how to get the Fourier series coefficients (fourier series engineering mathematics)

Summary

This educational script introduces the concept of Fourier series, a mathematical tool for representing functions as infinite sums of sines and cosines, which are particularly useful for periodic phenomena. The video explains the difference between Taylor and Fourier series, outlines the general form of a Fourier series, and delves into the process of calculating the coefficients 'a_n' and 'b_n' through integration over a 2π interval. It simplifies the complex-looking expression into a more digestible format, aiming to demystify the subject for students.

Takeaways

• 📊 The goal of the Taylor series is to write a function as a power series, resembling an infinite polynomial.
• 🔢 In the Taylor series, the function is expressed in terms of (x - a) raised to the power of whole numbers.
• 📐 The Taylor series uses coefficients, denoted as C's, for its power series representation.
• 🧮 The Fourier series aims to represent a function as a trigonometric series using sine and cosine functions.
• 🔄 Sine and cosine functions are used in the Fourier series because they are periodic and useful for representing repeating phenomena.
• 🔍 The general form of the Fourier series includes a constant term (a0), cosine terms (an * cos(nx)), and sine terms (bn * sin(nx)).
• ➕ The Fourier series combines these terms into a single summation for simplicity.
• ✏️ To find the coefficients an and bn in the Fourier series, integration is used instead of differentiation as in the Taylor series.
• 🧩 For calculating the coefficients, functions are integrated over an interval of length 2π, typically from -π to π.
• 🔧 Special considerations are made for integrals involving sine and cosine products, leading to specific formulas for an and bn.

Q & A

What is the main goal of the Fourier series?

The main goal of the Fourier series is to represent a function as a trigonometric series, specifically using sine and cosine functions as the building blocks.

How is the Fourier series different from the Taylor series?

The Fourier series uses sine and cosine functions to represent a function, focusing on periodicity, while the Taylor series represents a function as a power series using polynomial terms.

What are the building blocks of the Fourier series?

The building blocks of the Fourier series are sine and cosine functions, with coefficients a_n for cosine terms and b_n for sine terms.

Why are sine and cosine functions used in the Fourier series?

Sine and cosine functions are used because they are periodic, which makes them suitable for representing functions that exhibit repetitive behavior.

What is the general form of the Fourier series?

The general form of the Fourier series is a sum of cosine terms (a_0 plus a_n times cosine of n times x) and sine terms (b_n times sine of n times x), where n ranges from 1 to infinity.

How is the coefficient a_0 in the Fourier series calculated?

The coefficient a_0 is calculated by integrating the function over the interval from negative pi to pi and dividing by 2 pi, with a_0 being equal to 1 over 2 pi times the integral of the function.

What happens to the integral of the cosine terms in the Fourier series when n is not equal to m?

When n is not equal to m, the integral of the cosine terms in the Fourier series evaluates to 0 due to the orthogonality of the sine and cosine functions.

How is the coefficient a_n in the Fourier series determined?

The coefficient a_n is determined by multiplying the function by the cosine of n times x, integrating over the interval from negative pi to pi, and then dividing by pi.

What is the integral result of the sine terms in the Fourier series when n equals m?

When n equals m, the integral result of the sine terms in the Fourier series is pi, which contributes to the calculation of the b_n coefficients.

How do you calculate the coefficient b_n in the Fourier series?

The coefficient b_n is calculated by multiplying the function by the sine of n times x, integrating over the interval from negative pi to pi, and then dividing by pi.

Outlines

00:00

📚 Introduction to Fourier Series and Comparison with Taylor Series

The script begins with an introduction to the Fourier series, emphasizing its purpose to represent a function as an infinite series, akin to an infinite polynomial. The comparison is made with the Taylor series, highlighting the use of coefficients (c's and b's) and the role of sine and cosine as the building blocks for the Fourier series, due to their periodic nature. The general form of the Fourier series is presented, illustrating the combination of cosine and sine terms with varying frequencies.

05:00

🔍 Calculating the Fourier Coefficients: An and Bn

This paragraph delves into the process of determining the Fourier coefficients, starting with the calculation of the constant term a0, which is achieved by integrating the function over the interval from negative pi to pi. The explanation clarifies that the integration of cosine terms results in zeros due to their periodicity, except for the constant term. The formula for a0 is derived, showing that it is proportional to the integral of the function over the specified interval.

