Here is a proof of Euler's formula using Taylor series expansions
as well as basic facts about the powers of "i":

and so on. The functions "e", cos "x" and sin "x" of the (real) variable "x" can be expressed using their Taylor expansions around zero:

For complex "z" we "define" each of these functions by the above series, replacing the real variable "x" with the complex variable "z". This is possible because the radius of convergence of each series is infinite. We then find that

The rearrangement of terms is justified because each series is absolutely convergent. Taking "z" = "x" to be a real number gives the original identity as Euler discovered it.

Differential equation definition
अन्य उपयोग
Using calculus
Using Taylor series