For 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(c) 𝑓 (𝑥 + 𝑖𝑦) = 𝑥2 + 𝑖𝑦2(d) 𝑓 (𝑧) = Re(𝑧)

Sure, let's solve these step by step:

(a) 𝑓 (𝑧) = |𝑧|²

The function f(z) = |z|² is not differentiable, hence it does not satisfy the Cauchy-Riemann equations at any point. Therefore, it is not analytic at any point.

(b) 𝑓 (𝑥 + 𝑖𝑦) = 𝑥³ + 𝑖(𝑦 − 1)³

Here, u(x, y) = x³ and v(x, y) = (y - 1)³. The Cauchy-Riemann equations are:

du/dx = dv/dy and du/dy = - dv/dx

Calculating these, we get:

3x² = 3(y - 1)² and 0 = 0

The only solution to these equations is x = y - 1. Therefore, the function satisfies the Cauchy-Riemann equations on the line y = x + 1. However, since the Cauchy-Riemann equations are not satisfied everywhere, the function is not analytic.

(c) 𝑓 (𝑥 + 𝑖𝑦) = 𝑥² + 𝑖𝑦²

Here, u(x, y) = x² and v(x, y) = y². The Cauchy-Riemann equations are:

du/dx = dv/dy and du/dy = - dv/dx

Calculating these, we get:

2x = 2y and 0 = 0

The only solution to these equations is x = y. Therefore, the function satisfies the Cauchy-Riemann equations on the line y = x. However, since the Cauchy-Riemann equations are not satisfied everywhere, the function is not analytic.

(d) 𝑓 (𝑧) = Re(𝑧)

The function f(z) = Re(z) = x is not differentiable, hence it does not satisfy the Cauchy-Riemann equations at any point. Therefore, it is not analytic at any point.