Find nonzero complex numbers z such that z¯=iz2

To find the complex numbers z such that z̄ = iz², we can follow these steps:

1. Let's denote z = x + yi, where x and y are real numbers and i is the imaginary unit. The conjugate of z (z̄) is x - yi.

2. Substitute z and z̄ into the given equation: x - yi = i(x + yi)².

3. Expand the right side of the equation: x - yi = i(x² - y² + 2xyi).

4. Separate the real and imaginary parts on both sides: x = -2xy and -y = x² - y².

5. From the first equation, we get two possibilities: either x = 0 or y = -1/2.

6. If x = 0, then from the second equation we get -y = -y², which gives y = 0 or y = 1. But y cannot be 0 because we are looking for nonzero complex numbers. So, y = 1 is the only solution in this case, and the corresponding complex number is z = 0 + i.

7. If y = -1/2, then from the second equation we get -(-1/2) = x² - (-1/2)², which simplifies to 1/2 = x² - 1/4. Solving for x, we get x = ±√(3/4) = ±√3/2. So, the corresponding complex numbers are z = √3/2 - i/2 and z = -√3/2 - i/2.

So, the nonzero complex numbers z such that z̄ = iz² are z = 0 + i, z = √3/2 - i/2, and z = -√3/2 - i/2.