Re: riemann surface

On Oct 20, 9:28 pm, tommy1729 <tommy1...@xxxxxxxxx> wrote:
matt [(10^5)*e] wrote:
On Oct 20, 10:14 am, mike3 <mike4...@xxxxxxxxx>
wrote:
Hi.

I've noticed it is possible to integrate 1/z on the
complex plane.
However, I'm curious as to how this jives with the
multivalued nature
of the complex logarithm relationship (not
function, as functions are
single valued)? An integral is defined as a type of
infinite
summation, which by definition has only one sum (as
addition is a
function). What happens when you try out the
simplest possible
generalization of the real logarithm -- integrate
from z = 1 along a
straight line to some given point? The equations
get a little messy
when I try it out. Is this why the "principal"
logarithm is defined
the way it is, with a branch cut down the negative
real axis (since
the singularity at z = 0 "casts a shadow" along the
negative real axis
in the "rays" (straight lines) emanating out of z =
1)?

I hope I've remembered this right. I'm sure I'll be
corrected if I
haven't!

A visualisation that I once came across, that I found
very useful, is
to picture the "graph" of log(z) as a spiral surface
winding upwards
like a spiral staircase, or like the flanges on the
thread of a screw.
The axis of the spiral is perpendicular to the
complex plane, and
passes through the origin. The spiral extends
indefinitely above and
below the complex plane. Given a complex number, z,
draw a line
through z perpendicular to the complex plane, and the
points of
interesection of this line with the spiral surface
represent the
infinitely many complex values of log(z) (obviously
in three
dimensions this doesn't literally work -- it's just a
visualisation
aid).

this visualization is the riemann surface.

Yes indeed. Thanks for reminding me what the thing is called!

.

Relevant Pages

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