Solve the equation `2z= |z| + 2i` is complex numbers

Answer

To solve the equation \( 2z = |z| + 2i \), where \( z \) is a complex number, we will follow these steps:

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

### Step 2: Write the modulus of \( z \)
The modulus \( |z| \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]

### Step 3: Substitute \( z \) and \( |z| \) into the equation
Substituting \( z \) and \( |z| \) into the original equation:
\[ 2(x + iy) = \sqrt{x^2 + y^2} + 2i \]

### Step 4: Separate the real and imaginary parts
This gives us:
\[ 2x + 2iy = \sqrt{x^2 + y^2} + 2i \]
From this, we can separate the real and imaginary parts:
1. Real part: \( 2x = \sqrt{x^2 + y^2} \)
2. Imaginary part: \( 2y = 2 \)

### Step 5: Solve the imaginary part equation
From the imaginary part, we have:
\[ 2y = 2 \implies y = 1 \]

### Step 6: Substitute \( y \) into the real part equation
Substituting \( y = 1 \) into the real part equation:
\[ 2x = \sqrt{x^2 + 1^2} \implies 2x = \sqrt{x^2 + 1} \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides:
\[ (2x)^2 = x^2 + 1 \implies 4x^2 = x^2 + 1 \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 4x^2 - x^2 - 1 = 0 \implies 3x^2 - 1 = 0 \]

### Step 9: Solve for \( x \)
Solving for \( x^2 \):
\[ 3x^2 = 1 \implies x^2 = \frac{1}{3} \implies x = \pm \frac{1}{\sqrt{3}} = \pm \frac{\sqrt{3}}{3} \]

### Step 10: Write the solutions for \( z \)
Thus, the solutions for \( z \) are:
\[ z = \frac{\sqrt{3}}{3} + i \quad \text{and} \quad z = -\frac{\sqrt{3}}{3} + i \]

### Final Answer
The solutions to the equation \( 2z = |z| + 2i \) are:
\[ z = \frac{\sqrt{3}}{3} + i \quad \text{and} \quad z = -\frac{\sqrt{3}}{3} + i \]

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