Where does the function w=z^2 map the range D={z: Imz>0}?

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.

Explore how the complex function w = z^2 maps the upper half-plane Im(z) > 0 to the complex plane. Learn key concepts of complex analysis, including transformations and the behavior of complex numbers under squaring. Suitable for undergraduate students.

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