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.

  • $\begingroup$ 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 $\endgroup$ – oweydd May 1 '17 at 20:54

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

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    $\begingroup$ Oh I know! I know! $\endgroup$ – DanielSank May 1 '17 at 21:04
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    $\begingroup$ Thank you for the answer. I realise it was a simple question, but I'm not well versed with this kind of stuff, there was no need to be condescending @DanielSank $\endgroup$ – oweydd May 2 '17 at 19:21

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