or each of the given functions find the set of points on which the Cauchy-Riemann equations aresatisfied. Also, check whether the function is analytic or not on the points where Cauchy-Riemannequations are satisfied.(a) 𝑓 (𝑧) = |𝑧|2(b) 𝑓 (𝑥 + 𝑖𝑦) = 𝑥3 + 𝑖(𝑦 − 1)3

(a) 𝑓 (𝑧) = |𝑧|²

The function 𝑓 (𝑧) = |𝑧|² can be written as 𝑓 (𝑥 + 𝑖𝑦) = 𝑥² + 𝑦². The Cauchy-Riemann equations are:

1. ∂u/∂x = ∂v/∂y
2. ∂u/∂y = -∂v/∂x

Here, u = x² + y² and v = 0. So, ∂u/∂x = 2x, ∂u/∂y = 2y, ∂v/∂x = 0 and ∂v/∂y = 0.

From equation 1, 2x = 0 and from equation 2, 2y = 0. The only solution to these equations is x = y = 0. So, the Cauchy-Riemann equations are satisfied only at the point (0,0).

However, for a function to be analytic, it must satisfy the Cauchy-Riemann equations in a neighborhood of points, not just at a single point. Therefore, 𝑓 (𝑧) = |𝑧|² is not analytic.

(b) 𝑓 (𝑥 + 𝑖𝑦) = 𝑥³ + 𝑖(𝑦 − 1)³

Here, u = x³ and v = (y - 1)³. So, ∂u/∂x = 3x², ∂u/∂y = 0, ∂v/∂x = 0 and ∂v/∂y = 3(y - 1)².

From equation 1, 3x² = 3(y - 1)² and from equation 2, 0 = 0. The first equation simplifies to x² = (y - 1)². This equation is satisfied for all points on the line y = x + 1.

However, for a function to be analytic, it must also satisfy the condition that the partial derivatives are continuous. The partial derivatives of u and v are not continuous for all points on the line y = x + 1, so the function 𝑓 (𝑥 + 𝑖𝑦) = 𝑥³ + 𝑖(𝑦 − 1)³ is not analytic.