4 ms·
Even e^(ix) = cosx + i*sinx is an obvious result of the definition i^2= - 1.
by jleahy 5y ago
Even e^(ix) = cosx + i*sinx is an obvious result of the definition i^2= - 1.
- sobriquet9 5y agoIn hindsight, and if you know Mclaurin series. Complex numbers were known and used for many years before Euler found this obvious result.
- red_trumpet 5y agoIf you already know the power series expansions of cos and sin then yes. But then you put the burden of proof into showing that the power series of cos and sin indeed define the same functions as the trigonometric definition.
- japanuspus 5y agoI do not think this is true: you can introduce the complex numbers (including the imaginary unit `i`) without defining exponentiation or introducing `e`. As a consequence, no properties of `e` can be deduced from the definition of the imaginary unit. In particular, the result is not "obvious".