Re:Why does e to pi i equal -1? (2015) [video]
by cousin_it on Tuesday February 14, 2017 @03:01PM
(#13646613)
Sure, step 1 relies on the existence of e^z which is not obvious. Step 2 relies on the chain rule over C, step 4 relies on having a unique solution to a differential equation, etc. My comment can be expanded to a proof, but it was more aimed to explain why Euler's identity is natural.
Re:Why does e to pi i equal -1? (2015) [video]
by ouid on Tuesday February 14, 2017 @07:04PM
(#13648536)
step 1 isn't "not obvious"; it's nonsense. What, exactly, are you trying to prove exists? e^z? e^z is undefined at this point. If you're trying to prove that e^i\pi=-1, you need to start with a definition of e^z.
There exists no logical path to the statement from the previous definition of e^x because, up to this point, e^x is a function on real numbers. e^ix is nonsense until you define e^x in complex coordinates, at which point the proof needs to rely on properties of that definition.
what you have posted misses the nature of the insight.
If two analytic functions on the complex numbers agree on uncountably infinitely many points, then they agree everywhere they are defined. This means that there is a unique analytic function that maps (ix)[x \in R] to the circle and (x)[x \in R] to the natural exponential, and they are the same complex function. Without the knowledge that analytic continuations are unique, the statement is entirely (pun intended) meaningless.