Find all non zero complex numbers z satisfying `barz=iz^2`

Explanation

To find all non-zero complex numbers \( z \) satisfying the equation \( \bar{z} = iz^2 \), we can follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Write the conjugate and square of \( z \)
The conjugate of \( z \) is:
\[ \bar{z} = x - iy \]
The square of \( z \) is:
\[ z^2 = (x + iy)^2 = x^2 - y^2 + 2xyi \]

### Step 3: Substitute into the equation
Substituting \( \bar{z} \) and \( z^2 \) into the equation \( \bar{z} = iz^2 \):
\[ x - iy = i(x^2 - y^2 + 2xyi) \]
This simplifies to:
\[ x - iy = -2xy + i(x^2 - y^2) \]

### Step 4: Equate real and imaginary parts
From the equation, we can equate the real and imaginary parts:
1. Real part: \( x = -2xy \)
2. Imaginary part: \( -y = x^2 - y^2 \)

### Step 5: Solve the real part equation
From the real part equation \( x = -2xy \):
- If \( x \neq 0 \), we can divide both sides by \( x \):
\[ 1 = -2y \implies y = -\frac{1}{2} \]
- If \( x = 0 \), then we will check the imaginary part equation.

### Step 6: Substitute \( y = -\frac{1}{2} \) into the imaginary part equation
Substituting \( y = -\frac{1}{2} \) into the imaginary part equation:
\[ -y = x^2 - y^2 \implies \frac{1}{2} = x^2 - \left(-\frac{1}{2}\right)^2 \]
This simplifies to:
\[ \frac{1}{2} = x^2 - \frac{1}{4} \]
\[ x^2 = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \]
Thus,
\[ x = \pm \frac{\sqrt{3}}{2} \]

### Step 7: Write the solutions for \( z \)
Now we can write the solutions for \( z \):
1. For \( x = \frac{\sqrt{3}}{2} \) and \( y = -\frac{1}{2} \):
\[ z = \frac{\sqrt{3}}{2} - \frac{1}{2}i \]
2. For \( x = -\frac{\sqrt{3}}{2} \) and \( y = -\frac{1}{2} \):
\[ z = -\frac{\sqrt{3}}{2} - \frac{1}{2}i \]
3. For \( x = 0 \), we will check the imaginary part:
\[ -y = 0 - y^2 \implies -y = -y^2 \implies y(y - 1) = 0 \]
This gives \( y = 0 \) or \( y = 1 \). However, \( y = 0 \) gives \( z = 0 \) which is not allowed. Thus, we take \( y = 1 \):
\[ z = i \]

### Final Solutions
The non-zero complex numbers \( z \) satisfying the equation \( \bar{z} = iz^2 \) are:
\[ z = \frac{\sqrt{3}}{2} - \frac{1}{2}i, \quad z = -\frac{\sqrt{3}}{2} - \frac{1}{2}i, \quad z = i \]