When and WHY did mathematicians start turning their attention to imaginary exponents? I read on Wikipedia about Euler's correspondences with Bernouille and such, but it doesn't answer what exactly triggered this. What did they hope to achieve by exploring such things? Why did they not do this earlier?
I have recently read the history of fractional exponents and how they were a suggested notation according to the laws of exponents by Nicole Oresme for numbers which were not "nice" (natural number) powers of each other (e.g. 4 and 8, $\frac{3}{2}$; and expressions like $5^\frac{1}{3}$ were taken to mean the number whose cube is 5). I don't know whether Oresme's work had the same philosophy of exponents as Euler's though!