# 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),

*i.e.*, .

## Rational Exponents

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

*e.g.*, ). In the general case of th roots, there are distinct values, in general. After proving Euler's identity, it will be easy to find them all (see §3.11). As an example, , , , and , since .

## 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.^{3.1}An *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 as^{3.2}

*rational*number. That is, it can be rewritten as an integer divided by another integer. For example,

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

*limit*of as goes to infinity is .

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

*real*exponents.

## 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 (see Problem 8). We have , so the first term is no problem. But what is ? In other words, what is the derivative of at ? Once we find the successive derivatives of at , we will have the definition of for any complex .

## 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

^{3.3}where denotes the log-base- of .

^{3.4}Formally, the chain rule tells us how to differentiate a function of a function as follows:

*i.e.*,

##
Back to *e*

Above, we defined as the particular real number satisfying

or

Numerically, is a transcendental number (a type of irrational
number^{3.5}), so its decimal expansion never repeats.
The initial decimal expansion of is given by^{3.6}

## 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

so that

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

*e.g.*, . When , we get the same thing as when . When , we get the same thing as when , and so on, so there are only distinct cases. Thus, we may define the th th-root of as

## Roots of Unity

Since for every integer , we can write

*th roots of unity*. The special case is called a

*primitive th root of unity*, since integer powers of it give all of the others:

^{3.7}We may also call a

*generator*of the mathematical

*group*consisting of the th roots of unity and their products.

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:

*integer*using mathematical induction and elementary trigonometric identities.

*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

From trigonometry, we have the following sum-of-angle identities:

These identities can be proved using only arguments from classical
geometry.^{3.8}Applying 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`

**Next Section:**

Sinusoids and Exponentials

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Complex Numbers