I have a question about the famous identity of Euler $e^{i\pi}=-1$. I opened the other day this question about the number of roots of a complex number with irrational exponent.
Under this light and applying to Euler identity, it is the identity of Euler just a main value of infinity countable roots of $e^{i\pi}$? Its a convention or it is the unique value for the expression? Thanks in advance.
EDIT: I will add some thoughts in relation to the comment of @Travis... may exist a "secret relation" under the surface when you do exactly this kind of exponentiation $e^{i\pi x}$ that "evade" the existence of infinite countable roots for the expression.
My thoughts are based in this analogy: $a^b=e^{b\log a}$. When you do this change of base we are putting, nearly sure, a irrational number on the exponent ($\log a$ for mainly all cases will be irrational) so if $a^b$ (with $a,b \in \Bbb N$) is equivalent to $e^{b\log a}$ then we must assume one of three things:
1) This specific relation with irrational base $e$ and irrational exponent $b\log a$ dont create countable infinity roots, and analogously it doesnt happen with expressions derived from $e^{i\pi}$.
2) Or maybe that an irrational exponent doesnt means countable infinite roots, what is really counter-intuitive because you can decompose a irrational number in a countable infinite series of fractions (and a fraction as exponent is the expression for a finite number of roots).
3) Some arithmetic operations doesnt show the real nature of the operation, they have some degree of ambiguity for complex realm.
So these type of things that happen with exponentiation on the complex realm could show a "hide" relation between some specific irrational numbers.
It is the unique value. For an exponential expression we write
$$a^b = \exp(b \log a),$$
and this indeed depends on a choice $\log$ of branch cut.
On the other hand, $z \mapsto \exp z$ requires no choice of branch cut, so there is no ambiguity in expressions like $\exp (i \pi)$.