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.