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 ex, cos(x) and sin(x) (assuming x is real) can be written as: and for complex z we define each of these function by the above series, replacing x with iz. 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.