Today I mentioned the famous Euler’s formula briefly in my calculus class (when discussing hyperbolic functions, lecture notes here):

*

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).

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One of the usual ways to lớn derive this formula is by comparing the power nguồn series of the exponential function và the trigonometric functions

*
*
:

*

Putting

*
in the first expansion and comparing with the remaining two, it’s easy to see that

*

Here, I am going lớn 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 & 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 lớn be

*

From this it is easy lớn see that

*

It is then natural lớn 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 gọi
*
and
*
the modulus (or length) & the argument (or angle) of
*
respectively.

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*


The complex plane

The complex multiplication of

*
*
is then

*

i.e. The modulus of

*
is the product of the two moduli & the argument of
*
is the sum of the two arguments.

So now, let’s fix

*
và compute

*

which by definition would be

*
. We will argue that its length is
*
and its argument is
*
, i.e. (1) holds:

*


*


The geometry of the powers of 1+yi

lớn see this, let

*
. Then
*
và so

*

From this we have

*
^\frac12n\\ \rightarrow& \displaystyle \left(e^x^2\right)^0=1 \ \ \ \ \ (3)\endarray " class="latex" />

as

*
. On the other hand, the argument of
*
(which is well-defined up lớn a multiple of
*
) can be chosen to be

*

Then by the L’Hôpital’s rule,

*

So we have

*
= \lim_n\rightarrow \infty n\arg(z_n)= \lim_n\rightarrow \infty n\theta_n=x. \ \ \ \ \ (4)\endarray " class="latex" />

Combining (3) & (4), we have

*

Added Nov 12, 2017:I found a clip explaining

*
(but without giving the full mathematical details) in the above approach: