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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!

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    $\begingroup$ It was quite obvious for us in the school, about at 15-16 years old. There was practically no internet at the time. We heard about complex numbers and could multiply and divide them, but did not know about the imaginary exponents. We have thought about them, but absolute no result. Then, once one of my classmates got a paper, with the taylor expansion-based calculation. It was... an extreme feeling. But we were no mathematicians, only ordinary high schoolers specialized for math. So I think, the problem was very obvious for the Mathematicians of the time. $\endgroup$ – peterh May 14 '20 at 22:35
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    $\begingroup$ The function $e^x$ is well-known in real analysis. As soon as Euler, Gauss, Riemann, Cauchy, etc. started doing complex analysis, the function $e^z$ was a natural. Probably the logarithm came first. Roger Cotes (1714) noted $ix = \log(\cos x + i \sin x)$ $\endgroup$ – Gerald Edgar May 15 '20 at 10:50
  • $\begingroup$ Mathematicians by nature can't resist "extending" any new concept to see if they can either discover some new properties / theorems or break existing ones. Take a look at nonEuclidean Geometry, or Godel's Incompleteness Theorem. $\endgroup$ – Carl Witthoft May 15 '20 at 10:55
  • $\begingroup$ Related post, particularly in view of Alexandre Eremenko's answer: hsm.stackexchange.com/questions/11801/… $\endgroup$ – Danu♦ May 17 '20 at 14:59
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The original object of study was logarithm, rather then exponent. Logarithm (discovered by Napier) was an important function, both in pure and applied mathematics. So people started asking what the log of a negative number can be. Research on this question quickly lead to paradoxes. (Which still cause difficulties to students).

The solution of these paradoxes is mostly due to Euler, who explained that understanding of logarithm was impossible without considering complex values (of argument and of the function). Moreover, he understood that logarithm is a "multivalued function". To simplify the matter, he introduced the inverse which is an honest, single valued function of a complex variable. This is the genesis of the exponential function.

L. Euler, Sur les logatithms des nombres negatifs et imaginaires

L. Euler, De la controverse entre Mrs. Leibnitz et Bernoulli sur les logarithmes des nombres négatifs et imaginaires

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