Now I realized that

has a lot of singularities except for

where it is a linear function.

A singularity can only occur if the logarithm of 0 is taken.

For

no singularity occurs because

is not in the image of

. And we can simplify it to:

Because

iff

the only singularity can occur when

which is again not possible. So

will also not have any singularities. It can indeed be written as a affine function.

For

we determine the zeros of

:

So whenever

, then there is a singularity of

, this is for:

,

.

Thatswhy

has a singularity at each

.

Particularly at

which restricts the domain of definition of

to

.

Let us generalize this some more.

Let now

and

.

has singularities exactly if

for some

. As we assume that we take the real logarithm of real numbers (otherwise singularities could be avoided by choosing non-real branches).

In other words

has singularities at

.

These values are reached by

at

where

.

However

does not need to have singularities at all of these values, as some

may take a different branch

instead of returning 0 (or one of

) if the argument of

was non-real.

So if we always take the primary logarithm

we obtain a possible set of singularities which lies inside the set

. As the primary logarithm

maps

bijectively to

we conclude that

is bijective on

. More importantly any path

to

from

in the upper halfplane will be mapped to

in the upper halfplane, i.e. it will not wind around 0.

Hence by our construction we must take the primary logarithm

of the point

,

. But the primary logarithm

lies again in the upper halfplane and so on, that means we must always take the primary logarithm

of

,

.

Particularly this is true singular choices of

, they *must* yield singularities

can not escape to some other branch. Hence

Proposition. Every element of the set

is a singularity of

(which is defined via path-continuation).

Where we consider

to be the primary branch.

(0 and 1 are excluded because they have no logarithm for

.)

These singularities are not isolated but they are branch points.

So depending how the path to a point winds around these singularites we get different results of the

.

So we have to restrict ourselves to a simply connected neighborhood of the real axis where no singularities exist, there we have a unique continuation.

The interesting question is now how these singularities are distributed in the limit case.

Do some singularities converge to the real axis?