Exotic expansions of solutions to an ordinary differential equation

First we study series of pure imaginary powers of the variable with constant coefficients. We show that they can correspond to functions with very complicated singularities. Next we consider expansions over complex power exponents with coefficients, which are either constants or polynomials in logarithm of the variable, and their power exponents are in an angle of the complex plane and there are infinitely many terms with fixed real parts of power exponents. We call such expansions as exotic. We show a way of computing such expansions of solutions to ordinary differential equations of very general form. Some examples are considered as well.