Given a set of the k first eigenvalues $ (\lambda_i)_ 1 ^k $ of some operator , and a set of the first k orthomormal eigenfunctions for these eigenvalues : $ ( \phi_i ) $ . Define: $ \Phi(x,y) = \sum_{i=1}^k \phi_i(x) \phi_i(y) $ and then define the fourier transform of this function: $ \hat \Phi (z,y)= (2 \pi)^{-n/2} \int_{x \in \mathbb{R} ^ n } \Phi(x,y)e^{ix \cdot z} dx$.

Can someone explain me the second equality in the following: $ z_j \hat{\Phi} (z,y) = (2 \pi)^{-n/2} \int_{\mathbb{R}^n } \Phi(x,y)z_j e^{ixz} dx = (2 \pi)^{-n/2} \int_{\mathbb{R}^n } \Phi(x,y)(-i) \frac{\partial}{\partial x_j } e^{ix \cdot z } dx $

BTW- What does the notation $ z_j $ means in this context?

Hope someone will be able to help me

Thanks in advance