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Exponential Sum Formulas

sum_(n=0)^(N-1)e^(inx)=(1-e^(iNx))/(1-e^(ix))
(1)
=(-e^(iNx/2)(e^(-iNx/2)-e^(iNx/2)))/(-e^(ix/2)(e^(-ix/2)-e^(ix/2)))
(2)
=(sin(1/2Nx))/(sin(1/2x))e^(ix(N-1)/2),
(3)

where

sum_(n=0)^(N-1)r^n=(1-r^N)/(1-r)
(4)

has been used. Similarly,

sum_(n=0)^(N-1)p^ne^(inx)=(1-p^Ne^(iNx))/(1-pe^(ix))
(5)
sum_(n=0)^(infty)p^ne^(inx)=1/(pe^(ix)-1)
(6)
=(1-pe^(-ix))/(1-2pcosx+p^2).
(7)

By looking at the real and imaginary parts of these formulas, sums involving sines and cosines can be obtained.

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Cite this as:

Weisstein, Eric W. "Exponential Sum Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialSumFormulas.html

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