In most complex analysis textbooks it is written that the function

is 'non regular' along the negative real axis... according to my [very modest...] opinion that is false!...

To demostrate that it is sufficient consider the well known Taylor series expansion...

(1)

Operating tha change of variables

and taking into account that

the (1) becomes...

(2)

Now if we set in (2)

instead of

and consider that is

we obtain...

(3)

Observing (2) and (3) we derive that the first is the Taylor expansion of

in

and the second the Taylor expansion of

in

so that we can conclude that the function

is analytic both in

and

...

Kind regards