# Non-analyticity of the modulus function

I'm studying the Ginzburg-Landau theory of superconductors, and I have read that the free energy can not depend on odd powers of the modulus of the order parameter $\psi$ because $| \psi| = \sqrt{\psi_x^2 +\psi_y^2}$ is not analytic at $\psi = \psi_x +i \psi_y = 0$. How do you show this non-analyticity?

I might have thought you'd use Cauchy-Riemann, but its a real valued function of a complex variable, whereas Cauchy-Riemann only applies to complex valued functions.

• Sorry I realise that maybe this should have been asked on the mathematics stackexchange, but I don't have the fortitude for a mathematicians answer – oweydd May 1 '17 at 20:54

Hint: the function $$f:x\mapsto\sqrt x$$ is not (real) analytic at $x=0$ (why?).