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

The function given is f(z) = |z|^2.

First, let's express this function in terms of x and y, where z = x + iy. So, f(z) = |x + iy|^2 = x^2 + y^2.

The Cauchy-Riemann equations are two partial differential equations which are satisfied by a function if it is analytic. They are given by:

1. ∂u/∂x = ∂v/∂y
2. ∂u/∂y = -∂v/∂x

Here, u = x^2 + y^2 and v = 0 (since the function f(z) is purely real).

Let's calculate the partial derivatives:

∂u/∂x = 2x ∂u/∂y = 2y ∂v/∂x = 0 ∂v/∂y = 0

Now, let's check if the Cauchy-Riemann equations are satisfied:

1. ∂u/∂x = ∂v/∂y => 2x = 0
2. ∂u/∂y = -∂v/∂x => 2y = 0

From the above equations, it is clear that 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 not just at a point, but in a neighborhood of that point. Since the given function does not satisfy this condition, it is not analytic.