**Rotating Fourier Series**

**Chapter 2. Complex function exp (jωt) as a model of the real function cos (ωt)**

**Chapter 2.1 ****Introduction
**The purpose of this article is to treat

**sine/cosine**functionsc as

**rotating vectors**. For example, vector

**(1,0)**rotating with speed

**ω0=1/sec**, corresponds to the real function

**1*cos**(

**ω0t)=1cos(1t)**. I remind you that the angular velocity

**ω0=1**(more precisely

**ω0=1/sec**) is assigned to the period

**T=2πsec≈6.28sec**. This approach will greatly facilitate the intuive understanding of the formulas related to the

**Fourier Series**,

**Fourier**and

**Laplace Transforms**.

**Note**on

**ω0**pulsation

Why not

**ω**alone? Because this is an article about

**Fourier Series**and

**ω0**is the

**pulsation**of the

**fundamental harmonic**, while

**ω=n*ω0**refers to the

**n-th**harmonic of the periodic function

**f(t)**.

**Chapter 2.2 Euler Equation
**I assume you know the complex numbers and the complex function

**exp(jω0t)**, wich are used in many fields. For me, electrician are

**rotating vectors**. They show the phase shift

**ϕ**between the

**current**and

**voltage**better than in ordinary

**time graphs**. If you are not sure, I recomend the article

**“Complex Numbers”**from the top bar.

**Fig. 2-1**

Euler Equation

Treat

Euler Equation

**α**generally as good

**x**, that is, as a

**real number**related to the equation! And

**jα**is an

**imaginary number**on the

**Im z**axis. Only from this equation results its interpretation with a

**circle**and a

**right triangle**. Here you can cleaarly see that

**α**is the angle in

**radians**.

For any real number

**α**, the complex value

**z=exp(jα)**will be somewhere on the circle with radius

**A=1**. At first it seems as obvious as the definitions of

**sine**and

**cosine**. But what is the relationship between the complex exponential function

**exp (jα)**and

**trigonometry**? Mr.

**Euler**in his day could only use

**complex**,

**addition**,

**subtraction**,

**multiplication**and

**division.**What about the function

**exp (jω0t)**? How to calculate it using only the above-mentioned four actions?

**Euler**already knew, however, that the real function

**exp (x)**is the

**sum of infinitely**many decreasing polynomials, such as the real number

**1**is the

**infinite sum**of the series

**1/2 + 1/4 + 1/8 + 1/16 + …**

He treated the complex exponential function

**exp (jx)**in a similar way. Ie. He calculated its value as the sum of

**polynomials**using only

**four**operations on

**complex numbers**. And he probably was surprised when the point

**z = exp (jx)**began to

**spin**around a

**circle**with radius

**A = 1**. Only then did he associate

**x**with the angle

**α**. And he expected the function to go to

**infinity**somewhere. As for any honest real exponential function! Substitute in the

**Euler formula**in order

**α = 0**,

**0.25π**,

**0.5π**,

**0.75π**…

**2π**. You will see the point

**z = exp (jα)**

**spin**in a

**circle**!

**Chapter 2.3 Video Support
**The main advantage of the article is animation. It is much more imaginative than a simple chart.

**Fig. 2-2**

Description of the buttons and video indicators

**-Start**clicking starts the animation and changes to the

**Stop**button

**-Clock**the current experiment time, also shown as a yellow bar

**-Simulation time**differently – experiment duration, here

**13 sec**.

**-Full screen**clicking enlarges the screen, re-clicking reduces etc …

**-Stop**Clicking stops the simulation and changes to a

**Restart**button

**-Restart**as the name suggests.

**Chapter 2.4 Study of the various harmonic motions f(t) as their two-dimensional versions F (jω0t)
**This chapter is just a pretext to get acquainted with the complex function

