Euler Theorem

eulerAs indicated earlier when discussing the scope of numbers, Euler’s identity is very special example of an algebraic equation encompassing exponential, complex and negative numbers with some important geometric and trigonometric implications.

[1]     1

However, while many texts on electromagnetism and wave theory use a notation that stems from this equation, there is often little explanation of the mathematical rationale (and shorthand) adopted as a result of this theorem. Many discussions of wave theory represent the superposition of a sinusoidal wave via one of the following function:

[2]      2

Both these equations can be expanded as follows:

[3]      3

However, the expansion and manipulation of these functions can be both cumbersome and tedious. It is for this reason that mathematicians have devised an equivalent approach using complex numbers and, in this context, we could view the introduction of complex numbers, as many before, as a useful mathematical trick without any debate of the perceived philosophical implications associated with complex numbers. The correlation of Euler’s identity with the trigonometric functions begins with the generalised exponential series expansion:

[4]      e   = 1 + 1/1! + 1/2! + … 1/n!       » 2.718

[5]      ex = 1 + x1/1! + x2/2! + …xn/n!

Similar expansions exist for sine and cosine:

[6]      cos x = 1 – x2/2! + x4/4! - …xn/n!

[7]      sin x   = x – x3/3! + x5/5! - … xn/n!

It is tempting to compare equation [6] and [7] with [5], but we can see that the sign of some components differ, but if we introduce the complex number [i] into equation [5], we get:

[8]      eix = 1 + ( ix)1/1! + (ix)2/2! + …(ix)n/n!

However, by definition (i)2 = -1, which means we can arrive at Euler’s formula by sorting [8] into the form of [6] and [7]:

[9]      eix = [1 - (x)2/2! + (x)4/4! -...] + [ ( ix)1/1! - ( ix)3/1! +…]

We can now substitute [6] and [7] into [9] to get:

[10]    4

Although there are requirements for complex angles, there are generally more applications involving a wave function of the form A.cos(x+θ).  We can therefore use the basic form of equation [10] to represent this wave function as follows:

[11]    5

Where Re{} simply implies that only the real part of the expression is being used and in most cases this aspect of the notation is often dropped, which is often a fact that is not always realised, but which leads to:

[12]    6

When dealing with complex numbers, it is often convenient to transform between Cartesian and polar coordinates. Using Euler’s formula it is possible to transform a complex number, [a+ib], into the form [re], where a, b, r and θ are all real numbers. Note, in this example, we have constrained the discussion to 2 dimensions, rather than 3 dimensions, as it still retains the essence of the argument, while making diagrams a lot easier to draw:

[13]    re= r.cosθ + ir.sinθ = a + ib

coordinatesFrom the diagram, in conjunction with the basic equations of trigonometry, we can define the transforms as:

a = r.Cosθ
b = r.Sinθ
r = √(a2 + b2)

Note: The trigonometric value of [θ] depends on the sign of [a] and [b], which can place [a+ib] in any of the four quadrants.

Finally, it is worth just introducing the concept of the complex conjugate, which essentially amounts to changing the sign of the imaginary part of the complex number:

z   = a + ib             complex number
z* = a – ib              its complex conjugate

Again, the use of the complex conjugate can be seen as simply another mathematical trick that eliminates the imaginary component of a complex number, but leaves the real component, which can lead to a simplification of complex equations, e.g.

7

So, no matter how complicated the equation, the complex conjugate is obtained simply by negating the sign in front of the imaginary component of the complex number. However, this trick also leads to a general method of obtaining the real part of a complex expression, i.e. add the complex conjugate and divide by 2:

[14]    Re{z} = ( z + z* )/2

If we apply this method to Euler’s formula itself, we get:

[15]   8

In the context of this discussion, Euler’s formula and complex numbers are just seen as a convenient method of finding the real component of a wave function. As such, there is a case that the final presentation of any conclusion should also be descriptive and not predicated solely on the mathematical methodology used to derive the conclusion. However, this point may be strengthened by quoting Clerk Maxwell:

Mathematicians may flatter themselves that they possess new ideas which mere human language is as yet unable to express. Let them make the effort to express these ideas in appropriate words without the aid of symbols, and if they succeed they will not only lay us laymen under a lasting obligation, but, we venture to say, they will find themselves very much enlightened during the process, and will even be doubtful whether the ideas as expressed in symbols had ever quite found their way out of the equations into their minds.