is the definition of the complex exponential function arbitrary?

Question

If I cite my textbook the complex exponential function is defined as:

$$ e^{\theta i} = cos \theta + i \sin \theta $$

Is this just an arbitrary definition or is it possible to prove this somehow?

Answer 1

The actual definition, for a general complex number $z =x + i y$ would be $$ e^z = e^{x+iy} = e^x(\cos y + i \sin y ) $$ the formula you mention corresponds to the particular choice $x = 0$. Anyway, this is not an arbitrary choice, we can approach this from different directions, one of which is that this is the only way of extending the usual exponential to the complex plane as an analytical function.