$\newcommand{\Arg}{\operatorname{Arg}}$

$\newcommand{\Log}{\operatorname{Log}}$

The range is "sector shaped," consisting of two ray boundaries. The first one lies at the positive Re-axis. The second ray starts at the origin and makes the angle $\alpha\pi$ from the first, counterclockwise. Fill in the region bounded by these two rays starting from the positive Re-axis and continue counterclockwise, and dot the border.

The range will never "overlap" as $0 < \alpha < 2$, so the possible principal arguments of $z^\alpha$ is always $0 < \Arg (z) < 2\pi$.

The appropriate choice for $\log z$ is $\Log z$.

**Proving $f(z) = z^\alpha$ maps one-to-one to the sector range:** $f(z_1) = f(z_2) \Rightarrow z_1 = z_2$.

Let $z = re^{i\theta}$, such that $r$ is the magnitude, and $\theta$ is the principal argument. Note $0 < \theta < \pi$, the inequalities are strict.

Then $z^\alpha = r^\alpha e^{i\alpha\theta}$. Note $0 < \Arg (z^\alpha) < \alpha\pi$

Since $f(z_1) = f(z_2) = r^\alpha e^{i\alpha\theta}$, it follows that $r$ and $\theta$ for both $z_1, z_2$ are the same.

We conclude $z_1 = z_2 = re^{i\theta}$, so $f(z)$ is injective.

**Proving $f(z) = z^\alpha$ maps onto the range:** For all $f(z)$ on the "range" domain ${w: 0 < \Arg w < \alpha\pi}$.

Since $f(z) = r^\alpha e^{i\alpha\theta}$, it follows $z = re^{i\theta}$.

We note that $0 < \Arg f(z) < \alpha\pi$, and it follows that $0 < \Arg z < \pi$.

We conclude that $f(z)$ maps the upper half-plane onto all of the range domain.

**Proving that $f$ also carries the boundary to the boundary**:

The boundary of the domain domain consists of all the real numbers.

The boundary of the range domain consists of 0, the positive Re-axis, and the ray $\alpha\pi$ from that axis counterclockwise, which is $\{w: \Arg w = \alpha\pi\}$.

If $z$ is zero, $z^\alpha$ is also zero.

If $z$ is positive real, $z^\alpha$ is also positive real, with magnitude raised to the power of $\alpha$. $\Arg z = \Arg z^\alpha = 0$.

If $z$ is negative real, $\Arg z = \pi$, and it follows $\Arg z^\alpha = \alpha\pi$.The domain domain is the open upper-half plane, the set of all the all the complex imaginary numbers with positive imaginary part. Fill the upper half of the plane and dot the border.