Euler's Formula
Euler's Formula
The beautiful and perhaps mysterious formula of Euler which is the subject of this section is
Several questions might immediately come to mind.
What does an exponential function have to do with trigonometric
functions?
At the pre-calculus level we are familiar with 
as a function
which
increases rapidly as x grows, and with the oscillatory nature
of the trigonometric functions.
How do we make sense of raising a real number to an imaginary power? Our rules of arithmetic have only told us how to extend addition and multiplication from the real numbers to the complex numbers.
We will eventually give a complete and airtight answer to these
questions in
the section on complex functions(link), but we can get acquainted with
Euler's formula and strip away some of its mystery by extrapolating a
few simple properties of the real function .
First, if 
then the equation which we obtain
by replacing i with -i should also be true. After all
-i is as good a square root of -1 as i.
So we should also have 
. And if we multiply
these together,
so that 
So we know at least that
for some angle 
. But does 
?
Recall that the exponential function 
is well approximated
by the linear function 1+x when x is very small. (Try it on a
calculator.) We will now assume that the function 
retains this
property for complex z with small modulus.
so in particular
And now we use two more approximations which are very good for small
values of 
. (try it)
and
These approximations become increasingly good as 
decreases
and it is clear that Euler's formula holds when 
.
Putting all this together we boldly claim:
Notice that by de Moivre's formula from the previous section, this means that
And since any angle can be written as an integer multiplied by something ``sufficiently small!'', our claim is bold indeed.
What we should notice at this stage, is that if we want to extend the
function to complex values of x in a way consistent with
what we already know about the function, then it is very reasonable
to expect oscillatory behavior from 
and perhaps also
reasonable to accept Euler's lovely formula.
We will soon analyze and understand the formula completely.
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