To determine which function $f(z)$ satisfies the Cauchy-Riemann equations throughout the complex plane, let's analyze each function.

The Cauchy-Riemann equations are: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x},$ where $f(z) = u(x, y) + iv(x, y)$ with $u(x, y)$ as the real part and $v(x, y)$ as the imaginary part.

Let's examine each option:

1. $f(z) = x^2 - iy^2$:

   • $u(x, y) = x^2$, $v(x, y) = -y^2$.
   • Calculating partial derivatives: $\frac{\partial u}{\partial x} = 2x, \quad \frac{\partial v}{\partial y} = -2y.$ Since $\frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y}$, this function does not satisfy the Cauchy-Riemann equations.

2. $f(z) = (x^2 - y^2) - i \cdot 2xy$:

   • $u(x, y) = x^2 - y^2$, $v(x, y) = -2xy$.
   • Calculating partial derivatives: $\frac{\partial u}{\partial x} = 2x, \quad \frac{\partial v}{\partial y} = -2x,$ $\frac{\partial u}{\partial y} = -2y, \quad \frac{\partial v}{\partial x} = -2y.$ Both Cauchy-Riemann equations are satisfied: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.$ This function does satisfy the Cauchy-Riemann equations throughout the complex plane.

3. $f(z) = (x^2 - y^2) + i \cdot 2xy$:

   • $u(x, y) = x^2 - y^2$, $v(x, y) = 2xy$.
   • Calculating partial derivatives: $\frac{\partial u}{\partial x} = 2x, \quad \frac{\partial v}{\partial y} = 2x,$ $\frac{\partial u}{\partial y} = -2y, \quad \frac{\partial v}{\partial x} = 2y.$ Here, $\frac{\partial u}{\partial y} \neq -\frac{\partial v}{\partial x}$, so this function does not satisfy the Cauchy-Riemann equations.

4. $f(z) = x^2 + iy^2$:

   • $u(x, y) = x^2$, $v(x, y) = y^2$.
   • Calculating partial derivatives: $\frac{\partial u}{\partial x} = 2x, \quad \frac{\partial v}{\partial y} = 2y.$ Since $\frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y}$, this function does not satisfy the Cauchy-Riemann equations.

Conclusion

The function $f(z) = (x^2 - y^2) - i \cdot 2xy$ is the only one that satisfies the Cauchy-Riemann equations throughout the complex plane.

Would you like more details on the calculations?

Here are 5 related questions for further study:

1. What are the Cauchy-Riemann equations, and why are they important in complex analysis?
2. How do we interpret a function satisfying the Cauchy-Riemann equations geometrically?
3. Can a function satisfy the Cauchy-Riemann equations at a point but not be differentiable there?
4. What are the conditions for a complex function to be analytic?
5. How are harmonic functions related to complex differentiable functions?

Tip: To confirm that a complex function is differentiable, check that it satisfies the Cauchy-Riemann equations and that the partial derivatives are continuous.