Using Cauchy Riemann equations, check whether complex function
𝑓(𝑧) = ((𝑥 + 1)3 − 3(𝑥 + 1)𝑦2 − 3𝑥) + 𝑖(𝑦3 + 3𝑦 − 3(𝑥 + 1)2𝑦) is analytic in the
complex plane or not.

To determine whether the complex function \( f(z) = ((x + 1)^3 - 3(x + 1)y^2 - 3x) + i(y^3 + 3y - 3(x + 1)^2y) \) is analytic, we need to use the **Cauchy-Riemann equations**. These equations provide necessary conditions for a function to be analytic (i.e., differentiable) at a point in the complex plane.

### Step 1: Express the Function in Terms of \( u(x, y) \) and \( v(x, y) \)

Given \( f(z) = u(x, y) + iv(x, y) \), where \( z = x + iy \):

- \( u(x, y) = (x + 1)^3 - 3(x + 1)y^2 - 3x \)
- \( v(x, y) = y^3 + 3y - 3(x + 1)^2y \)

### Step 2: Compute the Partial Derivatives

We need to compute the partial derivatives of \( u(x, y) \) and \( v(x, y) \) with respect to \( x \) and \( y \):

#### Partial Derivatives of \( u(x, y) \):
\[
u_x = \frac{\partial}{\partial x} \left[(x + 1)^3 - 3(x + 1)y^2 - 3x\right]
\]
\[
u_y = \frac{\partial}{\partial y} \left[(x + 1)^3 - 3(x + 1)y^2 - 3x\right]
\]

Calculating these:

\[
u_x = 3(x + 1)^2 - 3y^2 - 3
\]
\[
u_y = -6(x + 1)y
\]

#### Partial Derivatives of \( v(x, y) \):
\[
v_x = \frac{\partial}{\partial x} \left[y^3 + 3y - 3(x + 1)^2y\right]
\]
\[
v_y = \frac{\partial}{\partial y} \left[y^3 + 3y - 3(x + 1)^2y\right]
\]

Calculating these:

\[
v_x = -6(x + 1)y
\]
\[
v_y = 3y^2 + 3 - 3(x + 1)^2
\]

### Step 3: Apply the Cauchy-Riemann Equations

The Cauchy-Riemann equations state that for \( f(z) \) to be analytic, the following must hold:

1. \( u_x = v_y \)
2. \( u_y = -v_x \)

Substitute the partial derivatives we calculated:

1. \( u_x = 3(x + 1)^2 - 3y^2 - 3 \) and \( v_y = 3y^2 + 3 - 3(x + 1)^2 \)

   Here, \( u_x \) and \( v_y \) are not equal unless specific conditions are met (i.e., \( 3(x + 1)^2 - 3y^2 - 3 = 3y^2 + 3 - 3(x + 1)^2 \)), which simplifies to \( 2(3(x + 1)^2 - 3y^2) = 0 \). This implies \( (x + 1)^2 = y^2 \).

2. \( u_y = -6(x + 1)y \) and \( -v_x = 6(x + 1)y \)

   Here, \( u_y = -v_x \) holds true.

### Step 4: Conclusion

The function \( f(z) \) satisfies the Cauchy-Riemann equations only when \( (x + 1)^2 = y^2 \), which defines a set of curves in the complex plane. Outside of this set, the function does not satisfy the Cauchy-Riemann equations.

Therefore, **the function \( f(z) \) is not analytic in the entire complex plane**. It is only analytic on the specific curves where \( (x + 1)^2 = y^2 \).

Would you like more details on any part of this solution, or do you have any questions?

### Relative Questions:
1. How can we determine the analyticity of a function using the Cauchy-Riemann equations?
2. What does it mean for a function to be analytic in the complex plane?
3. Can a function satisfy the Cauchy-Riemann equations at some points but still not be analytic?
4. How do the Cauchy-Riemann equations relate to the concept of differentiability in complex analysis?
5. Are there other methods besides the Cauchy-Riemann equations to check if a function is analytic?

### Tip:
Remember that satisfying the Cauchy-Riemann equations is necessary but not always sufficient for analyticity; the function must also be continuously differentiable.

Learn how to determine if the complex function 𝑓(𝑧) = ((𝑥 + 1)3 − 3(𝑥 + 1)𝑦2 − 3𝑥) + 𝑖(𝑦3 + 3𝑦 − 3(𝑥 + 1)2𝑦) is analytic using Cauchy-Riemann equations. Detailed step-by-step solution and conditions for analyticity provided.

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