# Fast Fourier Transform Series

Why is the following equality the same?

$\sqrt{1 \over 2N} \sum\limits_{j=-N}^N'' \exp({ijk\pi \over N})g({j\pi \over N})=\sqrt{1 \over 2N} \sum\limits_{j=0}^{2N-1} \exp({ijk\pi \over N})g({j\pi \over N}),$ where $\sum''$ means the first and the last value of the seies are halved and $i$ is a complex number.