Euler's Formula: A Complete Guide
I find that the most elegant way to understand exponential-related things is by defining the exponential as the function satisfying f&#x27;=f (with suitable normalization). In other words, it is the &quot;eigenfunction&quot; of the derivative, which is a linear operator. This is a very natural thing to ask when studying derivatives, ODEs, etc.<p>With this definition, you can define the constants e and pi in a natural way. In particular, 2 * pi * i is the period of e^x. It then makes sense to analyse the function e^ix, with period 2*pi, and separate its real and imaginary parts. These are sure to be periodic real functions, and we can name them cos(x) and sin(x).<p>This Quora answer talks about this approach pretty well [1].<p>[1] <a href="https:&#x2F;&#x2F;qr.ae&#x2F;pNy7qp" rel="nofollow">https:&#x2F;&#x2F;qr.ae&#x2F;pNy7qp</a>