The question requires converting a set of complex numbers into the rectangular form $x + jy$, where $j$ represents the imaginary unit.

Let's analyze the given complex numbers:

1. $\sqrt{2} e^{-j \pi/4}$
2. $\sqrt{2} e^{-j 9\pi/4}$
3. $\sqrt{2} e^{j 9\pi/4}$
4. $\sqrt{2} e^{j \pi/4}$
5. $e^{j 5\pi/2}$
6. $e^{-j \pi/2}$
7. $e^{j \pi/2}$
8. $\frac{1}{2} e^{-j \pi}$
9. $\frac{1}{2} e^{j \pi}$

To convert each complex number from polar form $re^{j\theta}$ to rectangular form $x + jy$, we use:

$x = r \cos(\theta)$ $y = r \sin(\theta)$

Now, let’s calculate each value:

1. $\sqrt{2} e^{-j \pi/4}$:

   • $r = \sqrt{2}$, $\theta = -\pi/4$
   • $x = \sqrt{2} \cos(-\pi/4) = \sqrt{2} \cdot \frac{\sqrt{2}}{2} = 1$
   • $y = \sqrt{2} \sin(-\pi/4) = \sqrt{2} \cdot \left(-\frac{\sqrt{2}}{2}\right) = -1$
   • Result: $1 - j$

2. $\sqrt{2} e^{-j 9\pi/4}$:

   • $r = \sqrt{2}$, $\theta = -9\pi/4$ (Equivalent to $-\pi/4$ because angles are periodic with $2\pi$)
   • Result: $1 - j$ (same as above)

3. $\sqrt{2} e^{j 9\pi/4}$:

   • $r = \sqrt{2}$, $\theta = 9\pi/4$ (Equivalent to $\pi/4$)
   • Result: $1 + j$

4. $\sqrt{2} e^{j \pi/4}$:

   • $r = \sqrt{2}$, $\theta = \pi/4$
   • Result: $1 + j$

5. $e^{j 5\pi/2}$:

   • $r = 1$, $\theta = 5\pi/2$ (Equivalent to $\pi/2$)
   • Result: $0 + j$

6. $e^{-j \pi/2}$:

   • $r = 1$, $\theta = -\pi/2$
   • Result: $0 - j$

7. $e^{j \pi/2}$:

   • $r = 1$, $\theta = \pi/2$
   • Result: $0 + j$

8. $\frac{1}{2} e^{-j \pi}$:

   • $r = \frac{1}{2}$, $\theta = -\pi$
   • $x = \frac{1}{2} \cos(-\pi) = \frac{1}{2} \cdot (-1) = -\frac{1}{2}$
   • $y = \frac{1}{2} \sin(-\pi) = 0$
   • Result: $-\frac{1}{2}$

9. $\frac{1}{2} e^{j \pi}$:

   • $r = \frac{1}{2}$, $\theta = \pi$
   • Result: $-\frac{1}{2}$

Let me know if you need further details or have any questions. Here are some related questions that might expand on this topic:

1. How is the polar form of a complex number different from its rectangular form?
2. What is the significance of the argument (angle) in polar form?
3. How can we convert rectangular form back to polar form?
4. Why are complex numbers often represented in polar form for certain calculations?
5. What are the real-world applications of using complex numbers in different forms?

Tip: Remember that angles in polar form can often be simplified by adding or subtracting multiples of $2\pi$, as they represent the same direction on the complex plane.