**Start*exp(jω0t)**. I emphasize that the parameters

**Start**and

**jω0t**are

**complex numbers**! Now that we are familiar with

**video handling**, we will examine

**harmonic motion**s for different

**initial states**of the

**rotating**vector

**Start**and pulsation

**ω0**. You will see that the time function

**x(t)**is the projection of the

**rotating**vector

**Start*exp (jω0t)**onto the real axis

**Re z**. So

**x(t)=Re z {Start * exp (jω0t)}**.

**We**will examine the

**5**harmonic

**motions**.

**1.1cos (1t)**as

**Start*exp(jω0t)**for

**Start=1**and

**ω0=1/sec**

**2.1sin (1t)**as

**Start*exp(jω0t)**for

**Start=1*exp(-jπ/2)=1*exp(-j90°=-j**and

**ω0=1/sec**

**3.1cos (1t-π/4)**as

**Start*exp(jω0t)**for

**Start=1*exp(-jπ/4)=1*exp(-j45 °)≈0.707-j0.707**and

**ω0=1/sec**

**4.1cos (2t)**as

**Start*exp(jω0t)**for

**Start=1**and

**ω0=2/sec**

**5.**

**0.5cos (1t)**as

**Start*exp (jω0t)**for

**Start=0.5**and

**ω0=1/sec**

**Chapter 2.4.1 f (t)=1*cos (1t) as F(jω0t)=1*exp (j1t) i.e. as Start*exp(jω0t) for Start=1 and ω0=1/sec**

**The most important conclusion:**

The rotating vector **F(jω0t)=1*exp(j1t)** is a **two-dimensional** version of the function **f(t)=x(t)=1*cos(1t)**. Here **ω0=1/sec**.

Later it will turn out that almost every periodic function **f(t)** on the period **T** corresponding to the pulsation **ω=2π/T**

has its **two-dimensional** version as the sum **F (jt)=c1*exp(j1ω0t)+c2*exp(j2ω0t)+c3*exp(j3ω0t)…**

**c1, c2, c3 …, cn** are complex numbers or vectors **an+jbn** as initial states (for**t=0**) of spinning vectors** cn*exp (jnω0t)**.

In the **two-dimensional** version of **F(jω0t)** of the function **f(t)**, some features are more visible and intuitive than in the **one-dimensional**.

**Fig. 2-3
**

**Fig. 2-3a***

**x(t)=1cos(1t)**

It is a

**one-dimensional**motion only along the real axis

**Re z**on the complex plane

**Re z, Im z**.

Click the

**“Start”**button or the drawing to see the

**2 T**periods of the

**harmonic motion**.

We know the amplitude

**A=1**and the time of

**2**periods, i.e.

**2T≈12 sec≈12.56 sec≈4πsec**shown on the video stopwatch, i.e.

**T=2πsec**->

**ω0=2/T=1/sec**.

The absolute values of speed are the highest in the

**middle**, and the smallest, that is,

**zero**at the ends “at turns”.

*****On the occasion. I’ll teach you Polish.

**Rys. 2-3a**it’s

**Fig.2-3a**

**Rys. 2-3b**

Complex function

**1exp (jω0t)**as a rotating vector with length

**1**and initial state

**(1,0)**

**Note:**

Since the

**Start**point

**=z=(1,0)=1+0j**

The complex function has the value

**Start*exp(j1t)=(1+0j)*exp(j1t)=1*exp(j1t)**and it is a

**rotating vector**of length

**1**and the initial state

**(1,0)**.

It can be seen that the animation from

**Rys. 2-3a**is a projection of the animation from

**Fig. 2-3b**onto the

**real**axis

**Re z**

Or otherwise

**Re 1exp (j1t) = 1cos (1t)**

The projection of the rotating vector onto the real axis

**Re z**, moves as in

**Rys. 2-3a**.

Some people almost equate

**exp(jω0t**) with

**cos (ω0t)**. It is not exact, but it is accurate and pictorial. Just like the famous phrase “I am for and even against”. We know what’s going on, although the logic is wrong.

