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
Question
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
Solution
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:
- ∂u/∂x = ∂v/∂y
- ∂u/∂y = -∂v/∂x
Here, u(x, y) = x^2 + y^2 and v(x, y) = 0.
Let's calculate the partial derivatives:
∂u/∂x = 2x ∂u/∂y = 2y ∂v/∂x = 0 ∂v/∂y = 0
Clearly, ∂u/∂x ≠ ∂v/∂y and ∂u/∂y ≠ -∂v/∂x. Hence, the Cauchy-Riemann equations are not satisfied for any point in the complex plane.
Therefore, the function f(z) = |z|^2 is not analytic at any point in the complex plane.
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