# Fourier series

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## INTRODUCTION

This post has become part of my Learning math serie. (Check the menu above)

I am looking for some reason at Fourier series.

I found a nice video from 3Blue1Brown, visualizing what a Fourier transform is but it never learned me how to make calculation with Fourier series.
I have found a few video in Youtube from Acadian Learning School, I will share here.

I studied math at University about 30 years ago and I seldom teached math at high school level. So  I had to refresh  some old math knowledges, like:

To make integral clculations and plots,I also had to install these programs on my Ubuntu linux laptop:

# INDEX

## Fourier serie lessons

Fouriers work and story is well described by Mark Newman:

#### Fourier transform:

Here is the 3Blue1Brown Fourier transform visualizing video

As said in my introduction, Acadiana Learning center offers a Fourier course that I think is the best to start understanding Fourier and how to make calculations with it.

For me it is important to take notes to think about , to better understand. I share with you my notes here on this page.

There is transcript like with Ted youtubes. So, if you have problems with math English and reading transcripts, you may find my  notes and images below useful so you can read the info with your language using the appropriate language button on my site.

All info is taken from this Acadiana Learning center Fourier youtube course.

I hope they dont mind me doing this.

We have this periodic step function This function f(x) is very similar to a sine function as seen in this image Fourier has came to a solution of how to model the periodical step  function (f(x) above, mathematically as a integral of a sine and cosine  function like this one.

He approximated f(x) as a sum of a constant a0, and a sum of of sine and cosine functions: Fourier was able to calculate  the coefficient a0, an and bn.  ,.i fyou know these constants I(coeficients) the you can model any function cwith this equation.

The sine function better match and approximation of f(x) as f(x) is a odd function.  For even functions cosine terns are used. This will be shown mathematically as an  will turn out to be 0 with odd funtctions.

## finding a0

JJean Baptiste Fourier found out how to calculate the value of the coefficient a0, an and bn.

to be He found out the a0 coefficient 1/2L where L= π.

For an and bn. he also found out formulas. So he developed for us 0these three equations: ## Calculating an and bn.

TheAcadiana Learning center teacher show in the next coming videos (part 2 to 5) how Fourier calculated the value of an and bn

#### Calculating a1 As f(x) is divided in two parts, the integral/area calculation has to be split up in a sum of two (2) parts, the first from -pi (-π) to 0(zero) and the second from 0 (zero) to pi (π)

In the image below shows  clearly thath the sum of the two areas from -π to 0, is equal to 0.

The same happens with the two areas from 0 to π -So

#### a1=0  In part 2, the Acadiana Learning center teacher demonstrates that

• for any value of n —-> a = 0 (zero).

#### Calculating bn

The teacher demonstrates that

• for n=even numbers  —->bn =0 (zero)
• For n= odd numbers —-> bn = 4c/nπ .

## Fourier in action

To see what Fouriers equation does you must get a cartesian graph plotter. I downloaded and installed KM plot for linux on my Ubuntu laptop. You find good plotter for sure also with other operative systems.

in the video information in part 1 (source: https://www.youtube.com/watch?v=x04dnqg-iPw )  we read:

“You can graph this with your calculator easily and watch the modeling in action. Make sure you’re in radian mode and let c=1

f(x) = 4/(pi)*sin(x) + 4/(3pi)*sin(3x) + 4/(5pi)*sin(5x) + 4/(7pi)*sin(7x) + 4/(9pi)*sin(9x) + 4/(11pi)*sin(11x)” (source: https://www.youtube.com/watch?v=x04dnqg-iPw )

This equation is equivalent to where

• a0 = 1/2n
• an=0
• bn for even numbers of n, is not calculated The integration goes from n=1 to n=11

I choosed to get a a cartesian plot and added the formual above

f(x) = 4/(pi)*sin(x) + 4/(3pi)*sin(3x) + 4/(5pi)*sin(5x) + 4/(7pi)*sin(7x) + 4/(9pi)*sin(9x) + 4/(11pi)*sin(11x)”

The plot  forf(x) = 4/(pi)*sin(x) + 4/(3pi)*sin(3x) + 4/(5pi)*sin(5x) + 4/(7pi)*sin(7x) + 4/(9pi)*sin(9x) + 4/(11pi)*sin(11x) showed up like this: ## I tried to integrate from n=1 to 29 with

f(x) = 4/(pi)*sin(x) + 4/(3pi)*sin(3x) + 4/(5pi)*sin(5x) + 4/(7pi)*sin(7x) + 4/(9pi)*sin(9x) + 4/(11pi)*sin(11x)+ 4/(13pi)*sin(13x)+ 4/(15pi)*sin(15x)+ 4/(17pi)*sin(17x)+ 4/(19pi)*sin(19x)+ 4/(21pi)*sin(121x)+ 4/(23pi)*sin(23x)+ 4/(25pi)*sin(25x)+ 4/(27pi)*sin(127x)+ 4/(29pi)*sin(29x)

and got this plot: The more intervalls the more precise the approximated plot. Its a plity,I dont know how to let my plotter plot with n=1 to n=999999999 with odd n.

