The question

Where is the function analytic?

My attempt

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

and

Therefore, not analytic when

Is this correct? Thank you.

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- Jun 15th 2011, 09:44 PMGlitchAnalyticity of Log(z)
**The question**

Where is the function analytic?

**My attempt**

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

and

Therefore, not analytic when

Is this correct? Thank you. - Jun 15th 2011, 11:13 PMFernandoRevillaRe: Analyticity of Log(z)
The principal value is analytic. Then, .

- Jun 20th 2011, 09:25 PMojonesRe: Analyticity of Log(z)
The principal value is defined everywhere except at 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 PMchisigmaRe: Analyticity of Log(z)
The complex function is composed by two terms...

(1)

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

Kind regards

- Jun 20th 2011, 10:51 PMFernandoRevillaRe: 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 on . - Jun 20th 2011, 11:04 PMchisigmaRe: Analyticity of Log(z)
The general question of the analyticity of the function in the whole complex plane with the only exception of has been discussed here...

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

Kind regards

- Jun 20th 2011, 11:17 PMFernandoRevillaRe: Analyticity of Log(z)
Yes, the analyticity of does not change trough the years.

- Jun 20th 2011, 11:26 PMojonesRe: Analyticity of Log(z)
Oops! I was talking about , not . I think for the latter we have to avoid points where lies on the negative real axis or is zero.

- Jun 20th 2011, 11:29 PMojonesRe: Analyticity of Log(z)
- Jun 20th 2011, 11:32 PMojonesRe: Analyticity of Log(z)
- Jun 21st 2011, 03:48 PMojonesRe: Analyticity of Log(z)