Fourier series Definition and 706 Threads

When is a function "equal to" its Fourier series? First of all - a bit unsure where this post fits in, there seems to be no immediately appropriate subforum. So I'm a physics student and currently looking at what it takes for a Fourier series to converge. I've looked at wiki...

I am trying to prove a theorem related to the convergence of Fourier series. I will post my proof below, so first check it and then my question will make sense. Is there any flaw in my proof? Also, here I proved it for integrable functions monotonic on an interval on the left of 0. But what if...

Where is the fallacy in this "proof" that the Fourier series of f(x) converges to f(x) if f is continuous at x and has period 2π? (I read in Wikipedia that a counterexample had been provided). Start with the Dirichlet integral for the N-th partial sum of the (trigonometric) Fourier series...

Suppose $f(\theta) = |\theta|$ for $-\pi < \theta < \pi$. Find the formal series solution of the corresponding heat problem in the disk. How many terms of the series will give $u(r,\theta)$ with an error $< 0.1$ throughout the disk? Evaluate $u\left(\frac{1}{2},\pi\right)$ to two decimals. Show...

Apply Theorem 1.4 to evaluate various series of constants. Theorem 1.4: Let $f$ be periodic and piecewise differentiable. Then at each point $\theta$ the symmetric partial sums $$ S_N(\theta) = \sum_{n=-N}^Na_ne^{in\theta} $$ converge to $\frac{1}{2}\left[f(\theta)+f(-\theta)\right]$; if $f$ is...

Calculate the Fourier series of the function $f$ defined on the interval [\pi, -\pi] by $$ f(\theta) = \begin{cases} 1 & \text{if} \ 0\leq\theta\leq\pi\\ -1 & \text{if} \ -\pi < \theta < 0 \end{cases}. $$ f is periodic with period 2\pi and odd since f is symmetric about the origin. So f(-\theta)...

Calculate the Fourier series of the function $f$ defined on the interval $[\pi, -\pi]$ by $$ f(\theta) = \begin{cases} 1 & \text{if} \ 0\leq\theta\leq\pi\\ -1 & \text{if} \ -\pi < \theta < 0 \end{cases}. $$ $f$ is periodic with period $2\pi$ and odd since $f$ is symmetric about the origin. So...

Homework Statement Hello, Check each function to see whether it is piecewise smooth. If it is, state the value to which its Fourier series converges at each point x in the given interval and the end points (a.) f(x)=|x|+x, -1<x<1 (it would be very helpful to see if i did this right, as the...

greetings, why do we use Euler equation that is e ^(jωt)=cos(ωt)+i sin(ωt) in Fourier series and what does it represent? advanced thanks.

If you write $$ e^{ik\theta} = \cos k\theta + i\sin k\theta, $$ then $\sum\limits_{k = 0}^ne^{ik\theta} = \frac{1 - e^{i(n + 1)\theta}}{1 - e^{i\theta}}$ yields two real sums $$ \sum\limits_{k = 0}^n\cos k\theta = \text{Re}\left(\frac{1 - e^{i(n + 1)\theta}}{1 - e^{i\theta}}\right) $$ and $$...

Homework Statement Use the Fourier series of f(x) = { 1 |x|<a { 0 a<|x|<\pi for 0<a<\pi extended as a 2-Pi periodic function for x \inR to find \sum Sin2(na)/n2 [b]2. Homework Equations [/b I got that the Fourier series of f(x) was a/\pi+\sum (2/(m\pi) sin(ma) sin(mx)...

Homework Statement Given the function f(x) = Acos(∏x/L), find its Fourier series Homework Equations Okay so, f(x) is even, so the Fourier series is given by: f(x) = a0 + \sumancos(nx) where a0 = 1/∏\int f(x).dx with bounds ∏ and -∏ and an = 1/∏\int f(x)cos(nx).dx with bounds ∏...

I am very confused on how to start this problem, would highly appreciate some help! Consider the function
f(x)=   0, -1≤x≤0 2x, 0≤x≤1. The Fourier series coefficients for this function are given by [a][/0]=0.5, and for  k=1, 2, 3,...  ...

We know that because \sin{nx} and \cos{nx} are degenerate eigenfunctions of a hermition operator(the SHO equation),and eachof them form a complete set so we for every f(x) ,we have: f(x)=\frac{a_0}{2}+\Sigma_1^{\infty} a_n \cos{nx} and f(x)=\Sigma_1^{\infty} b_n \sin{nx} But...

