Analyticity of Log(z)

**Glitch** · June 15th 2011, 09:44 PM

The question
Where is the function $Log(z^2 - 1)$ analytic?

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

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

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

Is this correct? Thank you.

**FernandoRevilla** · June 15th 2011, 11:13 PM

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

**ojones** · June 20th 2011, 09:25 PM

The principal value is defined everywhere except at $z=0$ 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.

**chisigma** · June 20th 2011, 10:06 PM

The complex function is composed by two terms...

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

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

Kind regards

$\chi$ $\sigma$

**FernandoRevilla** · June 20th 2011, 10:51 PM

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

**chisigma** · June 20th 2011, 11:04 PM

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

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

Kind regards

$\chi$ $\sigma$

**FernandoRevilla** · June 20th 2011, 11:17 PM

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

**ojones** · June 20th 2011, 11:26 PM

Oops! I was talking about $\text{Log}(z)$ , not $\text{Log}(z^2-1)$ . I think for the latter we have to avoid points where $z^2-1$ lies on the negative real axis or is zero.

**ojones** · June 20th 2011, 11:29 PM

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 $\log z$ on $\textrm{Re}\; z >0$ .

Interesting. I'll take a look at that.

**ojones** · June 20th 2011, 11:32 PM

Originally Posted by chisigma

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

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

Kind regards

$\chi$ $\sigma$

So you're saying that holomorphicity and analyticity don't necessarily coincide?

**ojones** · June 21st 2011, 03:48 PM

Originally Posted by chisigma

The complex function is composed by two terms...

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

$\chi$ $\sigma$

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

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