# About fourier transform in the PDE

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?

-
 What is the meaning of $(-\partial_x^2)^{\frac{\rho}{2}}u$ ? – doraemonpaul Oct 2 '12 at 9:42