10:04

📐 Advanced Integration Techniques for Fourier Coefficients

The script introduces a more advanced method for finding the Fourier coefficients an and bn by integrating the product of the function with cosine or sine functions over the interval from negative pi to pi. The discussion highlights the need to consider different cases based on whether the indices n and m are equal or not, leading to integral results of either zero or pi. The formulas for an and bn are derived, emphasizing the conditions under which non-zero values are obtained.

15:12

🎓 Conclusion and Final Thoughts on Fourier Series

The final paragraph wraps up the discussion on the Fourier series by summarizing the integral formulas for the coefficients an and bn, and how they relate to the original function. It reiterates the importance of the sine and cosine functions in the Fourier series and invites viewers to engage with the content by leaving comments and subscribing to the channel for more educational content.

Keywords

💡Fourier series

The Fourier series is a mathematical tool used to represent a function as an infinite sum of sines and cosines. It is central to the video's theme as it is the main concept being discussed. In the script, the Fourier series is introduced as an alternative to the Taylor series for representing functions, especially in cases where the function exhibits periodic behavior.

💡Taylor series

The Taylor series is a way to express a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point. It is mentioned in the script as a comparison to the Fourier series, highlighting the difference in their applications. The Taylor series is used for functions that can be approximated by a polynomial around a specific point.

💡Power series

A power series is an infinite series of terms involving powers of a variable. In the script, the power series is used to describe the general form of both the Taylor and Fourier series, emphasizing the idea of expressing functions as sums of terms that involve powers of the variable, whether polynomial or trigonometric.

💡Trigonometric functions

Trigonometric functions, such as sine and cosine, are periodic mathematical functions that describe the angles in a right-angled triangle. In the script, these functions are the building blocks of the Fourier series, which is used to represent periodic phenomena in various fields, such as physics and engineering.

💡Periodic function

A periodic function is one that repeats its values at regular intervals or periods. The concept is crucial in the script as it explains why sine and cosine are used in the Fourier series, given their inherent periodic nature, making them suitable for modeling repeating patterns.

💡Coefficients

In the context of the script, coefficients are the numerical factors that multiply the trigonometric functions in the Fourier series. They are determined through integration and are unique to each term of the series, allowing for the accurate representation of the original function.

💡Integration

Integration is a mathematical operation that finds the accumulated value of a function over an interval. In the script, integration is used to calculate the coefficients of the Fourier series, which involves integrating the product of the function and the trigonometric functions over a specified period.

💡Differentiation

Differentiation is the mathematical process of finding the rate at which a function changes. Although the script focuses on the Fourier series, it contrasts this with the Taylor series, where differentiation is used to find the coefficients of the series by taking derivatives of the function.

💡Convergence

Convergence in mathematics refers to the property of a sequence or series to approach a certain value or behavior as its terms increase in number. In the script, the convergence of the Fourier series is assumed for the integration process to work correctly and to accurately represent the original function.

💡Odd and even functions

Odd functions are symmetric with respect to the origin, while even functions are symmetric with respect to the y-axis. In the script, the properties of odd and even functions are used to determine the result of certain integrals, which simplifies the process of finding the coefficients of the Fourier series.

💡Sine and cosine coefficients

In the Fourier series, the sine and cosine coefficients (a_n and b_n) are the values that multiply the sine and cosine terms, respectively. The script explains how these coefficients are calculated through integration and are essential for constructing the Fourier series that represents the function.

Highlights

Introduction to the concept of Fourier series as a method to express functions as an infinite series using sine and cosine functions.

Differentiation between Fourier series and Taylor series, with the former using trigonometric functions and the latter using polynomials.

Explanation of the periodic nature of sine and cosine functions as the reason for their use in Fourier series.

General form of the Fourier series, including the representation of the function with cosine and sine terms.

Clarification on the absence of the term P0 in the Fourier series due to the constant value of cosine at zero.

Integration method to determine the coefficients of the Fourier series, contrasting with differentiation in Taylor series.

Definition of the integration interval for calculating Fourier coefficients, typically from -π to π.

Simplification of the integral calculation for a0 by showing that it results in a constant times 2π.

Demonstration that the integral of cosine terms for an ≠ m results in zero due to orthogonality.

Result of the integral for sine terms when n = m, showing that it equals π, which is crucial for determining the coefficients.

