2/@z@z* f(z,z*) dz dz*
(2D, or complex, divergence and laplacian)
I am turning everything to real variables,
@/@z -> (1/2)(@/@x -i @/@y), dz dz* -> 2dx dy
but I feel it lacks the power of complex tools.
Thanks for any hint
Here:
http://www.lemonparty.org
.
|
|
|
|
| User: "Igor" |
|
| Title: Re: Technical math question |
13 Jan 2004 12:58:33 PM |
|
|
"Iwonder" <yssual@yahoo.com> wrote in message news:<4002d121$0$7146$626a54ce@news.free.fr>...
Where can I find a readable account (not some differential form formulation
on complex varieties full of theorems without use) of how to tackle 2D
vector calculus, especially so as to integrate in the complex plane, namely,
integrate on the plane (not on a contour as one usually find it done when it
comes to integration in C) for a function f(z,z*) stuff of the like
(@/@z + @/@z*)f(z,z*)dz dz*
@^2/@z@z* f(z,z*) dz dz*
(2D, or complex, divergence and laplacian)
I am turning everything to real variables,
@/@z -> (1/2)(@/@x -i @/@y), dz dz* -> 2dx dy
but I feel it lacks the power of complex tools.
Thanks for any hint
Treat it as you would any other double integral without limits,
treating z and z* as independent integration variables. That's all
there is to it.
.
|
|
|
| User: "Edward Green" |
|
| Title: Re: Technical math question |
13 Jan 2004 07:59:25 PM |
|
|
(Igor) wrote in message news:<d434b6c6.0401131058.616d4010@posting.google.com>...
"Iwonder" <yssual@yahoo.com> wrote in message news:<4002d121$0$7146$626a54ce@news.free.fr>...
Where can I find a readable account (not some differential form formulation
on complex varieties full of theorems without use) of how to tackle 2D
vector calculus, especially so as to integrate in the complex plane, namely,
integrate on the plane (not on a contour as one usually find it done when it
comes to integration in C) for a function f(z,z*) stuff of the like
(@/@z + @/@z*)f(z,z*)dz dz*
@^2/@z@z* f(z,z*) dz dz*
(2D, or complex, divergence and laplacian)
Igor says he knows the answer, but I am uncomfortable that I know the
question. You seem to mean to treat functions of two complex
variables, whereas "the complex plane" which you cite is the space of
a _single_ complex variable z = x + iy. The preponderance of evidence
suggests that you mean the former, which is something like the
"bi-complex plane" ...
I am turning everything to real variables,
@/@z -> (1/2)(@/@x -i @/@y), dz dz* -> 2dx dy
... whereas the last arrow above implies that you have in mind
functions f(z,w) of two complex variables where the variables have the
enjoy special relationship w = conjugate(z).
I may not be the only one who is confused here! What are you trying
to do? I have a feeling you are trying to simultaneously use and not
use "the power of complex methods" applied to a single complex
variable z = x + iy.
If you want to treat f(z) as two real functions of two real variables,
R^2 -> R^2, and integrate the divergence, e.g., then you've tossed the
structure of complex numbers at the onset; if you want to treat an
analytic function w = f(z) from Z -> Z, then you're back in the land
of contour integrals and such. There are some mappings from one case
to the other, but it seems like you are trying to do both approachs at
once in a MixMaster.
<...>
.
|
|
|
| User: "Iwonder" |
|
| Title: Re: Technical math question |
20 Jan 2004 03:46:22 AM |
|
|
Igor says he knows the answer, but I am uncomfortable that I know the
question. You seem to mean to treat functions of two complex
variables, whereas "the complex plane" which you cite is the space of
a _single_ complex variable z = x + iy. The preponderance of evidence
suggests that you mean the former, which is something like the
"bi-complex plane" ...
Yes, I mean using functions of z and z* granted as independent variables
(that is @z/@dz* = 0). So that a generic function is f(z,z*). It is
typically a function of z, z^* and |z|^2.
I am not aware this is connected in any way to so-called "bi-complex" plane
!?
I am turning everything to real variables,
@/@z -> (1/2)(@/@x -i @/@y), dz dz* -> 2dx dy
... whereas the last arrow above implies that you have in mind
functions f(z,w) of two complex variables where the variables have the
enjoy special relationship w = conjugate(z).
It is an odd way to present things!
I may not be the only one who is confused here! What are you trying
to do? I have a feeling you are trying to simultaneously use and not
use "the power of complex methods" applied to a single complex
variable z = x + iy.
I am trying to derive some Fokker-Planck equation in complex space. Those
are functions of z, z^* and so on, independent. You usually want to make
some integration on the plane of such a function, not just some
contour-integrations, i.e., on a line. You want to integrate on the full
plane. I find that this topic is very poorly addressed both in the little
litterature I have and on internet. I do not understand why this rich and
interesting, and most of all, powerful formalism is so much neglected. I
could have seen it called "Grassman formalism", and so on. But then, it all
amount to say that when you have a complex x+iy with x and y independent,
then you can use x+iy as a single unit and x-iy as the other, independent,
variable. This is strenghtened by such things like analytic functions have
d/dz* = 0 and so on. Should be familiar to all. Why is it not?
Thanks for your interest.
.
|
|
|
|
|
|
|
Related Articles |
|
|
