Re: The function x^x

From: Ioannis (morpheus_at_olympus.mons)
Date: 11/28/04 

Date: Sun, 28 Nov 2004 11:56:25 +0200

mike3 wrote:
> Hi.
>
> Does anyone know what the graph of the function f(x) = Imag(x^x) for x
> < 0, that is, the imaginary part of x^x (e^(x ln(x)) with principal
> branch of
> ln(x)), looks like? I haven't really been able to get a good idea of
> it's behavior from looking at a few points.

The negative axis, taken as a branch cut for the principal branch of the
complex function log, can be parametrized as:

x=r*exp(i*pi), 0<=r<+inf.

Plug x in exp(x*log(x)) to get:

exp(x*log(x))=exp(r*(-1)*(r+i*pi))=
exp(-r^2)*exp(-i*r*pi)

Separate real and imaginary parts,

Im(x^x)=exp(-r^2)*sin(-r*pi)

Now graph that for 0<=r<+inf. 

-- 
I. N. G. --- http://users.forthnet.gr/ath/jgal/

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