info prev up next book cdrom email home

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)

\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.

© 1996-9 Eric W. Weisstein