Artsy chaos: the secret life of a class of trigonometric sums

ONU McIntosh Center; Activities Room

We start from classical trigonometric sums (of terms such as k^n*cos(k), k^n*sin(k) - where n is a positive integer). These classical sums allow fairly straightforward closed form representations. In our work we considered changing the arguments of the trigonometric factors to powers (so that they get replaced by cos(k^a) and sin(k^a) - for a positive real exponent that may or may not be an integer), while also introducing in any term of such a sum a "rotational" factor of the form omega^k, where "omega" is a complex number of modulus 1 (that may or may not be a root of unity). As an interesting outcome, we discover that our modified sums reveal surprisingly esthetic (”artsy”) chaotic complex plots (courtesy of the MAPLE computer algebra system), especially around the exponent a~1.29!