definite integral

. This is seen directly be using complex numbers, by the Euler formula, then this is the imaginary part. The sum over i is potencing the exponential function like the q in the geometric series, but inifinte up to now. There is a closed formula for the finite geometric series. This quotient is rewritten and the imaginary part taken.

 

. This is analoguous to  making use of the approximated trunketed Taylor or MacLaurin series expansion if the trigonometric funstions sine and cosine at half argument.

 

The integral interpretation is much more simple:

It is the integral:  in the limit n->∞. Perhaps a Riemannian sum definition with a equisdistant partition π/n for each n, summing over n+1 intervals. Taking into account sin(0)=0..