Homework Statement The Attempt at a Solution I don't understand where that 2 comes from in the denominator in cos nπ/2

Homework Statement The Attempt at a Solution I don't understand the step above. It has something to do with this equation I think. I'm supposed to expand it into an appropriate Fourier series.

Homework Statement The Attempt at a Solution I don't understand this equation. 2pi, 4pi, 6pi only = 0 when there is a sine function before it, so I don't see how the evens = 0. I don't see why the e vanishes. I also can't get the i's to vanish since one of them is in exponential...

Homework Statement The Attempt at a Solution Obviously brackets mean something other than parentheses because .5[0 + 0] ≠ .5

Homework Statement This series is what dictates the graph above. The Attempt at a Solution I don't understand what's going on. If they're using the series that i pasted below then why aren't they multiply each value in the brackets by -2/pi? I also don't get why terms...

Homework Statement The Attempt at a Solution I don't see why the n is in the denominator

Homework Statement The Attempt at a Solution I don't see how if n = 0 then the answer is 1/2. By my reckoning 1/pi * sin nx/n = 1/pi * 0 = 0 I also don't see where the 2 comes from when sin(npi/2) first shows up.

Homework Statement this comes from a problem with the Fourier series The Attempt at a Solution I don't get the above step.

Homework Statement f(x) = |sin x|, -pi < x < pi, f(x) = f(x + 2pi) Determine the Fourier series of f(x) The attempt at a solution I am unsure how to evaluate an integral with absolute signs in it, however, I am wondering if I could reduce the bounds to 0<x<pi and and f(x) = sin x and assume...

Hi everyone, So I was trying to calcule the Fourier Series of x(t) = cos4t + sin8t, but I'm a little bit confused. What would be ω0 in this case since I have a combination of two functions with different frequencies? Thank you in advance.

I'm trying to show that the Fourier series of f(x)=x^2 converges and I can't. Does anybody know if it actually does converge? (I'm assuming that f(x)=x^2 for x\in [-\pi,\pi]). The Fourier Series itself is \displaystyle\frac{\pi^2}{3}+4\sum_{n=1}^\infty \frac{(-1)^n}{n^2}\cos nx I tried...

Question about the "start" of a cosine Fourier series Hey. I was just looking through Paul's Online Notes http://tutorial.math.lamar.edu/Classes/DE/FourierCosineSeries.aspx to teach myself Fourier Series and I had a question about the a_{0} term of the cosine series. In the online lesson...

Let f(k) be the Fourier transform of F(x). Prove that the Fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the Fourier transform is defined to have a factor of 1/2pi.

Hi, I don't understand why does n goes from -∞ to +∞ in the complex Fourier series, but it goes from n=1 to n=+∞ in the real Fourier series?

Half wave Fourier series help ! I am trying desperately to figure out a Fourier series question i have been given and have hit a big mental wall can someone please help and maybe point me in the right direction. The problems i have been given are:- a) f(t)=e[^2t] 0<t<1 (cosine series)...

Are there general formulas for Fourier coefficients on an integral [a, a + T], where T is the period. There is a general formula for the coefficients of exponential Fourier series. Are there general formulas for the coefficients of the trigonometric Fourier series that would work on any interval...

A book I'm reading says that the set of continuous functions is an Euclidean space with scalar product defined as <f,g> = \int\limits_a^bfg and then defines Fourier series as \sum\limits_{i\in N}c_ie_i where c_i = <f, e_i> and e_i is some base of the vector space of continuous functions. What...

Homework Statement Let f(x) = x^2 on [-\pi,\pi]. Computer the Fourier Coefficients of the 2π-periodic extension of f. Use Dirichlet's Theorem to determine where the Fourier Series of f converges. Use the previous two conclusions to show that \sum_{n=1}^\infty \frac{1}{n^2} =...

Hey, I have to show: Should I try to find the Fourier series from -2pi to 2pi? I have tried this already but I can't seem to get rid of the cos(nx/2) and sin(nx/2) to turn them into just sin(nx) and cos(nx) and the denominator stays as (n^2+4 instead of n^2+1. Any suggestions would be...

In Fourier series we have small waves on the top of big waves (the function seems like that), but the small waves do not have the same amplitude. Does somebody know how to get a function with waves and small waves on the top but with the same amplitude.

How you get Fourier transform from Fourier series? Do Fourier series becomes Fourier transform as L --> infinity? http://mathworld.wolfram.com/FourierTransform.html I don't understand where discrete A sub n becomes continuous F(k)dk ( where F(k) is exactly like A sub n in Fourier series)...