Introduction of a method to calculate the coefficients an by multiplying the function by cos(nx) and integrating.

Similar approach for calculating bn coefficients by multiplying the function by sin(nx) and integrating.

Explanation of why the integral of sine times cosine results in zero, simplifying the calculation of bn coefficients.

Final formulas for calculating the Fourier coefficients an and bn using integration over the defined interval.

Summary of the three main components needed to determine the Fourier series of a function.

Encouragement for viewers to ask questions and engage with the content, promoting further learning and discussion.

Transcripts

ladies and gentlemen today is the day we will  be talking about the Fourier series however how  

can I leave out the tailor right so let's talk  about this again remember the goal is to write  

a function as a power series and this right here  it's just like a infinite polynomial and look at  

this right here we have the X minus a to a past  the whole numbers and this is how can look at it  

you can just think about this right here they are  like the building blocks for that function maybe a  

left-hand side you have e to the X well you are  going to have this right here as your building  

blocks along with the coefficients namely the CS  and to freaking the students out we all thanks to  

Taylor which are showed in the previous video  we have this formula for that and whenever you  

use this formula to figure out the power series  for f of X we are going to be respectful we will  

call this the Taylor series that's all now let's  talk about what Fourier series is first of all  

this right here it's just the name of a person  just like Taylor so nothing too fancy it's just  

the name of person right and the goal right here  it's pretty similar we want to write a function  

into something else as well but this time we will  actually like to write f of X as a trick series  

so it's not polynomial like earlier this right  here is like a polynomial but this time we'll  

be using sine cosine to help us out namely is  like a trick series why do we use sine cosine we  

all because they are periodic another in a lot of  situations you know a lot of things keep repeating  

so it might be a good idea to have sine cosine  as the building blocks and here is the general  

form we will have a zero and then next year's a-1  times that's right on the cosines first cosine of  

one x and then next we have a two cosine of 2x and  the next we have a three cosine of 3x and so on so  

on so on this time though as you can see we have  the cost of one x cosine two x cosine 3x and so  

on these right here are our building blocks where  all not yet we also need to have the signs to help  

us out here for the size in fact you don't have  P 0 why because this is us setting a zero times  

cosine of zero times X cosine of zero times X is  constant zero which is one which is just a 0 B 0  

times sine of zero times X times zero times X is  just 0 which is 0 anyway so you don't have the p  

0 so you start with B 1 times sine of 1 X and yes  you just pretty much keep on going B 2 times sine  

of two x and then plus B 3 sine of 3x and so  on and so on so as you can see this is a trick  

series and thanks to full year we'll figure out  the formulas body and if you wrote the a s and  

also the PS after all that your respect for the  code is the Fourier series all right now let's  

see first of all this is a lot right so let's  put this in this summation 1 and you see that  

we have the a n PI stuff so that I mean a sear  on the outside right a is 0 oh this right here  

I was just ready says the sum s and goes from 1 to  infinity and then the coefficients are the AR and  

down the building blocks are cosine and then we  have n times X so it's just like that now for the  

signs yes this right here and this is just like  we add the sum as n goes from 1 to infinity be n  

times sine of X like this so this right here it's  a pretty scary-looking expression it's actually  

not that bad and another small remark is that  because the summations are pretty much the same  

so you can actually put this and that getting  1 summation that's fine by anyway this is what  

we have first of all let's go ahead and figure  out whom below for a zero well look back to the  

Taylor series this right here the tale of Ohm's  law we differentiate it huh if you differentiate  

this equation what the roofless you're just zero  so cannot figure it out in fact the roughly is not  

going to be helped because we differentiate cosine  you can make this sound so long right no maybe you  

guessed it yes instead of doing differentiations  and like the Taylor series right here we will  

actually be doing integrations to figure out the  ANS and also the piers and here is the deal as I  

mentioned it sine cosines they are periodic so  first of all we want to define an interval with  

length 2 pi and we want to just integrate over  that interval so I will just write this down  

right here for you guys perhaps we can choose an  interval and it's pretty easy to work with from  

negative PI to PI you can also do the following  computation from 0 to 2 pi up to you this is good  

because 0 it's right in the middle so why not  ok so first of all what we would like to do is  

let me just look at this equation and integrate  each and every one from 0 from false right from  

negative PI to PI and hope for the best it will  be the best you'll see so let me just write this  

