The function $w = z^2$ maps complex numbers from the domain $D = \{ z : \text{Im}(z) > 0 \}$, which is the upper half-plane of the complex plane, to another region in the complex plane.

Let $z = x + iy$ where $x$ and $y$ are real numbers, and $y > 0$ (since $\text{Im}(z) > 0$).

Now, applying the transformation $w = z^2$:

$w = (x + iy)^2 = x^2 - y^2 + 2ixy$

This gives the real and imaginary parts of $w$ as:

$\text{Re}(w) = x^2 - y^2, \quad \text{Im}(w) = 2xy$

Understanding the Image of $D$:

1. Imaginary part: Since $y > 0$, the imaginary part $\text{Im}(w) = 2xy$ can be positive, negative, or zero depending on the value of $x$. If $x = 0$, then $\text{Im}(w) = 0$; otherwise, the sign of $\text{Im}(w)$ depends on the sign of $x$.

2. Real part: The real part $\text{Re}(w) = x^2 - y^2$ can be either positive or negative depending on the relationship between $x^2$ and $y^2$.

3. Overall mapping: The function $w = z^2$ maps the upper half-plane (where $\text{Im}(z) > 0$) onto the entire complex plane except for the negative real axis (since on the negative real axis, $\text{Im}(w) = 0$ but the original point $z$ would have $y = 0$, which isn't in the upper half-plane).

In conclusion, the map $w = z^2$ sends the upper half-plane $\{ z : \text{Im}(z) > 0 \}$ to the complex plane minus the non-positive real axis $\mathbb{C} \setminus (-\infty, 0]$.

Would you like further details, or have any questions?

Here are 5 related questions:

1. What happens to the lower half-plane under the same transformation $w = z^2$?
2. How does the function $w = z^2$ map the unit circle in the complex plane?
3. What are the fixed points of the transformation $w = z^2$?
4. How does the function $w = z^3$ map the upper half-plane?
5. What is the inverse function of $w = z^2$, and what are its properties?

Tip: Visualizing the transformation of different regions in the complex plane can help deepen your understanding of complex functions.