To solve this problem, let's break it down step-by-step.

Problem Analysis

1. We are given the set $D = \{ x + yi : x^2 - y^2 = 1, x, y \in \mathbb{R} \}$, which represents a hyperbola in the complex plane $\mathbb{C}$.
2. We are given a function $f : D \rightarrow \mathbb{C}$ defined by $f(z) = z^2$.
3. The task is to find $f(D)$, which is the image of $D$ under the function $f$, and express this in set notation.

Solution

1. Parameterize the Points on $D$:

   Since $D$ is defined by the equation $x^2 - y^2 = 1$, this is the equation of a hyperbola. Let $z = x + yi$ be a point on $D$, where $x$ and $y$ satisfy $x^2 - y^2 = 1$.

2. Compute $f(z) = z^2$:

   For $z = x + yi$, $f(z) = z^2 = (x + yi)^2 = x^2 - y^2 + 2xyi.$

   Since $x^2 - y^2 = 1$ for points on $D$, we can substitute this to get: $f(z) = 1 + 2xyi.$

3. Determine the Range of $2xy$:

   Since $x^2 - y^2 = 1$, $x$ and $y$ must vary in such a way that this equation holds. As $x$ and $y$ range over all real numbers satisfying $x^2 - y^2 = 1$, the product $2xy$ can take any real value. This is because for any real number $t$, we can find $x$ and $y$ satisfying $x = \cosh u$ and $y = \sinh u$ (a standard hyperbolic parametrization), which allows $xy$ to cover all real numbers.

4. Express $f(D)$ in Set Notation:

   Since $2xy$ can take any real value, the image $f(D)$ represents all complex numbers of the form $1 + bi$ where $b \in \mathbb{R}$. Thus, we can write: $f(D) = \{ 1 + bi : b \in \mathbb{R} \}.$

Final Answer

$f(D) = \{ 1 + bi : b \in \mathbb{R} \}.$

This is a line in the complex plane parallel to the imaginary axis, shifted one unit to the right.

Would you like further clarification on any part of this solution, or do you have any additional questions?

Related Questions to Explore

1. What would $f(D)$ look like if $D$ were defined by a different hyperbola, such as $x^2 - y^2 = k$ for $k \neq 1$?
2. How would the solution change if the function were $f(z) = z^3$ instead of $z^2$?
3. Can we generalize this approach to other types of conic sections, like ellipses?
4. What would the image look like if we instead used polar coordinates?
5. How does the choice of hyperbolic parameterization affect the form of $f(D)$?

Tip

When working with transformations in the complex plane, analyzing the behavior under squaring or other powers can reveal interesting geometrical properties and mappings.