Exponential Sum Formulas

$\displaystyle \sum_{n=0}^{N-1} e^{inx}$	$\textstyle =$	$\displaystyle {1-e^{iNx}\over 1-e^{ix}} = {-e^{iNx/2}\left({e^{-iNx/2}-e^{iNx/2}}\right)\over -e^{ix/2}\left({e^{-ix/2}-e^{ix/2}}\right)}$
	$\textstyle =$	$\displaystyle {\sin({\textstyle{1\over 2}}Nx)\over \sin({\textstyle{1\over 2}}x)} e^{ix(N-1)/2},$	(1)

where

$\begin{displaymath} \sum_{n=0}^{N-1} r^n = {1-r^N\over 1-r} \end{displaymath}$

(2)

has been used. Similarly,

$\displaystyle \sum_{n=0}^{N-1}p^ne^{inx}$	$\textstyle =$	$\displaystyle {1-p^Ne^{iNx}\over 1-pe^{ix}}$	(3)
$\displaystyle \sum_{n=0}^\infty p^ne^{inx}$	$\textstyle =$	$\displaystyle {1\over e^{ipx}-1} = {1-pe^{-ix}\over 1-2p\cos x+p^2}.$	(4)

By looking at the Real and Imaginary Parts of these Formulas, sums involving sines and cosines can be obtained.