I was looking my material on Fourier transforms and I came across the following statements. In my notations, $L > 0$, $I$ denotes the identity matrix and $\mu$ a self-adjoint matrix. The sum: $$\sum_{p \in \frac{2\pi}{L}(\mathbb{Z} + \frac{1}{2})}\bigg{(}(-ipI + \mu)^{-1} + \frac{1}{ip}I\bigg{)} = \sum_{p \in \frac{2\pi}{L}(\mathbb{Z}+\frac{1}{2})}(-ipI + \mu)^{-1}$$ where, in the right hand side, the sum is understood as its principal part: $$\sum_{p\in \frac{2\pi}{L}(\mathbb{Z}+\frac{1}{2})}(-ipI+\mu)^{-1} = \sum_{p \in \frac{2\pi}{L}(\mathbb{Z}+\frac{1}{2})}\mu(p^{2}I+\mu^{2})^{-1}$$
What does this principal part mean? Why is the left hand side being interpreted as the right hand side in this last expression?