1. ## 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.

2. ## Re: Analyticity of Log(z)

The principal value $\displaystyle \log:\textrm{Re}\;t>0\to \mathbb{C}$ is analytic. Then, $\displaystyle \textrm{Re}\;(z^2-1)>0\Leftrightarrow x^2-y^2-1>0$ .

3. ## 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.

4. ## 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

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

5. ## 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 $\displaystyle \textrm{Re}\; z >0$ .

6. ## 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

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

7. ## Re: Analyticity of Log(z)

Yes, the analyticity of $\displaystyle \log z$ does not change trough the years.

8. ## 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.

9. ## Re: Analyticity of Log(z)

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.

10. ## Re: Analyticity of Log(z)

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?

11. ## Re: Analyticity of Log(z)

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.