## conclusion I find it extraordinary that Jean-Baptiste Joseph Fourier born 21 March 1768 during his lifetime (he died – 16 May 1830)  had such a brilliant mathematical talent to create the Fourier equation that today 200 years late  in the digital era, is essential for  sharing music, speech recognition and building engines and bridges.

I like to know more about such talents. It is remarkable Jean-Baptiste Joseph Fourier story reminds me a little bit of Einstein.

Jean-Baptiste Joseph Fourier was a son of a tailor and soon lost his father. He learned math in a Benedectine order. He served in the Revolutionary committe during the French Revolution as well as a scientific adviser for Napoleon in his Egypt expedition 1798 How the h…. did he came up to this equation?

Was it that by studying structures looking like this that his math brain lift off and trigometry knowledges came to use? Maybe he devloped and found use of Fourier series in this expedition.

The developing of his math must have been done during sparetime a little like Einstein did his math work during sparetime in the Swiss patent office. Read more in https://en.wikipedia.org/wiki/Joseph_Fourier#Biography

Obvioulsy, not all great mathematician matured their genius in schools and Universities.

Studying Fourier, I learned again about integration and trigonometry something I did more than 30 years ago at Chalmers in Sweden. A great brain exercise.

# Feynman lectures

## Introduction

As a math teacher I like when people try to simplify math with eg illustrations. I am struggleing to understand quantum physics math.

I were happy that  PBS Space Time informed be about Richard Feynman.

# The language of physics is mathematics

“The language of physics is mathematics. In order to study physics seriously, one needs to learn mathematics that took generations of brilliant people centuries to work out.” (Source:  www.superstringtheory.com  )

To understand the math of superstringtheory you have to understand the math  of

• Ordinary Differential Equations
• Partial Differential Equations
• Methods of approximation
• Probability and statistics

You may be intrested Matlab.

Math explained

string_theory

(Source: math.berkeley.edu  )

# the Origin of Zero

I knew that the first Zero was found in a building in India  according to the BBS documentary  “The Story of one” ( now deleted in youtube but available at http://www.imdb.com/title/tt0482651/ ) .

# Bra matteundervisning i grundskolan!

NÃ¤sta Ã¥r vill jag arbeta pÃ¥ ett annat sÃ¤tt. Det har jag skrivit om i tidigare post. Denna gÃ¥ng efterlyser jag dina ideer. Du som nu gÃ¥r ut nian hur skulle du vilja ha undervisningen om din syster eller broder skall bÃ¶rja i sexan?

En kommentar jag fick var denna:

NÃ¤r du bÃ¶rjar arbeta nÃ¤sta Ã¥r med sexor sÃ¥ ska du bÃ¶rja med att ha roliga lektionen genom att lÃ¥ta alla elever komma fram till tavlan och visa de dem kan som tex gÃ¥ngetabelen mm… Sedan sÃ¥ ska du dela in dem i grupper och ge dom ett papper som de ska lÃ¶sa tal.Â  Efter att de har haft det sÃ¥ ska de fÃ¥ gÃ¥ ner till data salen och bÃ¶rja sammarbeta med sin grupp och hitta fakta och mattetal pÃ¥ nÃ¤tet. FÃ¶r att de ska tycka att matte Ã¤r roligt sÃ¥ ska de gÃ¥ till hemkunskap fÃ¶r att visa hur vikt fungerar sedan de som ni har vÃ¤gt ska de fÃ¥ Ã¤ta upp tex.. Ã¤pple, banan… Slutet av terminen sÃ¥ ska ni ner till syslÃ¶jds salen fÃ¶r att klippa i tyg och mÃ¤ta mÃ¥tt… SÃ… SKA DET VARA FÃ–R ATT HA EN ROLIG LEKTION !!! (Redzifa)

Skriv Ã¤ven du en kommentar.

# Meaningful math  This book was written 1995! It has been available for several years in our school math library but undiscovered by me. It is of course written in Swedish.

Two pages talkes about the Pythagorean Theorem and how to experiment with it. Reading about it I thought it would be perfect to let the kids work with the theorem while learning how to calculate the area of the rectangle and the triangle.

The drawing on the right shows how to create:

• 4 triangles
• 1 square

and how to combine them to demonstrate the Theorem.

To create the demonstration shown on the second image below, one should also have:

• 1 rectangle for the hypotenuse area (c2)
• 2 rectangles for the two legs area (a2 and b2)
• 1 rectangle with the area (a+b)2

If the students have woodwork or art lessons it is maybe an idea to cooperate with those colleagues to let the pupils create these 5 shapes in some adequate material and then maybe do math experiments. I would like to find other good reasons to learn geometry but at least, these two experiments would maybe show some of the magic of math.

