I was looking at this video, and since I've been recently studying Complex Analysis, where the number to the power of an imaginary number tend to appear, there is the main doubt that I would like to ask: What is the most complete definition of what a number to the power of something (either number of matrix or whatever) is?
It may sound trivial, but since when you study maths in high school you study that a number to the power of another is that number multiplying itself the number of times defined by the second number, and this changes completely when you study more linear algebra, complex analysis, ... I started to see the case of a real number to the power of an integer as a particular case of the definition the general way of calculating numbers to the power of others, but I haven't been able to find a formal definition of this mathematical operation that makes compatible all of the previously mentioned ways of calculating exponents.
Another way I thought of seeing this is that the operation is just defined differently for each one of the possibilities (this is, a number to the power of an imaginary number just has a different definition than the operation of a real number to the power of another real number, although they are called the same). This would bring up more doubts to me, since combining different definitions in the case where you have a number to the power of a complex number, with its real and imaginary part, seems a bit strange to me.
