Apply Cauchy-Riemann equations on $f(z)=z+|z|$? I am trying to check if the function $f(z)=z+|z|$ is analytic by using the Cauchy-Riemann equation.
I made 
$z = x +jy$ 
and therefore 
$$f(z)= (x + jy) + \sqrt{x^2 + y^2}$$
put into $f(z) = u+ jv$ form:
$$f(z)= x + \sqrt{x^2 + y^2} + jy$$
where  
$u = x + \sqrt{x^2 + y^2}$
and that
$v = y$
Now I need to apply the Cauchy-Riemann equation, but don't know how would I go about doing that.
Any help would be much appreciated.

In order for your function to be analytic, it must satisfy the Cauchy-Riemann equations (right?  it's good to think about why this is true).  So, what are the equations?
Well, du/dx = dv/dy.
Does this hold?  Or you could consider du/dy = -dv/dx.
If either of these equations do not hold, then the function is not analytic.
