- #1
madness
- 815
- 70
- TL;DR Summary
- Sum rather than integral versions of orthogonality relations
Hi all,
I've come across some problem where I have terms such as ##\sum_{j=1}^N \cos(2 \pi j k /N) \cos(2 \pi j k' /N)##, or ##\sum_{j=1}^N \cos(2\pi j k/ N)##, or ## \sum_{j=1}^N \cos(2\pi j k/ N) \cos(\pi j) ##. In all cases we have the extra condition that ##1 \le k,k' \le N/2-1## (and ##k,k'## are integers of course, and ##N## is even.).
If these were integrals rather than sums I would be able to apply standard orthogonality relations. In fact, they seem to obey orthogonality relations when I compute them in Matlab (e.g., ##\sum_{j=1}^N \cos(2 \pi j k /N) \cos(2 \pi j k' /N) = N/2 \delta_{k,k'}##). However, if I plug them into Mathematica I get quite nasty expressions out involving functions like sec, cosec, etc. Does anyone know if there are any simple relations for these things?
I've come across some problem where I have terms such as ##\sum_{j=1}^N \cos(2 \pi j k /N) \cos(2 \pi j k' /N)##, or ##\sum_{j=1}^N \cos(2\pi j k/ N)##, or ## \sum_{j=1}^N \cos(2\pi j k/ N) \cos(\pi j) ##. In all cases we have the extra condition that ##1 \le k,k' \le N/2-1## (and ##k,k'## are integers of course, and ##N## is even.).
If these were integrals rather than sums I would be able to apply standard orthogonality relations. In fact, they seem to obey orthogonality relations when I compute them in Matlab (e.g., ##\sum_{j=1}^N \cos(2 \pi j k /N) \cos(2 \pi j k' /N) = N/2 \delta_{k,k'}##). However, if I plug them into Mathematica I get quite nasty expressions out involving functions like sec, cosec, etc. Does anyone know if there are any simple relations for these things?