You can see the initial state of the circulating vector

**Start**

**=(1+0j)=+1**in

**Rys 2-3b**before the animation.

**Remember**. I will not repeat it a second time!

Each

**Start**vector is an

**initial state**and “stands still”. Here

**Start**

**=z=1+0j**, but it could be, for example,

**Start**

**=z=2-3j**.

When

**Start**is multiplied by

**exp (jω0t)**, the vector

**Start***

**exp(jω0t)**starts to spin at

**ω0**angular speed. You will find out about it in the next animations as well.

**Rys. 2-3c**

Time plot

**x(t)=1cos (1t)**(ω0 = 1 / sec)

**-The horizontal axis (“x”**) is the time

**t**in seconds.

Characteristic points

**0sec 0.5πsec≈1.57sec πsec≈3.14sec 0.75πsec≈2.36sec 2πsec≈6.28 sec**

**– the vertical axis (“y”)**is

**x(t)**in units

Try to stop the simulation “around” at these times and compare

**Rys. 2-3a**,

**Rys. 2-3b**and

**Rys. 2-3c**.

**Chapter 2.4.2 f(t)=1*sin(1t) jako F(jt)=Start*exp(jω0t) dla Start=-1j and ω0=1/sek**

**Fig. 2-4
Rys. 2-4a
x(t)=x(t)=1sin(1t)
**This is the projection of the circulating vector from

**Rys. 2-4b**onto the real axis

**Re z**.

**Rys.2-4b**

Complex function

**-1j*exp (j1t**) for

**ω0=1/sec**, otherwise

**exp(-jπ/2)* exp (j1t)**because

**-1j =exp(-jπ/2)**

**Note:**

Since the

**Start**point

**=0-1j=-1j**

The complex function has the value

**Start*exp(j1t)=-1j*exp (j1t)**.

You can see the initial state of the circulating vector

**Start=-1j**in

**Rys.2-4b**before the simulation. Compare with

**Rys. 2-3b**. You can clearly see the truth, known for centuries, that

**sin(ω0t)**lags by

**π/2=90º**with respect to

**cos (ω0t)**.

**Rys.2-4c**

Time plot

**x(t)=1sin (1t)**

**Chapter 2.4.3 f(t)=1*cos(1t-π/4) as F(jt)=Start*exp(jω0t) for Start=1*exp(-jπ/4)=0.707-j0.707 and ω0=1/sek
**

**Fig. 2-5
**

**Rys. 2-5a**

**x (t)=1cos(1t-π / 4)**i.e.

**x (t)=Acos (ω0t-ϕ)**for

**A=1 ω0=1/sec**and

**ϕ=-π/4=-45º**

This is the projection of the circulating vector from

**Rys. 2-4b**onto the real axis

**Re z**.

**Rys 2-5b**

Complex function

**1*exp(j1t-π/4) as 1*cos (1t-π / 4**).

Notice that you can see the delay

**ϕ=-π/4=-45º**beautifully here. Check with pythagoras that amplitude

**A = 1**and that

**ϕ=-π/4=-45º**.

We can write a circulating vector in different ways.

**Start*exp(j1t)=exp(-jπ/4)*exp(j1t)=(1/√2-j1/√2)*exp(j1t)≈(0.707-j0.707)*exp (j1t).**

The initial state of the circulating vector

**Start=exp(-jπ/4)**is shown in

**Rys. 2-5b**.

**Rys. 2-5c**

Time plot

**x(t)=1cos(1t-π/4)**

**Chapter 2.4.4 f(t)=1*cos(2t) as F(jt)=Start*exp(jω0t) for Start=1and ω0=2/sek
**We increased the speed to

**ω0=2/sec**in relation to

**Rys. 2-3**

**Fig. 2-6
**The course is

**2**times faster than in

**Fig. 2-3**. Besides, the initial state

**Start**is the same, i.e.

**Start=(1+0j)=+1**

Compare with the corresponding animations in

**Fig. 2-3**.

**Rys 2-6a**

**x (t) = 1cos (2t).**

**Rys 2-6b**

Complex function

**1exp (j2t)**as

**1cos (2t)**

**Rys. 2-6c**

Time plot

**x(t)=1cos (2t)**

**Chaptwer 2.4.5 f(t)=0.5cos(1t) as F(jω0t)=Start*exp(jω0t) for Start=0.5 and ω0=1/sek
**

**Fig. 2-7
Rys. 2-7a
x(t)=0.5cos(1t).
**

**2**times smaller

**A=0.5**amplitude compare with

**Fig. 2-**4!

**Rys 2-7b**

Complex function

**0.5exp (j1t)**as

**0.5cos (1t)**

**Rys. 2-7c**

Time

**plot x(t)=0.5cos (1t)**.

**Chapter 2.5 Functions Start*exp(jω0t)
**Compare once again the previously discussed complex functions for different parameters

**Start**and

**ω0**.

**1-**

**Start=1**and

**ω0=1/sek**

2- Start=1*exp(-jπ/2)=-j1and

2- Start=1*exp(-jπ/2)=-j1

**ω0=1/sek**

**3- Start=1*exp(-jπ/4)**and

**ω0=1/sek**

**4- Start=1**and

**ω0=2/sek**

5- Start=0.5and

5- Start=0.5

**ω0=1/sek**

**Fig. 2-8
**

**5**versions

**Start*exp(jω0t)**

These functions are shown in

**Fig. 2-4b … Fig. 2-7b**as rotating vectors. Their ends indicate the points z whose coordinates are precisely the complex functions