down right here for you guys first of all on the  left hand side we have f of X and as I said it we  

are going to integrate this from negative PI to  PI and I'm going to leave a space here you guys  

will see why so I will have the DX right here  and this is equal to I will also integrate this  

from negative PI to PI and this is a knot voice  you up to you however you want to say it leave a  

space right here this right here don't forget this  right here it's just a bunch of the adings right  

here right when you integrate this you can just  integrate this term by term under the assumption  

that this right here converges and of course you  has to converge me order to for the following to  

work so what we are going to do is we will just  add and instead of putting the integral here we  

can actually to the integral here because it's  just like we're adding the integrals so we have  

the sum as n goes from 1 to infinity integrating  it from negative PI to PI of this expression which  

is a and cosine of an X and if a little space you  guys will see why and then put a DX same thing  

here so we are going to add the sum as n goes from  1 to infinity I'm just integrate this term by term  

so we have to integrate this for negative PI to  PI and we have B and sine of an X leave a space  

and then put a DX right here okay so that's  see that's figure things out well if you have  

a function this right here is no you have got to  work it out so that's there let's see this right  

here is just a constant if you integrate this you  get 8 9 times X and if you go from negative PI to  

PI this right here it's actually really easy  because the answer is just 2 pi times a naught  

or a 0 right so you can just work that out it's  not that bad no and now let's look at this well a  

and in the X words just this is just a constant so  you can know that let's focus on the integral so  

let me just put note right here the integral from  negative PI to PI cosine or an X Y X and let's do  

this real quick well cosine of an X in style when  you integrate that you get positive sign of an X  

and don't forget that you can do the derivative  of e^x is n but you are integrating so you have  

to divide it by that so it's one of n right here  and then you are going from negative PI to PI and  

you just do the usual thing you have the 1 over N  sine off and to the pike here so we open pi here  

and then you are going to subtract 1 over N sine  of and and I put the negative pi here or that well  

remember and it is one two three four five so long  dangly party phone numbers so when you have sine  

of n PI in fact this is just 0 why because sine  of 1 PI 0 to PI 0 sound 3 PI 0 and so on this  

is 0 and this right here it's also 0 therefore  all this right here is just 0 in other work oh  

this right here is just 0 cool huh so this is  gone now this right here if you integrate just  

focus on negative PI to PI of just the sine of X  like this this right here it's also 0 why because  

science an odd function and when you integrate our  function from negative PI to PI this is going to  

be 0 so this right here it's just a bunch of zeros  right so in another work all this to account you  

have this it's equal to 2 pi times a now we can  justify 2 pi on both sides so the first little  

formula right here is a nap equals 1 over 2 pi  times the integral hold touch times the integral  

from negative PI to PI of whatever the function  is DX this is the first piece of the formula that  

we have so this right here it's pretty cool huh  now one of those three things done we still have  

to figure out the ANS and also the PS you see  the a s and B ends earlier that didn't matter  

because you know the integral part there was 0  so now this time we cannot just integrate this we  

actually have to introduce something else to help  us out huh so what can we do hmm ok I'd like to  

Turkish this in an integral sine cosine they get  along with each other really well right so I want  

to have maybe the cosine to help us out let's see  so when you look at this not only we are going to  

integrate this from negative PI to PI but before  that we actually have to multiply everybody here  

by a new function and it's not that new it's just  cosine of something right but if you just simply  

multiply by cosine it's not going to work so  actually I will tell you we will have to multiply  

everybody by cos amp MX what's em though M it's  just a past default number so I'll just write M is  

belonging to the positive whole number so yeah so  you can just write cosine M X here and then this  

right here you also have to multiply by cosine M  X and you might be wondering how do I know two x  

close up at max I don't forget it so ask him don't  ask me anyway here is what we will do this right  

here let me just read it down better for you guys  yeah this is a no times cosine M X and then here  

is the DX that's why I left the space all right  let's see this right here once again to be done  

right so how did that this right here it's if we  choose in order comes to a now when you integrate  

from negative PI to PI of course up MX that's the  inter quota with the earlier for the concept an X  

from negative PI to PI so don't let the m1 bother  you in fact this right here will give you 0 just  

like earlier right now we have to figure this out  well leave the end right here okay so I'll just  

focus on this integral so I will just write this  down we are going to examinate negative PI to PI  

cosine of an x times cosine of M X DX so there's  two right here hmm sometimes maybe that M can be  

the same as n so be careful with that so we have  two situations to consider what if the N am at the  

same and what if they are different well well  can't just work this out on your own in fact I  

would like to tell you the following all right  this right here I will just do a separate video  

because otherwise this video would be over like  you'll be overly long so work this out you can  

use the English on for now whatsoever whatever  value you prefer but I will tell you if N and  

