Show that f(z) = z |z| is not analytic anywhere in the complex plane

To show that f(z) = z |z| is not analytic anywhere in the complex plane, we can use the Cauchy-Riemann equations. A function f(z) = u(x, y) + iv(x, y) is analytic if and only if the Cauchy-Riemann equations are satisfied and the partial derivatives of u and v are continuous.

First, express f(z) = z |z| in terms of z = x + iy:

|z| = sqrt(x2 + y2)
f(z) = (x + iy) sqrt(x2 + y2)

Let u(x, y) and v(x, y) be the real and imaginary parts of f(z):

u(x, y) = x sqrt(x2 + y2)
v(x, y) = y sqrt(x2 + y2)

The Cauchy-Riemann equations are:

∂u/∂x = ∂v/∂y
∂u/∂y = -∂v/∂x

Now, compute the partial derivatives of u and v:

∂u/∂x = ∂/∂x (x sqrt(x2 + y2))
Using the product rule:
∂u/∂x = sqrt(x2 + y2) + x ∙ ∂/∂x (sqrt(x2 + y2))
∂/∂x (sqrt(x2 + y2)) = x/sqrt(x2 + y2)
Thus,
∂u/∂x = sqrt(x2 + y2) + x ∙ x/sqrt(x2 + y2)
∂u/∂x = sqrt(x2 + y2) + x2/sqrt(x2 + y2)
∂u/∂x = (x2 + y2 + x2)/sqrt(x2 + y2)
∂u/∂x = (2x2 + y2)/sqrt(x2 + y2)

Next, compute ∂v/∂y:

∂v/∂y = ∂/∂y (y sqrt(x2 + y2))
Using the product rule:
∂v/∂y = sqrt(x2 + y2) + y ∙ ∂/∂y (sqrt(x2 + y2))
∂/∂y (sqrt(x2 + y2)) = y/sqrt(x2 + y2)
Thus,
∂v/∂y = sqrt(x2 + y2) + y ∙ y/sqrt(x2 + y2)
∂v/∂y = sqrt(x2 + y2) + y2/sqrt(x2 + y2)
∂v/∂y = (x2 + y2 + y2)/sqrt(x2 + y2)
∂v/∂y = (x2 + 2y2)/sqrt(x2 + y2)

For the Cauchy-Riemann equations to hold, we must have:

∂u/∂x = ∂v/∂y

However,

(2x2 + y2)/sqrt(x2 + y2) ≠ (x2 + 2y2)/sqrt(x2 + y2)

Since the Cauchy-Riemann equations are not satisfied, f(z) = z |z| is not analytic anywhere in the complex plane.