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)

          e^x = 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 e^iϑ, 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

          e^iϑ = 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, e^iϑ, 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) ae^iϑ 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 ae^iωt. 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 = ae^iωt

could be used to specify the position of a simple harmonic oscillator. Also, the expression

          ψ = ae^iωt

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 ae^iωt 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.

 

Related posts

18.11 Motion in a circle…
18.10 Waves
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
16.14 Aliasing