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To determine which function satisfies the Cauchy-Riemann equations throughout the complex plane, let’s analyze each function one by one.

Given:
- \( z = x + iy \)
- \( f(z) = u(x, y) + iv(x, y) \), where \( u \) is the real part and \( v \) is the imaginary part.

The Cauchy-Riemann equations state that for \( f(z) \) to be differentiable, the following must hold:
\[
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
\]

### 1. \( f(z) = (x^2 - y^2) - i2xy \)
   - Here, \( u(x, y) = x^2 - y^2 \) and \( v(x, y) = -2xy \).
   - Compute the partial derivatives:
     \[
     \frac{\partial u}{\partial x} = 2x, \quad \frac{\partial u}{\partial y} = -2y
     \]
     \[
     \frac{\partial v}{\partial x} = -2y, \quad \frac{\partial v}{\partial y} = -2x
     \]
   - Since \( \frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} \neq -\frac{\partial v}{\partial x} \), the Cauchy-Riemann equations are not satisfied.

### 2. \( f(z) = x^2 - iy^2 \)
   - Here, \( u(x, y) = x^2 \) and \( v(x, y) = -y^2 \).
   - Compute the partial derivatives:
     \[
     \frac{\partial u}{\partial x} = 2x, \quad \frac{\partial u}{\partial y} = 0
     \]
     \[
     \frac{\partial v}{\partial x} = 0, \quad \frac{\partial v}{\partial y} = -2y
     \]
   - Since \( \frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} \neq -\frac{\partial v}{\partial x} \), the Cauchy-Riemann equations are not satisfied.

### 3. \( f(z) = (x^2 - y^2) + i2xy \)
   - Here, \( u(x, y) = x^2 - y^2 \) and \( v(x, y) = 2xy \).
   - Compute the partial derivatives:
     \[
     \frac{\partial u}{\partial x} = 2x, \quad \frac{\partial u}{\partial y} = -2y
     \]
     \[
     \frac{\partial v}{\partial x} = 2y, \quad \frac{\partial v}{\partial y} = 2x
     \]
   - Since \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \), the Cauchy-Riemann equations are satisfied.

### 4. \( f(z) = x^2 + iy^2 \)
   - Here, \( u(x, y) = x^2 \) and \( v(x, y) = y^2 \).
   - Compute the partial derivatives:
     \[
     \frac{\partial u}{\partial x} = 2x, \quad \frac{\partial u}{\partial y} = 0
     \]
     \[
     \frac{\partial v}{\partial x} = 0, \quad \frac{\partial v}{\partial y} = 2y
     \]
   - Since \( \frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} \neq -\frac{\partial v}{\partial x} \), the Cauchy-Riemann equations are not satisfied.

### Conclusion
The only function that satisfies the Cauchy-Riemann equations throughout the complex plane is:
\[
f(z) = (x^2 - y^2) + i2xy
\]

Let me know if you want more details on any part of the solution or if you have further questions.

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Here are 5 related questions to deepen your understanding:
1. How are the Cauchy-Riemann equations derived from complex differentiability?
2. Can a function satisfy the Cauchy-Riemann equations at a point but not be analytic?
3. How does differentiability in complex analysis differ from real analysis?
4. What is the significance of harmonic functions in relation to the Cauchy-Riemann equations?
5. How do Cauchy-Riemann equations connect to conformal mappings?

**Tip:** The Cauchy-Riemann equations provide a necessary condition for a function to be analytic, but satisfying them alone doesn’t guarantee analyticity unless the function is also continuous.

Let z = x + iy. Which one of the following functions satisfy Cauchy-Riemann equations throughout the complex plane?

Explore which functions satisfy the Cauchy-Riemann equations throughout the complex plane. This problem involves key concepts in complex analysis, specifically focusing on differentiability and the Cauchy-Riemann equations.

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