Using Cauchy Riemann equations, check whether complex function
𝑓(𝑧) = ((𝑥 + 1)
3 − 3(𝑥 + 1)𝑦
2 − 3𝑥) + 𝑖(𝑦
3 + 3𝑦 − 3(𝑥 + 1)
2𝑦) is analytic in the
complex plane or not

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.

Learn how to determine the analyticity of complex function 𝑓(𝑧) = ((𝑥 + 1)^3 − 3(𝑥 + 1)𝑦^2 − 3𝑥) + 𝑖(𝑦^3 + 3𝑦 − 3(𝑥 + 1)^2𝑦) using Cauchy-Riemann equations. Detailed step-by-step solution provided.

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