We are given the function $f(z) = (x^2 + y) + i(y^2 - x)$, where $z = x + iy$. To determine the analyticity of $f(z)$, we need to verify if the Cauchy-Riemann equations are satisfied.

For a function $f(z) = u(x, y) + iv(x, y)$, where $u(x, y)$ is the real part and $v(x, y)$ is the imaginary part, the Cauchy-Riemann equations are:

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Here, the real part $u(x, y) = x^2 + y$ and the imaginary part $v(x, y) = y^2 - x$.

Let's calculate the partial derivatives:

1. $\frac{\partial u}{\partial x} = 2x$
2. $\frac{\partial u}{\partial y} = 1$
3. $\frac{\partial v}{\partial x} = -1$
4. $\frac{\partial v}{\partial y} = 2y$

Now, check the Cauchy-Riemann equations:

• First equation: $\frac{\partial u}{\partial x} = 2x$ and $\frac{\partial v}{\partial y} = 2y$. For this to hold, $2x = 2y$, which implies $x = y$.
• Second equation: $\frac{\partial u}{\partial y} = 1$ and $-\frac{\partial v}{\partial x} = 1$. This equation is always satisfied.

Thus, the Cauchy-Riemann equations hold only along the line $x = y$. Therefore, $f(z)$ is analytic only along this line, but not generally analytic everywhere.

From the given options, the correct answer is:

• f is not analytic at any point.

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the significance of the Cauchy-Riemann equations in complex analysis?
2. How can we classify singularities of complex functions?
3. What conditions make a function holomorphic?
4. How does conformal mapping relate to analytic functions?
5. Can a function be partially analytic in a region?

Tip: Always check both Cauchy-Riemann equations carefully to ensure a function is analytic.