Re: Analyticity of Log(z)

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

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.

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

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

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

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

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

Re: Analyticity of Log(z)

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

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.

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.

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?

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.