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Question 1

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Why Fourier sine series of $F(x,t)$ is $\sum_{n=1}^{\infty}b_n(t)\sin(\frac{n\pi}l x)$ with $b_n(t)=\frac2l \int_{s=0}^l F(s,t)\sin(\frac{n\pi}ls)ds$?

For a single-variable function $f(x)$, the Fourier sine series is:$$f(x)=\sum_{n=1}^\infty b_n\sin(\frac{n\pi}lx)\quad\text{where}\quad b_n=\frac2l\int_0^lf(x)\sin(\frac{n\pi}lx)dx$$However, I ...

User

8,489

askedNov 25 at 21:12

Question 2

0votes

2answers

71views

Having problem understanding the answer to the $1$D Heat Equation with the initial condition $U(x,0)=u_0-u_0\sin x$

Here is a problem and answer from my notes, where I have issues understanding parts of the provided answer.Problem:Solve the P.D.E. $U_t=U_{xx}$ with the following initial and boundary conditions:$...

User

8,489

askedNov 25 at 16:47

Question 3

3votes

2answers

93views

Problem with understanding the technique used to calculate $B_n=\frac{2u_0}{\pi}\int_{x=0}^\pi\sin(nx)(\sin x+\sin 3x)dx$ for heat equation.

This is a problem and answer from my notes:Solve heat equation for $l=\pi$ and with the initial and boundary conditions:$U(0,t)=U(\pi,t)=0;\;\;U(x,0)=u_0(\sin x+\sin 3x)$The answer to the above ...

User

8,489

askedNov 22 at 20:10

Question 4

0votes

0answers

90views

Decomposition of a Schwartz function by Lacey and Thiele

Suppose we have a Schwartz function $\varphi\colon \mathbb{R}\to \mathbb{R}$ supported in (0,1) such that $||\varphi||_{L^2}\leq 1$ satisfying that for all $\xi\in \mathbb{R}$,\begin{align}\sum_{l\...

wuentis

45

askedNov 19 at 6:32

Question 5

1vote

1answer

76views

Inverse discrete Fourier transform and Fourier series : contradiction ? What's wrong in my proof?

I was reading stuff about discrete Fourier transform (DFT) and its inverse when I produced the following proof which seems to lead to a contradiction for me. I don't know where the error is. Could ...

anthomedal

11

askedNov 11 at 12:49

Question 6

-2votes

1answer

53views

From where does $-\frac{\pi}{10}$ come in the Fourier series for the half-wave rectified cosine with period $T=4p$ and amplitude $A=1$?

As mentioned, the problem is:Derive the Fourier series of a half-wave rectified cosine with period T=4p and amplitude A=1 as you can see in Desmos graph (orange color).The solution given in textbook ...

stormer Phenom

3

askedNov 10 at 16:06

Question 7

1vote

1answer

61views

The Fourier coefficients are zero for all but finitely many integers $k$

For an appropriate function $g$ and for an integer $k$, the $k^{\text{th}}$ Fourier coefficient $\widehat{g}(k)$ of $g$ is defined as$$ \widehat{g}(k):=\int_0^1g(x)e^{-2\pi ikx}dx. $$$\textbf{...

A. Bond

453

askedNov 6 at 6:54

Question 8

1vote

1answer

97views

Gnuplot gave me some trouble plotting the Fourier series of $f(x) = e^{-|x|}$. Anyone else have this experience when plotting a Fourier series? [closed]

I wanted a plot of:\begin{equation}f(x) = e^{-|x|}\end{equation}and I wanted to compare $f(x)$ to its Fourier series ($n = 1,3,20$):\begin{equation}F(x) = \frac{e^{\pi}-1}{\pi e^{\pi}} + \frac{2}...

Rockmechanic

39

askedNov 3 at 22:40

Question 9

1vote

0answers

91views

Why is there a $(-1)^{n+1}$ in the answer for this Fourier series derivation?

I’m working on a problem from Asmar’s PDE textbook (2nd ed, 2004). The question comes from §2.2 #13:In Exercises 5-16, the equation of a 2$\pi$-periodic function is given on an interval of length 2$\...

