# 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

(Euler's Identity)

To prove'' this, we will first define what we mean by  ''. (The right-hand side, , is assumed to be understood.) Since is just a particular real number, we only really have to explain what we mean by imaginary exponents. (We'll also see where comes from in the process.) Imaginary exponents will be obtained as a generalization of real exponents. Therefore, our first task is to define exactly what we mean by , where is any real number, and is any positive real number.

## Positive Integer Exponents

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

where is real.

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)
Note that property (1) implies property (2). We list them both explicitly for convenience below.

## The Exponent Zero

How should we define in a manner consistent with the properties of integer exponents? Multiplying it by gives

by property (1) of exponents. Solving for then gives

## Negative Exponents

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

Dividing through by then gives

Similarly, we obtain

for all integer values of , i.e., .

## Rational Exponents

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

Applying property (2) of exponents, we have

Thus, the only thing new is . Since

we see that is the th root of . This is sometimes written

The th root of a real (or complex) number is not unique. As we all know, square roots give two values (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.3.1An 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

Every truncated, rounded, or repeating expansion is a rational number. That is, it can be rewritten as an integer divided by another integer. For example,

and, using to denote the repeating part of a decimal expansion, a repeating example is as follows:

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

That is, the limit of as goes to infinity is .

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

We have now defined what we mean by real exponents.

## A First Look at Taylor Series

Most smooth'' functions can be expanded in the form of a Taylor series expansion:

This can be written more compactly as

where ' is pronounced  factorial''. An informal derivation of this formula for is given in Appendix E. Clearly, since many derivatives are involved, a Taylor series expansion is only possible when the function is so smooth that it can be differentiated again and again. Fortunately for us, all audio signals are in that category, because hearing is bandlimited to below kHz, and the audible spectrum of any sum of sinusoids is infinitely differentiable. (Recall that and , etc.). See §E.6 for more about this point.

## 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

 (3.1)

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

For , we thus have .

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

and use the chain rule,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:

Evaluated at a particular point , we obtain

In this case, so that , and which is its own derivative. The end result is then , i.e.,

## Back to e

Above, we defined as the particular real number satisfying

which gave us when . From this expression, we have, as ,

or

This is one way to define . Another way to arrive at the same definition is to ask what logarithmic base gives that the derivative of is . We denote by .

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

Any number of digits can be computed from the formula by making sufficiently small.

## 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:

The simplicity comes about because for all and because we chose to expand about the point . We of course define

Note that all even order terms are real while all odd order terms are imaginary. Separating out the real and imaginary parts gives

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

where and are real numbers. Since, by Euler's identity, for every integer , we also have

Taking the th root gives

There are different results obtainable using different values of , 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

These are the th-roots of the complex number .

## Roots of Unity

Since for every integer , we can write

These are the 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:

The th roots of unity are so frequently used that they are often given a special notation in the signal processing literature:

where denotes a primitive th root of unity.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

where , , and is the sampling interval in seconds.

## Direct Proof of De Moivre's Theorem

In §2.10, De Moivre's theorem was introduced as a consequence of Euler's identity:

To provide some further insight into the mechanics'' of Euler's identity, we'll provide here a direct proof of De Moivre's theorem for 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.,

 (3.2)

Since it is true by hypothesis that

multiplying both sides by yields
 (3.3)

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

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Sinusoids and Exponentials
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Complex Numbers