where is a solution to (usually denoted by “”, but indeed there is no single-valued square root for complex numbers, or even negative real numbers).
Putting in the first expansion and comparing with the remaining two, it’s easy to see that
Here, I am going to give another approach which does not require any knowledge of series (thus avoid the problem of convergence), but only basic knowledge in complex number (complex addition and multiplication). I am sure this approach must has been taken before but I couldn’t find a suitable reference, especially an online one.
Let us agree that we define the Euler’s number to be
From this it is easy to see that
It is then natural to define the complex exponential function by
Here I am cheating a little bit because I have implicitly assumed that this limit exists.
Now recall the geometry of the complex plane. We can identify a complex number with the point on the plane. We can write a complex number in its polar form , which is identified with in polar coordinates. We call and the modulus (or length) and the argument (or angle) of respectively.
i.e. the modulus of is the product of the two moduli and the argument of is the sum of the two arguments.
So now, let’s fix and compute
which by definition would be . We will argue that its length is and its argument is , i.e. (1) holds:
To see this, let . Then and so
as . On the other hand, the argument of (which is well-defined up to a multiple of ) can be chosen to be
Then by the L’Hôpital’s rule,
Added Nov 12, 2017:
I found a video explaining (but without giving the full mathematical details) in the above approach: