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Gary

Posted: Thu Dec 18, 2003 7:57 am

Guest

Could someone give the integral formula for the natural log of a complex number ? In another group , it was mentioned to be the integral from 1 to z of 1 / z , z = complex number .
Is that the correct and complete definition ?
What is the best way to relate that to :

ln z = ln (modulus) + i * ( argument +- 2npi ) , n=integer

Thanks much in advance.
This seems to be the best math group for such a query.

Gary

phil

Posted: Thu Dec 18, 2003 11:40 am

Guest

On 18 Dec 2003, Gary wrote:

Quote:

Well here's a start:

z = exp [ln z] = exp [ln (modulus)] + i * ln (argument)]

Gary

Posted: Thu Dec 18, 2003 7:28 pm

Guest

On 18 Dec 2003, phil wrote:

Quote:

On 18 Dec 2003, Gary wrote:
Could someone give the integral formula for the natural log of a complex number ? In another group , it was mentioned to be the integral from 1 to z of 1 / z , z = complex number .
Is that the correct and complete definition ?
What is the best way to relate that to :

ln z = ln (modulus) + i * ( argument +- 2npi ) , n=integer

Thanks much in advance.
This seems to be the best math group for such a query.

Gary

Well here's a start:

z = exp [ln z] = exp [ln (modulus)] + i * ln (argument)]

And what would be the rest ? Thanks .

Gary

Eur Ing Panagiotis Stefan

Posted: Fri Dec 19, 2003 3:39 am

Guest

On 18 Dec 2003, Gary wrote:

Quote:

ln z = ln [{modz}e^i(theta)]=ln[mod z] +i(theta)
for z=e^[theta/90] ,then ln z = theta/90 =[2theta rads/pi rads]
ref:http://www.stefanides.gr/why_logarithm.htm
http://www.stefanides.gr/nautilus.htm
http://www.stefanides.gr/logarithm.htm

David C. Ullrich

Posted: Fri Dec 19, 2003 8:07 am

Guest

On Thu, 18 Dec 2003 17:57:14 +0000 (UTC), gflom@prodigy.net (Gary)
wrote:

Quote:

Yes, except that you're using the letter "z" for two different things.

If a <> 0 then there are many log(a)'s. The possible values of log(a)
are the same as the possible integrals of 1/z over a path from 1 to a
(where the path avoids the origin, of course). Different paths from
1 to a give different log(a)'s.

Quote:

What is the best way to relate that to :

ln z = ln (modulus) + i * ( argument +- 2npi ) , n=integer

GIven z <> 0 and an integer n, define a path from 1 to z as
follows: follow the straight line from 1 to |z|, then run along
the circle of radius |z| from |z| to z, and finally make n more
loops around that circle. If you integrate dw/w along that
path you get the value of log(z) you mention here.

(Then the slightly trickier part: show that any path from 1
to z is homotopic to one of the paths described in the
previous paragraph, so that those log(z)'s are the only
values you can get for the integral.)

Quote:

Thanks much in advance.
This seems to be the best math group for such a query.

Gary

************************

David C. Ullrich

G. A. Edgar

Posted: Fri Dec 19, 2003 8:20 am

Guest

In article <200312181747.hBIHlOx07846@proapp.mathforum.org>, Gary
<gflom@prodigy.net> wrote:

Quote:

Let s be a complex number. then log(s) is the
line integral of dz/z along a curve from 1 to s.
Of course the curve may not pass through 0.
Depending on what curve is used, the answer may differ
by an integer multiple of 2 pi i.

Quote:

What is the best way to relate that to :

ln z = ln (modulus) + i * ( argument +- 2npi ) , n=integer

Write dz/z in polar coordinates... if z=r*exp(i*theta), then
dz/z = dr/r + i*d theta.

Quote:

Thanks much in advance.
This seems to be the best math group for such a query.

Gary

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

Gary

Posted: Fri Dec 19, 2003 5:07 pm

Guest

On 19 Dec 2003, David C. Ullrich wrote:

Quote:

On Thu, 18 Dec 2003 17:57:14 +0000 (UTC), gflom@prodigy.net (Gary)
wrote:

Could someone give the integral formula for the natural log of a complex number ? In another group , it was mentioned to be the integral from 1 to z of 1 / z , z = complex number .
Is that the correct and complete definition ?

Yes, except that you're using the letter "z" for two different things.

If a <> 0 then there are many log(a)'s. The possible values of log(a)
are the same as the possible integrals of 1/z over a path from 1 to a
(where the path avoids the origin, of course). Different paths from
1 to a give different log(a)'s.

What is the best way to relate that to :

ln z = ln (modulus) + i * ( argument +- 2npi ) , n=integer

GIven z <> 0 and an integer n, define a path from 1 to z as
follows: follow the straight line from 1 to |z|, then run along
the circle of radius |z| from |z| to z, and finally make n more
loops around that circle. If you integrate dw/w along that
path you get the value of log(z) you mention here.

(Then the slightly trickier part: show that any path from 1
to z is homotopic to one of the paths described in the
previous paragraph, so that those log(z)'s are the only
values you can get for the integral.)

Thanks much in advance.
This seems to be the best math group for such a query.

Gary

************************

David C. Ullrich

Thanks to everyone for their well thought out
responses. Have been very helpful and informative.

Gary

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