I wasn't really sure where to post this because I am covering this in 2 classes (Math and Physics). Figured this would be my best bet. The Fourier series of some Function is a_{0}/2+etc.... I've looked in several textbooks but none explain why the 1/2 is there, and not in any of the other...

Let $f(x)=-x$ for $-l\le x\le l$ and $f(l)=l.$ a) Study the pointwise convergence of the Fourier series for $f.$ b) Compute the series $\displaystyle\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)}.$ c) Does the Fourier series of $f$ converge uniformly on $\mathbb R$ ? ------------- First I need to...

Two questions, actually. These just come from me doing a couple of problems on Fourier analysis from Rudin's text (I haven't actually taken a full course in the subject; we just spent about a week studying the topic in my real analysis class). (1) The Weierstrass Approximation Theorem...

Homework Statement -0.5\leq{t}\leq{1.5}, T=2 The wave is the attached picture. I need to determine the Fourier Series of the wave in the picture. I know that f(t)=a_0+{\sum}_{n=1}^{\infty}a_ncos(n\omega_0t)+{\sum}_{n=1}^{\infty}b_nsin(n\omega_0t) where a_0=\bar{f}=0 due to being an even...

Hi all, I am just trying to prove to myself the Fourier series representation of a periodic rectangular pulse train. The pulses have some period T, and each pulse has magnitude equal to 1 over a duration of T/4, and 0 the rest of the cycle. Using trignometric Fourier series, I get the...

The expressions for the coefficients of a Fourier series are valid for all integers [0;n]. Though sometimes when I evaluate the Fourier series of an even function (composed only of cosines) I get an expression for the r'th coefficient, which has r in the demoninator. It could be for instance...

Homework Statement Does anyone know of a website, or a book, where I can see a worked example of the Fourier Series f(t) = [t] -∏<t<∏ T=2∏ Finding a0 and an Of course, it doesn't have to be t, it could be x or any other variable. Thank you. Homework Equations The...

Homework Statement f(t) is an odd, periodic function with period 1 and: f(t) = -5.5 + 22*t2 for -0.5 ≤ t < 0 i) find the Fourier coefficient bn ii) find the Fourier coefficient b5 Homework Equations bn = (2/T) * ∫ f(t) *sin((2*n*∏*t)/T) dt between T/2 and -T/2...

Homework Statement I have an example were they determine the Cn coefficient for a Fourier series: My problem is i don't follow what happens in the following moment: neither am i sure how the two e^-inwt comes to be cos(nwt) in the last part. Homework Equations The Attempt...

Homework Statement Put cos^{4}(x) into a Fourier series. Homework Equations cos^{4}(x)=(\frac{e^{ix}+e^{-ix}}{2})^{4} a_0+\sum^{\infty}_{n=1}(a_{n}cos(nx)+b_{n}sin(nx)) The Attempt at a Solution I don't get what I'm supposed to use as a_{0}, a_{n} and b_{n} so I'm stuck...

ok i know that sin n*(pi/2) = 1 if n=1,5,9,13... = -1 if n=3,7,11,15... = 0 if n is even cos n*(pi/2) = 0 if n is odd = -1 if n=0,4,8,12 = 1 if n=2,6,10,14... is there a simpler way of expressing this? for example simple way to express cos(n*pi)=cos(-n*pi)=(-1)^n is there a...

The Fourier series can also be written as: f(x) = Ʃcr*exp(r*2π*i*x/L) where sum if from -∞ to ∞ My book says this at least, but I can't really determine the realitionship between the coefficients of an ordinary Fourier and the complex one. How do you get rid of the i that would appear in...

I have been trying to understand the Fourier series and the relationship between the energy in the original function and its Fourier representation. The example function: y = 3t has a period of 2∏. The Fourier coefficients are: The Fourier representation has a dc average of 3∏, it has no...

Homework Statement This question tests your ability to find, and evaluate, a sine Fourier series of a function. f(x) = 3 Find the Fourier series for this function in the form Ʃbnsin(nx∏/3) from n = 1 to infinity Where bn = 2/3∫f(x)sin(nx∏/3).dx where the integral is from 0 to 3 What is the...

1) Find the sine Fourier series of $f(x)=x(\pi-x),\,0\le x\le\pi$ and show that $\displaystyle\sum_{k\ge1}\frac{(-1)^k}{(2k-1)^3},\,\sum_{k\ge1}\frac1{(2k-1)^6}$ and $\displaystyle\sum_{k\ge1}\frac{\sin\big((2k-1)\sqrt5\pi\big)}{(2k-1)^3}$ and to show that...