In the PDE as below$$ \partial_t u - \frac{i}{\rho} ( - \partial_x^2 )^{\rho /2} u = 0 \;\;\;(t,x) \in \Bbb R^2 $$ How can I prove that $$ (- \partial_x ^2 )^{\rho/ 2} = \scr F ^{-1} | \xi |^\rho \scr F $$ where $\scr F$ is Fourier transform. Is this just definition?
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