M are different the result of this integral will  be 0 so let me just write this down here and it's  

not equal to M and if an M they happen to be the  same the result right here will be nicely equal  

to PI I'll just write this down if and it's equal  to M right so this right here is actually really  

really great and why do I mean by that well if  n m and and then M are the same this intergroup  

becomes just pi so let's see if you just focus  on integral part the whole thing here well we  

have a and right technically a.m. because if when  you have n is equal to 1 is equal to and so on if  

n happens to be a same as n so that's why is a M  right so that's the constant multiple and then for  

the integral part you end up with PI so it's a M  times pi all right so this right here will be just  

all that and it's just one term because that's the  only one situation that you enter it and number  

beside zero the other terms are just zeros so this  is when n is equal to M you have this result so  

that's very nice now let's talk about this right  here well let's focus on the integral of sine of  

n x times cosine M X in fact sign it's art cosines  even an even function times an odd function when  

you work this tiles you can verify this on your  own yet you actually still end up with an odd  

function and when you integrate this for negative  PI to PI guess what body will be zero so that's  

very nice perhaps I'll put on a happy face happy  zero thank mice this right here it's also a happy  

zero so why are we talking about it this time  on the left hand side this is the integral that  

we have to do and then on the right hand side we  actually only end up with a n times pi am times  

pi this is the subscript M right and of course we  can just now divide it by pi on both sets that's  

pretty much it but usually we don't write em we  still use that so we are going to replace the M  

pi and again so another thing that we know right  now is a and equals once again we're just fixing  

the index right here I would divide the pie here  so we have 1 over pi the integral from negative PI  

to PI and we will have to integrate we will have  to integrate this right so it's cosine it's f of x  

times and because I'm using n right here right I'm  using n so also replace the M by n so multiplied  

by cos of an X like that so once again this right  here at the end likewise with this M but let me  

just replace the end with end so this is the  second formula that we have kuhar now one more  

thing and then we'll be done we'll have to figure  out the B ends earlier I multiply everything by  

cosine of MX take a guess what are we going to do  this time well you raise the boy you guys can take  

guess what's the best breadth of cosine who's  the best friend of cosine yes sign so instead  

of multiplying everybody by cosine of M X we  will be multiplying everybody by sine of M X and  

once again M is just positive whole numbers so a  positive whole number and this right here is sine  

of M X and right here this is going to be sine of  M X and then this right here it's going to be sine  

of not simple sine of MX this right here is to be  done so leave that this right here thank God this  

is the odd function times ain't now which is still  hard this right here it gives us the happy 0 and  

mean that's making up here this right here just  focus on integration cosine times sine yes it's  

odd isn't it so you'll get 0 so same thing huh now  sign of an x times sine of MX well we are going to  

focus on that integral so let me just write this  down the negative PI to PI integral right this is  

sine of an x times sine of M X DX let me tell you  it's really similar to the other situation that we  

had we have two possibilities 0 or PI if n m M are  different you can see rough for the whole integral  

so this is when and it's not equal to n and when  they do happen to equal to each other the resolve  

this in the quote is nicely equal to PI aha so for  this part right here and goes from 1 to infinity  

so somewhere it's going to hit N and we'll just  have B M here times the integral part let me just  

write this down BM shopping read just like earlier  and then the integral part integral of that it's  

going to be pi yeah so look at this we have this  integral and inside us with the sine of MX here  

we have the BM replace the end with and again so  I will tell you BN it's equal to 1 over pi still  

integrated from negative PI to PI of the function  f of X and you have to multiply by sine of an X  

like this so this are these three main ingredients  for the Fourier series Oh feels so good listen  

it come and next that's why the Fourier series  of e to the X by the moment this is it leave a  

comment down below and let me know if you get any  questions and if you guys are new to my channel  

be sure to subscribe and as always that's it I'll  make this closer so you guys can see it yeah bye