To determine if the given complex function $f(z) = u(x, y) + iv(x, y)$ is analytic, we need to verify whether the Cauchy-Riemann equations are satisfied.

Given: $f(z) = \left( (x+1)^3 - 3(x+1)y^2 - 3x \right) + i\left( y^3 + 3y - 3(x+1)^2 y \right)$

Step 1: Express $f(z)$ in terms of $u(x, y)$ and $v(x, y)$

• The real part $u(x, y)$ is: $u(x, y) = (x+1)^3 - 3(x+1)y^2 - 3x$

• The imaginary part $v(x, y)$ is: $v(x, y) = y^3 + 3y - 3(x+1)^2 y$

Step 2: Cauchy-Riemann Equations

The Cauchy-Riemann equations are: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

We need to compute the partial derivatives of $u(x, y)$ and $v(x, y)$.

Step 3: Compute partial derivatives of $u(x, y)$

• $\frac{\partial u}{\partial x}$: $\frac{\partial u}{\partial x} = 3(x+1)^2 - 3y^2 - 3$

• $\frac{\partial u}{\partial y}$: $\frac{\partial u}{\partial y} = -6(x+1)y$

Step 4: Compute partial derivatives of $v(x, y)$

• $\frac{\partial v}{\partial x}$: $\frac{\partial v}{\partial x} = -6(x+1)y$

• $\frac{\partial v}{\partial y}$: $\frac{\partial v}{\partial y} = 3y^2 + 3 - 3(x+1)^2$

Step 5: Check the Cauchy-Riemann equations

• First Cauchy-Riemann equation: $\frac{\partial u}{\partial x} = 3(x+1)^2 - 3y^2 - 3 \quad \text{and} \quad \frac{\partial v}{\partial y} = 3y^2 + 3 - 3(x+1)^2$ Clearly, $\frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y}$.

Since the first Cauchy-Riemann equation is not satisfied, we conclude that the function is not analytic.

Conclusion:

The given function is not analytic in the complex plane because the Cauchy-Riemann equations are not satisfied.

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Would you like any further details or clarifications?

Here are some related questions to expand on this topic:

1. What are the conditions for a function to be analytic in a region?
2. How are the Cauchy-Riemann equations derived from the definition of differentiability in the complex plane?
3. What are some common examples of analytic functions?
4. How do harmonic functions relate to the Cauchy-Riemann equations?
5. Can you provide an example of a function that is not analytic and explain why?

Tip: A function is analytic at a point if it is differentiable at that point and in some neighborhood around it.