# Analyticity of Log(z)

• Jun 15th 2011, 09:44 PM
Glitch
Analyticity of Log(z)
The question
Where is the function $\displaystyle Log(z^2 - 1)$ analytic?

My attempt
The branch cut is on the negative real axis, so:

$\displaystyle z^2 - 1 = (x^2 - y^2 - 1) + i(2xy)$
$\displaystyle x^2 - y^2 - 1 \le 0$
and
$\displaystyle xy = 0$

Therefore, not analytic when $\displaystyle x = \pm 1$

Is this correct? Thank you.
• Jun 15th 2011, 11:13 PM
FernandoRevilla
Re: Analyticity of Log(z)
The principal value $\displaystyle \log:\textrm{Re}\;t>0\to \mathbb{C}$ is analytic. Then, $\displaystyle \textrm{Re}\;(z^2-1)>0\Leftrightarrow x^2-y^2-1>0$ .
• Jun 20th 2011, 09:25 PM
ojones
Re: Analyticity of Log(z)
The principal value is defined everywhere except at $\displaystyle z=0$ and is discontinuous on the negative real axis. Hence it can't be analytic at these points. The Cauchy-Riemann condition tells us that the function is anaytic everywhere else.
• Jun 20th 2011, 10:06 PM
chisigma
Re: Analyticity of Log(z)
The complex function is composed by two terms...

$\displaystyle \ln (z^{2}-1) = \ln (z-1) + \ln (z+1)$ (1)

As ojones said, the first term is analytic everywhere except in $\displaystyle z=1$ , the second term everywhere except in $\displaystyle z=-1$. The points $\displaystyle z=-1$ and $\displaystyle z=1$ are then the two 'brantch points' of the function (1)...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Jun 20th 2011, 10:51 PM
FernandoRevilla
Re: Analyticity of Log(z)
Some authors, see for example Elementary Theory of Analytic Functions of One or Several Complex Variables by Henry Cartan define the principal determination of $\displaystyle \log z$ on $\displaystyle \textrm{Re}\; z >0$ .
• Jun 20th 2011, 11:04 PM
chisigma
Re: Analyticity of Log(z)
The general question of the analyticity of the function $\displaystyle \ln z$ in the whole complex plane with the only exception of $\displaystyle z=0$ has been discussed here...

http://www.mathhelpforum.com/math-he...st-167358.html

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Jun 20th 2011, 11:17 PM
FernandoRevilla
Re: Analyticity of Log(z)
Yes, the analyticity of $\displaystyle \log z$ does not change trough the years.
• Jun 20th 2011, 11:26 PM
ojones
Re: Analyticity of Log(z)
Oops! I was talking about $\displaystyle \text{Log}(z)$, not $\displaystyle \text{Log}(z^2-1)$. I think for the latter we have to avoid points where $\displaystyle z^2-1$ lies on the negative real axis or is zero.
• Jun 20th 2011, 11:29 PM
ojones
Re: Analyticity of Log(z)
Quote:

Originally Posted by FernandoRevilla
Some authors, see for example Elementary Theory of Analytic Functions of One or Several Complex Variables by Henry Cartan define the principal determination of $\displaystyle \log z$ on $\displaystyle \textrm{Re}\; z >0$ .

Interesting. I'll take a look at that.
• Jun 20th 2011, 11:32 PM
ojones
Re: Analyticity of Log(z)
Quote:

Originally Posted by chisigma
The general question of the analyticity of the function $\displaystyle \ln z$ in the whole complex plane with the only exception of $\displaystyle z=0$ has been discussed here...

http://www.mathhelpforum.com/math-he...st-167358.html

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

So you're saying that holomorphicity and analyticity don't necessarily coincide?
• Jun 21st 2011, 03:48 PM
ojones
Re: Analyticity of Log(z)
Quote:

Originally Posted by chisigma
The complex function is composed by two terms...

$\displaystyle \ln (z^{2}-1) = \ln (z-1) + \ln (z+1)$ (1)

$\displaystyle \chi$ $\displaystyle \sigma$

You need to be careful here; the usual log laws don't necessarily hold in the complex case.