We are tasked with finding all complex numbers $z$ that satisfy the equation:

$\overline{z} = \frac{2}{z} + i$

Step 1: Express $z$ in terms of real and imaginary parts

Let $z = x + yi$, where $x$ and $y$ are real numbers, and $i$ is the imaginary unit. The complex conjugate of $z$, denoted by $\overline{z}$, is:

$\overline{z} = x - yi$

Thus, the equation becomes:

$x - yi = \frac{2}{x + yi} + i$

Step 2: Simplify the right-hand side

Now, we simplify $\frac{2}{x + yi}$. To do this, multiply both the numerator and the denominator by the complex conjugate of $x + yi$, which is $x - yi$:

$\frac{2}{x + yi} = \frac{2(x - yi)}{(x + yi)(x - yi)} = \frac{2(x - yi)}{x^2 + y^2}$

Therefore, the equation becomes:

$x - yi = \frac{2(x - yi)}{x^2 + y^2} + i$

Step 3: Separate real and imaginary parts

Now, expand the right-hand side:

$x - yi = \left( \frac{2x}{x^2 + y^2} \right) - \left( \frac{2y}{x^2 + y^2} \right)i + i$

Now, equate real and imaginary parts on both sides:

• Real part: $x = \frac{2x}{x^2 + y^2}$
• Imaginary part: $-y = -\frac{2y}{x^2 + y^2} + 1$

Step 4: Solve the real part equation

From the real part equation:

$x = \frac{2x}{x^2 + y^2}$

If $x \neq 0$, we can divide both sides by $x$:

$1 = \frac{2}{x^2 + y^2}$

This implies:

$x^2 + y^2 = 2$

Step 5: Solve the imaginary part equation

From the imaginary part equation:

$-y = -\frac{2y}{x^2 + y^2} + 1$

Using $x^2 + y^2 = 2$, substitute into the equation:

$-y = -\frac{2y}{2} + 1$

$-y = -y + 1$

This simplifies to:

$0 = 1$

This is a contradiction, meaning our assumption that $x \neq 0$ must be incorrect.

Step 6: Case when $x = 0$

If $x = 0$, the equation simplifies. Substituting $x = 0$ into the original equation, we get:

$z = yi$ $\overline{z} = -yi = \frac{2}{yi} + i$

Simplifying $\frac{2}{yi}$:

$\frac{2}{yi} = -\frac{2i}{y}$

Thus, the equation becomes:

$-yi = -\frac{2i}{y} + i$

Equate imaginary parts:

$-y = -\frac{2}{y} + 1$

Multiply through by $y$ to clear the denominator:

$-y^2 = -2 + y$

Rearranging:

$y^2 + y - 2 = 0$

Solve this quadratic equation using the quadratic formula:

$y = \frac{-1 \pm \sqrt{1^2 - 4(1)(-2)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 8}}{2} = \frac{-1 \pm 3}{2}$

Thus:

$y = 1 \quad \text{or} \quad y = -2$

Step 7: Conclusion

The solutions are:

$z = i \quad \text{or} \quad z = -2i$

These are the complex numbers that satisfy the given equation.

Would you like to see more details or ask any questions about the steps? Here are five related questions:

1. How do you calculate the complex conjugate of a complex number?
2. Can all quadratic equations have complex solutions?
3. How can we use the quadratic formula to solve other complex problems?
4. Why does the assumption $x \neq 0$ lead to a contradiction?
5. What are some other methods to solve equations involving complex numbers?

Tip: Always check for contradictions when solving complex equations.