Find all complex numbers z that satisfy the equation z*=(2/z)+i

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.

Find all complex numbers z that satisfy the equation z* = (2/z) + i

Explore the step-by-step solution to the complex equation z* = (2/z) + i. Learn how to express complex numbers in terms of real and imaginary parts, use algebraic manipulation, and solve a quadratic equation. Suitable for advanced high school and undergraduate students.

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