Let's solve this step by step.

Part (a)

1. Given Function: $f(z) = u(x, y) + iv(x, y)$, where $f(z) = -x + xy^2 - y^3 + (-x^2y + 3x - x^3)i$ Here, we need to identify the real part $u(x, y)$ and the imaginary part $v(x, y)$ in terms of $x$ and $y$.

2. Separate $u(x, y)$ and $v(x, y)$:

   • Real part $u(x, y)$: $u(x, y) = -x + xy^2 - y^3$
   • Imaginary part $v(x, y)$: $v(x, y) = -x^2y + 3x - x^3$

3. Differentiability Condition: For $f(z)$ to be differentiable at a point $z_0 = x_0 + iy_0$, the Cauchy-Riemann equations must hold, and the partial derivatives of $u$ and $v$ must be continuous at $z_0$.

   The Cauchy-Riemann equations are: $u_x = v_y \quad \text{and} \quad u_y = -v_x$

4. Calculate Partial Derivatives:

   • $u_x = -1 + y^2$
   • $u_y = 2xy - 3y^2$
   • $v_x = -2xy + 3 - 3x^2$
   • $v_y = -x^2$

5. Apply Cauchy-Riemann Equations:

   • For $u_x = v_y$: $-1 + y^2 = -x^2$
   • For $u_y = -v_x$: $2xy - 3y^2 = 2xy - 3 + 3x^2$

   Solving these equations will help identify the points where $f(z)$ is differentiable.

Part (b)

To find points where $f$ is analytic, it must satisfy the Cauchy-Riemann equations over a region, not just a point. This means that the conditions derived from the equations above must hold throughout a neighborhood for $f$ to be analytic.

Would you like me to proceed with solving the system of equations derived from the Cauchy-Riemann conditions for differentiability, or do you have any specific questions?

Related Questions

1. What are the Cauchy-Riemann equations, and why are they essential for complex differentiability?
2. How can we determine the points where the function is analytic after finding where it is differentiable?
3. What are the implications of the function being analytic in the complex plane?
4. How do we interpret the real and imaginary parts of a complex function geometrically?
5. What are some common techniques for solving systems of equations like the ones derived from Cauchy-Riemann equations?

Tip

When checking differentiability for complex functions, always verify the continuity of partial derivatives at the point, as it is a critical criterion along with the Cauchy-Riemann equations.