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: