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 (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
The Exponent Zero
How should we define in a manner consistent with the properties of integer exponents? Multiplying it by gives
What should be? Multiplying it by gives, using property (1),
A rational number is a real number that can be expressed as a ratio of two finite integers:
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
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:
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
Back to e
Above, we defined as the particular real number satisfying
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
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
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:
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.3.8Applying 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).
Sinusoids and Exponentials