Rockmechanic

39

askedOct 24 at 21:39

Question 10

3votes

1answer

75views

Is there a name for the Gibbs' Phenomenon-like oscillations when plotting the Fourier Series of a Dirac Train?

Below is a plot of $\sum_{k=-20}^{k=20}e^{2\pi ix}$ plotted using desmos. I know when you have a differentiable function with jump discontinuities in it, you get oscillations near the discontinuity ...

Arthur Prudius

55

askedOct 23 at 1:25

Question 11

0votes

0answers

39views

Modular integrals and Poisson resummation

Consider the integral\begin{equation}\int_0^\infty \dfrac{dT}{T} \sum_{(m,n)\in\mathbb{Z}^2}e^{-\alpha Q(m,n) T}\end{equation}This integral is divergent both at $T\to0$ logarithmically and $T\to\...

Fr6

43

askedOct 17 at 12:49

Question 12

3votes

1answer

196views

Fourier coefficients of $\sin(x)$

So obviously the Fourier series of $\sin(x)$ is just $\frac{1}{2i}\left(e^{ix} - e^{-ix}\right)$ almost by definition. However, I was wondering if there is a way to compute the Fourier coefficients of ...

Klaus

11.3k

askedSep 25 at 14:52

Question 13

0votes

0answers

34views

A question about Fourier Series from [Zill] Section 12.3, Example 3 [duplicate]

I am teaching Engineering Mathematics (II) in my school. I use the textbook [1]. In [1], the Fourier series is defined in DEFINITION 12.2.1:DEFINITION 12.2.1 Fourier SeriesThe Fourier series of a ...

Wei-Cheng Liu

704

askedSep 25 at 6:08

Question 14

-1votes

1answer

99views

Approximate unknown function with Fourier series

Say I have a monotonically increasing function $f : [0,1] \to {\Bbb R}$. I only know the values of $f$ for a finite set of points $x_1, \dots, x_n$. Can I use a Fourier series to approximate the ...

JF Meier

526

askedSep 23 at 18:06

Question 15

0votes

0answers

84views

Express Legendre functions in terms of Bessel functions

The solution to the Legendre differential equation$$ (1-x^2) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 2x\frac{\mathrm{d}y}{\mathrm{d}x} + n(n+1) y = 0 $$is a linear combinations of the Legendre ...

Jonathan Huang

469

askedSep 21 at 16:12

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Questions tagged [fourier-series]

Why Fourier sine series of $F(x,t)$ is $\sum_{n=1}^{\infty}b_n(t)\sin(\frac{n\pi}l x)$ with $b_n(t)=\frac2l \int_{s=0}^l F(s,t)\sin(\frac{n\pi}ls)ds$?

Having problem understanding the answer to the $1$D Heat Equation with the initial condition $U(x,0)=u_0-u_0\sin x$

Problem with understanding the technique used to calculate $B_n=\frac{2u_0}{\pi}\int_{x=0}^\pi\sin(nx)(\sin x+\sin 3x)dx$ for heat equation.

Decomposition of a Schwartz function by Lacey and Thiele

Inverse discrete Fourier transform and Fourier series : contradiction ? What's wrong in my proof?

From where does $-\frac{\pi}{10}$ come in the Fourier series for the half-wave rectified cosine with period $T=4p$ and amplitude $A=1$?

The Fourier coefficients are zero for all but finitely many integers $k$

Gnuplot gave me some trouble plotting the Fourier series of $f(x) = e^{-|x|}$. Anyone else have this experience when plotting a Fourier series? [closed]

Why is there a $(-1)^{n+1}$ in the answer for this Fourier series derivation?

Is there a name for the Gibbs' Phenomenon-like oscillations when plotting the Fourier Series of a Dirac Train?

Modular integrals and Poisson resummation

Fourier coefficients of $\sin(x)$

A question about Fourier Series from [Zill] Section 12.3, Example 3 [duplicate]

Approximate unknown function with Fourier series

Express Legendre functions in terms of Bessel functions

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