**Start*exp(jω0t)**.

The

**“beetroot”**vector is the initial state of each function. This is the

**Start**parameter that occurs before

**exp(jω0t)**. Carefully analyze each of the

**5**waveforms taking into account the

**Start**and

**ω**parameters. Compare, for example,

**Rys. 2-8b**and

**Rys. 2-8a**. Here it is better to see that

**sin(1t)->-j1exp (j1t)**is delayed by

**90°**in relation to

**cos(1t)->1exp (j1t)**.

**Chapter 2.6 Complex function F jt)=(a-jb)*exp (jω0t) as a two-dimensional version of f(t)=a*cos(ω0t)+b*sin (ω0t)
**

**Chapter 2.6.1 General Description**

Dessert at the end. So how to build a trajectory

**F(jω0t)**for any sinusoidal function

f (t)=a*cos (ω0t)+b*sin(ω0t)=c*cos(ω0t-ϕ)

f (t)=a*cos (ω0t)+b*sin(ω0t)=c*cos(ω0t-ϕ)

From

**Chapter 2.4.1**we know that the complex function

**1*exp (jω0t)**corresponds to the time function

**1*exp (ω0t)**.

Similarly, according to

**Chapter 2.4.2**of the complex function

**-1j*exp (jω0t)**corresponds to

**1*sin (ω0t)**.

We will write it like this:

**1 * exp (jω0t) <==> 1 * cos (ω0t)**

**-1j * exp (jω0t) <==> 1 * sin (ω0t)**

This is the case with amplitudes

**a=1**for cosine and b = 1 for sine.

It works for any amplitudes a and b:

**a * exp (jω0t) <==> a * cos (ω0t)**

**-jb * exp (jω0t) <==> b * sin (ω0t)**

Instead of writing that the function corresponds to something, you can more closely, like the following

**Fig. 2-9
**

**Re (a-jb)*exp(jω0t)=a*cos(ω0t)**+

**b*sin(ω0t)**

The vector

**(a-jb)**is spinning at speed

**ω0**.

**The projection**of the rotating vector

**(a-jb)*exp(jω0t)**onto the real axis is:

**real part (a-jb) * exp (jω0t)**

i.e.

**Re (a-jb)*exp(jω0t)**

that is

**a*cos(ω0t)+b*sin(ω0t).**

This is a generalization of the simplest case from

**Rys. 2-3**where

**a=1 and b=0**.

**a**

Complex function

**(a-jb)*exp(jω0t)**as vector rotating at speed ω0

**(a-jb)**

It has

**2**components: –

–

**real**(cosine)

**a*exp (jω0t)**

**-imaginary**(sinus)

**-jb*exp (jω0t)**

What you see in

**Rys. 2-9a**is the initial state of the rotating vector, i.e. for

**t = 0**.

I emphasize that “lonely”

**a**and

**b**are

**real**numbers!

**b**

The projection of the rotating

**a*exp(jω0t)**vector onto

**the real axis**is

**a*cos(ω0t)**

**c**

The projection of the rotating

**-jb*exp(jω0t)**

**vector onto the**

**imaginaryl axis**is

**b*sin(ω0t)**

**d**

The projection of the

**sum**of the

**rotating vectors**, i.e. the

**red**vector, onto the

**real axis**is

**a*cos(ω0t)**+

**b*sin(ω0t)=c*cos(ω0t-ϕ)**

It’s not obvious, but that’s trigonometry! Instead of proof*, let’s substitute specific values, e.g.

**a=0.75**and

**b=1.25**and check the animation.

* This can be done by a high school math class student.

**Chapter 2.6.2 Concrete example F (jω0t) = (0.75-j1.25) * exp (jω0t) as a two-dimensional version of
f (t) = 0.75 * cos (ω0t) + 1.25 * sin (ω0t)**

**Fig. 2-10
**

**Rotating**vector

**(0.75-j1.25)*exp(1jt)**as

**f(t)=0.75*cos(1t)+1.25*sin(1t)=1.458*cos(1t-59.04°)**

**Rys. 2-10a**

**Rotating**vector

**0.75 * exp (1jt)**and its projection onto the

**Re z**axis as a function of time

**0.75*cos(1t)**

**Rys. 2-10b**

**Rotating**vector

**-j1.25*exp(1jt)**and its projection on the

**Re z**axis as a function of time

**1.25*sin(1t)**

**Rys. 2-10c**

Sum of the

**2 left vectors**as a

**spinning**vector

**(0.75-j1.25)*exp(1jt)**. Its projection on the

**Re z**axis is a function of time

**f(t)=0.75*cos(1t)+1.25*sin(1t)=1.458*cos(1t-59.04°)**and its timing diagram is shown in the animation below.

**Fig. 2-11
**

**f(t)=1.458*cos(1t-59.04°)=0.75*cos(1t)+1.25*sin(1t)**

The angular shift of

**59.04 °≈1.03**radians is shown in the diagram.