In post 18.15, we saw that the number e raised to the power x (see post 18.2) is defined by the series (post 17.11)
ex = 1 + x +x2/2! + x3/3! + x4/4! + x5/5! + …
Here, for example, 5! = 5 × 4 × 3 × 2 × 1.
In post 18.6, we saw that the sine and cosine of the angle ϑ, measured in radians (post 17.11) could also be expressed as the series
sinϑ = ϑ – ϑ3/3! + ϑ5/5! – ϑ7/7! + ϑ9/9! -…
cosϑ = 1 – ϑ2/2! + ϑ4/4! – ϑ6/6! + ϑ8/8! – ϑ10/10! +…
Now let’s think about the number eiϑ, where i represents the square root of -1, as discussed in post 18.16; this means that, by definition, i2 = -1. It follows, from equation 1, that
eiϑ = 1 + iϑ + (iϑ)2/2! + (iϑ)3/3! + (iϑ)4/4! + (iϑ)5/5! + (iϑ)6/6! + (iϑ)7/7! + (iϑ)8/8! …
= 1 + iϑ – ϑ2/2! – iϑ3/3! + ϑ4/4! + iϑ5/5! – ϑ6/6! – iϑ7/7! + ϑ8/8! …
= (1 – ϑ2/2! + ϑ4/4! – ϑ6/6! + ϑ8/8! …) + (iϑ – iϑ3/3! + iϑ5/5! – iϑ7/7! + …)
= (1 – ϑ2/2! + ϑ4/4! – ϑ6/6! + ϑ8/8! …) + i(ϑ – ϑ3/3! + ϑ5/5! – ϑ7/7! + …)
= cosϑ + isinϑ.
This result is called “Euler’s relation”, named after the Swiss mathematician Leonhard Euler (his last name is pronounced like the English word “oiler” https://forvo.com/word/leonhard_euler/#de) who lived from 1707 to 1783.
According to post 18.16, we can represent cosϑ + isinϑ and, therefore, eiϑ, in two dimensions by the diagram below.
Its modulus (post 18.16) is then given by cos2ϑ + sin2ϑ = 1 (see appendix 4 of post 16.50) and its argument (post 18.16) is ϑ. As ϑ increases, the point P moves round in a circle of radius 1. So the complex number (post 18.16) aeiϑ represents the position of a point P that moves around a circle of radius a as ϑ increases.
The picture below is copied from post 18.11; it represents a point, initially at P moving around a circle, of radius a, with an angular speed (post 17.12) ω.
After a time t has elapsed, it reaches P’. From the previous paragraph, we can see that, at any given time t, we can represent the position of P’ by aeiωt. Sometimes this is written aexp(iωt) to avoid writing so much information in the superscript iωt. This method of representing the position of an object going round in a circle is an alternative to the trigonometric representation given in post 18.11.
We also saw, in post 18.11, that the projections of OP’ can represent a simple harmonic oscillator or a wave. The projection on OX gives the cosine form of the time-dependence of the position of the oscillator or amplitude of the wave; the projection on OY gives the sine form. So, the expression
x = aeiωt
could be used to specify the position of a simple harmonic oscillator. Also, the expression
ψ = aeiωt
could be used to represent a wave. These two equations contain both the sine and cosine forms of post 18.11, in a single equation for a simple harmonic oscillator and a single equation for a wave.
In post 18.11, we saw that the sine and cosine forms were solutions of a differential equation. If you think about it, you will see that if acos(ωt) and asin(ωt) are solutions of the differential equation then nacos(ωt) + masin(ωt), where m and n are any numbers, must be a solution too. Putting n = 1 and m = i, we can see why aeiωt is a solution too.
The picture below is also copied from post 18.11.
In this picture, there is a phase difference, ϑ, between Q and Q’. So our general expression for the position of an object going round in a circle, a simple harmonic oscillator or a wave has the form aexpi(ωt + ϑ) where ϑ is a phase term.
So a very abstract idea, like the square root of minus – 1, enables us to reduce the number of equations we need to describe everyday observations.
18.11 Motion in a circle…
18.8 Natural frequency and resonance
18.7 The simple pendulum
18.6 The pendulum: a simple harmonic oscillator
17.13 Centripetal force
17.12 Going round in circles