# En bÃ¤ttre matematik

Svenska ungdomar

• Ã¤r bra pÃ¥ att lÃ¤sa och skriva
• Ã¤r bra pÃ¥ engelska
• Ã¤r bra pÃ¥ statistik
• Ã¤r bÃ¤st pÃ¥ att komma sent till lektioner
• tycker att det kvittar med dÃ¥liga matteresultat

Samtidigt Ã¤r de dÃ¥liga i

• geometri
• aritmetik
• ekvationer
• problemlÃ¶sning

Samtidigt Ã¤r det mÃ¥nga duktiga tjejmatematiker som vÃ¤ljer bort en naturvetenskaplig eller teknisk utbildning.

Med “vÃ¤l valda” ord, svenska skolors matteundervisning suger!
Och jag Ã¤r en svensk mattelÃ¤rare… 🙁

Jag har fÃ¥tt uppfattningen att matematik

1. mÃ¥ste kÃ¤nnas motiverande
2. mÃ¥ste vara en utmaning
3. rÃ¤knas bÃ¤st tillsammans i mattedialog med bra bÃ¤nkkamrater

1. Matematik i ett sammanhang

Som en matematiker frÃ¥n Bengtsfors, sa, sÃ¥ trÃ¤nas vÃ¥ra ungdomar veckovis i att “sÃ¥ga” (rÃ¤kna brÃ¥k) fÃ¶r att sedan Ã¶vergÃ¥ till “hamrande” (rÃ¤kna procent) och fÃ¶r att avsluta skolÃ¥ret, med “skruvande” (lÃ¶sa ekvationer). Trots att vÃ¥ra elever fÃ¥r “skruva” i hela mÃ¥nader, sÃ¥ Ã¤r vÃ¥ra elever bland de sÃ¤mre i just “skruvande”. Vad beror detta pÃ¥?

Jag tror att matematik mÃ¥ste sÃ¤ttas in i ett sammanhang. T ex matte mÃ¥ste anvÃ¤ndas i andra Ã¤mnen Ã¤n matematik, nÃ¤r t ex enkÃ¤ter skall gÃ¶ras, halter skall berÃ¤knas i NO-laborationer, mÃ¤ngder skall bestÃ¤mmas i hemkunskapen eller mÃ¥tt skall mÃ¤tas i slÃ¶jden. Men matematikundervisningen skall Ã¤ven den styras av det Ã¶vergripande strÃ¤vansmÃ¥let “Undervisningen skall skapa interesse!” och inte av de nationella uppnÃ¥endemÃ¥len i matematik eller av matematikbÃ¶ckerna, dÃ¤r man drillas i brÃ¥krÃ¤kning utan att se eller kÃ¤nna dess betydelse.

2. Matematik som utmaning

Jag har hÃ¶rt att mattetalen i en bok Ã¤r sÃ¥ upplagda att de fÃ¶rsta 80% av uppgifterna kan lÃ¶sas mekaniskt utan nÃ¥gon som helst fÃ¶rstÃ¥else fÃ¶r lÃ¶sningsmetoden. Endast de sista uppgifterna, som de flesta eleverna aldrig hinner med, krÃ¤ver en djupare fÃ¶rstÃ¥else av matteavsnittet.

Jag funderar om det inte Ã¤r sÃ¥ att mattetalen Ã¤r felkonstruerade i vÃ¥ra skolbÃ¶cker. Och var finns charmen i matten? Gyllene snittet, mÃ¤rkliga brÃ¥klÃ¶sningar Ã¥tergivna pÃ¥ Rindhpapyrusen, andra talsystem, mm fÃ¥r endast ett fÃ¥tal sidor i slutet av kapitlen.

Oavsett kvaliteten i bÃ¶ckernas Ã¶vningsuppgifter, sÃ¥ tror jag att jag mÃ¥ste lÃ¤gga ner mer tid pÃ¥ att hitta uppgifter vars lÃ¶sningar visar pÃ¥ matematikens charm och inte Ã¤r uppenbara, utan krÃ¤ver reflektion och dialog.

3. Mattedialog

Endast i VG och MVG mÃ¥len, dyker dialog upp som krav. FÃ¶r dessa betyg mÃ¥ste eleven diskutera olika lÃ¶sningsmetoder, analysera skillnader mellan dessa och kunna se specifika och generella lÃ¶sningsmetoder. LikasÃ¥ krÃ¤vs kunskaper om matematikens anvÃ¤ndning genom historien och i nutid.

En fÃ¶relÃ¤sare som jag lyssnade pÃ¥ i mÃ¥ndags, sa att en elev som sitter och rÃ¤knar ensamt fÃ¶rlorar 2 tredjedelar av koncentrationen jÃ¤mfÃ¶rt med om denne rÃ¤knade tillsammans med en bÃ¤nkkamrat han/hon trivs med. Samtidigt har jag sett att elevernas glÃ¤dje i matte Ã¶kade markant nÃ¤r de fick tid att jÃ¤mfÃ¶ra varandras lÃ¶sningar.

Och nog Ã¤r det viktigt att hjÃ¤lpa eleverna att hitta bra mattekamrater.