where is Ln(z+10) NOT analytic?

would i be correct in thinking this is for

z\leq-10

i.e. on the non-positive real axis?

(since for Ln(z+10) , domain is |z+10|>0 and -pi<Arg(z+10)<pi

??

Reply 1

DFranklin

Assuming by Ln you mean the principal branch of the log function, then yes. (I think)!

Reply 2

Hasufel

Original post by DFranklin

Assuming by Ln you mean the principal branch of the log function, then yes. (I think)!

I did, yes - Thanks!

Reply 3

atsruser

Original post by Hasufel

would i be correct in thinking this is for

z\leq-10

i.e. on the non-positive real axis?

(since for Ln(z+10) , domain is |z+10|>0 and -pi<Arg(z+10)<pi

??

You mean

\operatorname{Re} z \leq -10

. It's not analytic at those points since it's discontinuous there.

Reply 4

RichE

Original post by atsruser

You mean

\operatorname{Re} z \leq -10

. It's not analytic at those points since it's discontinuous there.

No, I think by

z \leq -10

he means the real numbers

z \leq -10

. For the branch he's discussing, the complex logarithm with have a

2 \pi i

discontinuity across that cut. But it will be analytic everywhere else.

Reply 5

atsruser

Original post by RichE

No, I think by

z \leq -10

he means the real numbers

z \leq -10

It looks simply wrong to me to write

z \leq -10

.

Whenever I see

z

, I read "complex variable", so this makes no sense at all to me.

I guess it would have clearer if I'd written

\{ z \in \mathbb{C} : \operatorname{Re} z \le -10, \operatorname{Im} z = 0 \}

or whatever.

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