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Some slog stuff
#14
(05/14/2015, 05:58 PM)tommy1729 Wrote:
(05/14/2015, 03:13 PM)sheldonison Wrote:
(05/14/2015, 02:28 PM)tommy1729 Wrote: ... Unless the functional equations no longer hold there.
Consider that slog(0)=-1, slog(1)=0
given the 2pi i periodicity, then
slog(2pi i)=-1; slog(1+2pi i)=0
slog(2npi i)=-1; slog(1+2npi i)=0 for any value of n

sexp is the inverse of slog. Therefore, somewhere on the sexp(z) Riemann surface, as you circle around the singularity at -2:
sexp(-1)=0; sexp(0)=1
sexp(-1)=2pi i; sexp(0)=1+2pi i
sexp(-1)=2npi i; sexp(0)=1+2npi i for any value of n

Well approximately yes.
Analytic continuation forces perfect periodicity to extend everywhere.

Everywhere until you hit the singularity. The singularity is at L,L*. To the left of those singularities, slog(z) is both analytic and exactly 2pi i periodic. To the right, slog is no longer 2pi i periodic. A similar 2pi i periodic would be ; it is exactly 2pi i periodic in the left half of the complex plane, but not the right half.
- Sheldon
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Messages In This Thread
Some slog stuff - by tommy1729 - 05/12/2015, 09:55 PM
RE: Some slog stuff - by tommy1729 - 05/12/2015, 10:16 PM
RE: Some slog stuff - by tommy1729 - 05/12/2015, 10:45 PM
RE: Some slog stuff - by sheldonison - 05/12/2015, 11:45 PM
RE: Some slog stuff - by tommy1729 - 05/13/2015, 12:25 PM
RE: Some slog stuff - by tommy1729 - 05/13/2015, 11:49 PM
RE: Some slog stuff - by sheldonison - 05/14/2015, 12:37 PM
RE: Some slog stuff - by tommy1729 - 05/14/2015, 01:56 PM
RE: Some slog stuff - by tommy1729 - 05/14/2015, 02:28 PM
RE: Some slog stuff - by sheldonison - 05/14/2015, 03:13 PM
RE: Some slog stuff - by tommy1729 - 05/14/2015, 05:58 PM
RE: Some slog stuff - by sheldonison - 05/14/2015, 08:06 PM
RE: Some slog stuff - by tommy1729 - 05/14/2015, 02:43 PM
RE: Some slog stuff - by tommy1729 - 05/14/2015, 06:55 PM
RE: Some slog stuff - by tommy1729 - 05/14/2015, 08:58 PM
RE: Some slog stuff - by tommy1729 - 05/14/2015, 09:25 PM

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