Proof of Euler's Identity
This chapter outlines the proof of Euler's Identity, which is an important tool for working with complex numbers. It is one of the critical elements of the DFT definition that we need to understand.
Euler's Identity
Euler's identity (or ``theorem'' or ``formula'') is








Positive Integer Exponents
The ``original'' definition of exponents which ``actually makes sense'' applies only to positive integer exponents:


Generalizing this definition involves first noting its abstract mathematical properties, and then making sure these properties are preserved in the generalization.
Properties of Exponents
From the basic definition of positive integer exponents, we have
- (1)
-
- (2)
-
The Exponent Zero
How should we define in a manner consistent with the
properties of integer exponents? Multiplying it by
gives




Negative Exponents
What should be? Multiplying it by
gives, using property (1),






Rational Exponents
A rational number is a real number that can be expressed as a ratio of two finite integers:







![$\displaystyle \zbox {a^{\frac{1}{M}} \isdef \sqrt[M]{a}.}
$](http://www.dsprelated.com/josimages_new/mdft/img250.png)




![$ \sqrt[4]{1}=1$](http://www.dsprelated.com/josimages_new/mdft/img252.png)




Real Exponents
The closest we can actually get to most real numbers is to compute a
rational number that is as close as we need. It can be shown that
rational numbers are dense in the real numbers; that is,
between every two real numbers there is a rational number, and between
every two rational numbers is a real number.3.1An irrational number can be defined as any real
number having a non-repeating decimal expansion. For example,
is an irrational real number whose decimal expansion starts
out as3.2



![\begin{eqnarray*}
x &=& 0.\overline{123} \\ [5pt]
\quad\Rightarrow\quad 1000x &=...
...999x &=& 123\\ [5pt]
\quad\Rightarrow\quad x &=& \frac{123}{999}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img261.png)
Other examples of irrational numbers include

Their decimal expansions do not repeat.
Let
denote the
-digit decimal expansion of an arbitrary real
number
. Then
is a rational number (some integer over
).
We can say




Since
is defined for all
, we naturally define
as the following mathematical limit:

A First Look at Taylor Series
Most ``smooth'' functions can be expanded in the form of a
Taylor series expansion:








Imaginary Exponents
We may define imaginary exponents the same way that all
sufficiently smooth real-valued functions of a real variable are
generalized to the complex case--using Taylor series. A
Taylor series expansion is just a polynomial (possibly of infinitely
high order), and polynomials involve only addition, multiplication,
and division. Since these elementary operations are also defined for
complex numbers, any smooth function of a real variable may be
generalized to a function of a complex variable
by simply
substituting the complex variable
for the real variable
in the Taylor series expansion of
.
Let
, where
is any positive real number and
is
real. The Taylor
series expansion about
(``Maclaurin series''),
generalized to the complex case is then
This is well defined, provided the series converges for every finite









Derivatives of f(x)=a^x
Let's apply the definition of differentiation and see what happens:

Since the limit of
as
is less than
1 for
and greater than
for
(as one can show via direct
calculations), and since
is a continuous
function of
for
, it follows that there exists a
positive real number we'll call
such that for
we get



So far we have proved that the derivative of is
.
What about
for other values of
? The trick is to write it as












Back to e
Above, we defined as the particular real number satisfying





or







Numerically, is a transcendental number (a type of irrational
number3.5), so its decimal expansion never repeats.
The initial decimal expansion of
is given by3.6



e^(j theta)
We've now defined for any positive real number
and any
complex number
. Setting
and
gives us the
special case we need for Euler's identity. Since
is its own
derivative, the Taylor series expansion for
is one of
the simplest imaginable infinite series:






Comparing the Maclaurin expansion for
with that of
and
proves Euler's identity. Recall
from introductory calculus that
![\begin{eqnarray*}
\frac{d}{d\theta}\cos(\theta) &=& -\sin(\theta) \\ [5pt]
\frac{d}{d\theta}\sin(\theta) &=& \cos(\theta)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img321.png)
so that
![\begin{eqnarray*}
\left.\frac{d^n}{d\theta^n}\cos(\theta)\right\vert _{\theta=0}...
...} \\ [5pt]
0, & n\;\mbox{\small even}. \\
\end{array} \right.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img322.png)
Plugging into the general Maclaurin series gives

Separating the Maclaurin expansion for
into its even and odd
terms (real and imaginary parts) gives

thus proving Euler's identity.
Back to Mth Roots
As mentioned in §3.4, there are different numbers
which satisfy
when
is a positive integer.
That is, the
th root of
, which is
written as
, is not unique--there are
of them. How do
we find them all? The answer is to consider complex numbers in
polar form.
By Euler's Identity, which we just proved, any number,
real or complex, can be written in polar form as























Roots of Unity
Since
for every integer
,
we
can write











We will learn later that the th roots of unity are used to generate
all the sinusoids used by the length-
DFT and its inverse.
The
th complex sinusoid used in a DFT of length
is given by




Direct Proof of De Moivre's Theorem
In §2.10, De Moivre's theorem was introduced as a consequence of Euler's identity:
![$\displaystyle \zbox {\left[\cos(\theta) + j \sin(\theta)\right] ^n =
\cos(n\theta) + j \sin(n\theta), \qquad\hbox{for all $n\in{\bf R}$}}
$](http://www.dsprelated.com/josimages_new/mdft/img348.png)

Proof:
To establish the ``basis'' of our mathematical induction proof, we may
simply observe that De Moivre's theorem is trivially true for
. Now assume that De Moivre's theorem is true for some positive
integer
. Then we must show that this implies it is also true for
, i.e.,
Since it is true by hypothesis that
![$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^n =
\cos(n\theta) + j \sin(n\theta),
$](http://www.dsprelated.com/josimages_new/mdft/img352.png)
![$ [\cos(\theta) + j \sin(\theta)]$](http://www.dsprelated.com/josimages_new/mdft/img353.png)
From trigonometry, we have the following sum-of-angle identities:

These identities can be proved using only arguments from classical
geometry.3.8Applying these to the right-hand side of Eq.(3.3), with
and
, gives Eq.
(3.2), and
so the induction step is proved.
De Moivre's theorem establishes that integer powers of
lie on a circle of radius 1 (since
, for all
). It
therefore can be used to determine all
of the
th roots of unity
(see §3.12 above).
However, no definition of
emerges readily from De Moivre's
theorem, nor does it establish a definition for imaginary exponents
(which we defined using Taylor series expansion in §3.7 above).
Euler_Identity Problems
See http://ccrma.stanford.edu/~jos/mdftp/Euler_Identity_Problems.html
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Sinusoids and Exponentials
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Complex Numbers