Fourier Series The following sum is a finite Fourier series. f(x)=∑i=1^N ai sini x =a1 sinx+a2

step 1 Okay, what we want to do is we want to show multiple things. And we're going to start with what

Fourier Series The following sum is a finite Fourier series....

Key Concepts

Fourier Series

A Fourier series is a method for representing a periodic function as a sum of sinusoidal components, typically sines and cosines, each with a specific amplitude and frequency. This series allows complex periodic signals to be analyzed in terms of their frequency content and is a fundamental tool in areas such as signal processing and differential equations.

Sine Fourier Series

A sine Fourier series is a specialized Fourier series involving only sine terms, which is particularly useful for representing functions that are odd over a symmetric interval around zero. This representation simplifies the problem by taking advantage of the inherent symmetry in the function.

Orthogonality of Sine Functions

The sine functions form an orthogonal set on the interval [-?, ?], meaning that the integral of the product of two different sine functions over this interval is zero. This property is critical in isolating each Fourier coefficient when projecting a function onto the sine basis, ensuring that each coefficient captures only the contribution of its corresponding frequency.

Fourier Coefficient Calculation

The calculation of Fourier coefficients involves projecting the function onto the corresponding sine or cosine basis function, typically by integrating the product of the function and the basis function over the interval of interest. This process utilizes the orthogonality property to extract the individual coefficient for each term in the series.

Integration Techniques in Fourier Analysis

Evaluating Fourier coefficients often requires the use of integration techniques such as substitution and integration by parts, especially when dealing with products of polynomial and trigonometric functions. These techniques facilitate the computation of the definite integrals that define the coefficients in the Fourier series.

Transcript

00:01 Okay, what we want to do is we want to show multiple things.

00:07 And we're going to start with what is called a finite four -year series.

00:11 It's given by f of x is equal to the summation of a -sive -i, sine of i of x is equal to a1, sine of x plus a2, sine of 2x plus a3, sine of 3 -x plus, and it goes on to ace of n is equal to sign -n.

00:30 Of x.

00:31 And the first thing we want to do is to show that the nth coefficient is going to be given by 1 over pi times the integral from pi to negative pi of f of x sine of x d x so that nth coefficient is given by that integration.

00:49 Okay, so what we want to start with is that we know that f of x is equal to this summation, right? and so what we want to start with is that f of x times sine of n of x is going to be equal to this summation, and i guess this should be an n here, uppercase, of this ace of i, sine of i of x, and then this is going to be times a sine of n of x, which is going to be a1, sign of x times sign of n of x, plus a2 sign of 2 of x times sign of n of x plus dot, dot, dot, we have ace of n of x times another sign of n of x.

01:57 Okay, and we also know, so this then will be, now will be actually equal to the integral, and so we're going to kind of do the integral, so this will be if we do the integral from negative pi to pi of f of x, sine of n of x, dx.

02:27 And so this is going to be the integral from negative pi to pi of this summation times this sign n of x d x which we then will have these individual and the first one would be this um oops um this first one would be kind of this ace of one um sine of x times sine of n of x dx plus the integral from negative pi to pi of ace of two sine of two x sine of n of x dx plus and then we can keep going and then we have plus and then of course this is going to be a sine squared um the integral from negative pi to pi of a.

03:45 Of n sine squared of n of x d x um and we know we know previously um we know that the integral from negative pi to pi of sine of mx times sine of n x d x is equal to zero for m not equal to n so all of these all of these will go to zero the only one that is not going to go to zero is this last one right here so what we have is the integral from negative pi to pi of f of x times sine n of x d x is actually going to be equal to the integral from negative pi to pi of ace of n, which i actually can pull him out.

05:04 So let's go ahead and pull him out of the integral of sine squared n of x dx.

05:20 Okay.

05:21 So now all i have to do is find that integral.

05:26 That's the only thing i have to now do.

05:29 And so we're actually going to be using a trig identity.

05:34 I think it's going to be a double angle formula, i think, is what we're going to do, is that sine squared of n of x is actually going to be equal to one half.

05:47 And then it's going to be a minus cosine of 2 n of x.

05:59 Over so we're going to be actually using this identity to help us out and so this will actually be equal to ace of n times and let me do over two because i'm going to pull that two out of there and then it's going to be the integral from negative pi to pie of dx minus the integral from negative pi to negative to pi of cosine of 2 n of x d x and so this will be equal to ace of n over 2 times x and we're going to evaluate that at pi and negative pi and then of course this guy actually if i do a u sub on let you be 2 n of x then this becomes um, sign of 2n of x divided by 2n.

07:20 And we're going to evaluate at upper limit of pi and the lower limit of negative pie.

07:25 And so this will actually become ae of n of pi.

07:30 And so that tells me now that if i divide both sides by pi, ace of n is 1 over pi times the integral from negative pi to pi of f of x, sign of n of x, dx.

07:56 And so i've gone ahead and prove that out.

08:00 And now what we're going to do is we're actually going to find some of the coefficients, and we are actually going to let, now we're going to let, f of x equal x...

Please give Ace some feedback