0
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1answer
47 views

Infinite trigonometric series, find the constant C_n

Hi this is my first post :) I am not sure how to do part b. You get the infinite series of $\displaystyle c_n\cdot \sin(\frac{n\cdot \pi\cdot x}{L})$ from $n=1$ to infinity And this is equal to ...
1
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0answers
51 views

Proving periodicity of sine and cosine [duplicate]

If we define the sine and cosine functions by their Maclaurin expansions, how do we prove they are periodic with period $2\pi$?
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2answers
58 views

Find complex Fourier coefficients

let $f(x) = \sum^{10}_{m=1}(-1)^m \sin(2^m x)$. denote complex Fourier coefficients of $f(x)$ over $[-\pi, \pi]$ as $c_n = \frac{1}{2\pi} \int _{-\pi}^\pi f(x) e^{-inx}\,dx.$ ...
0
votes
1answer
51 views

Integral computation involving $\sin^2 x$

I am attempting to find the Fourier series for $\sin^2x$ but am getting stuck. For the value of $a_0$, I am trying to do it as follows: $$ \frac 1{2\pi}\int_{-\pi/2}^{\pi/2}(\frac 12 -\frac ...
1
vote
1answer
43 views

Fourier Series doubt

I have a doubt regarding the Fourier series usage in terms of the Fourier series formula, which has multiple variants and is quite complicated. EDIT: I would like to mention that this question (of ...
1
vote
0answers
35 views

Confusion regard Fourier Series formulae

EDIT: I need help today, please. It's very important for my homework. I need to understand this concept. Thank you! I have a doubt regarding the Fourier series formula. In one of my notes, it is ...
1
vote
1answer
36 views

CT Fourier Transform

I need to find the Fourier Transform of the given signal below; $$ x(t) = \frac{\sin(\pi t)}{\pi t} \frac{\sin(2\pi t)}{\pi t}.$$ I know that if $ x(t) = \frac{\sin(Wt)}{\pi t} $ , then $ X(w) = ...
0
votes
3answers
179 views

Fourier Series coefficients/Trigonometric functions

I need some help about finding the Fourier Series coefficient of the given signal; $$ x(t) = \sin(10\pi t + \frac {\pi}{6} ) $$ I know that, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
3
votes
1answer
163 views

Basic Fourier Series Question

Let $f$ be a $2π$ periodic function where $$f(x) = \frac{π - x}2$$ over $[0, π]$. It is known that the Fourier series of $f$ is $$\sum_{n=1}^{\infty}\frac{\sin nx}n$$ At which points in $[-π, π]$ ...
0
votes
1answer
68 views

Sinusoid sum of cosine and sine

I am studying Fourier series right now. I asked a question before of math.statckexchange regarding Fourier series. This question is related and hopefully quite simple: Generally Fourier series works ...
12
votes
1answer
308 views

Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$

let $\{a_{n}\}$ is non-increasing postive sequence;show that if for $n\ge 2001,na_{n}\le 1$, then for any positive integer numbers $m\ge 2001,x\in R$, we have ...
0
votes
2answers
127 views

Conceptual question about Discrete Fourier Transform

On the wikipedia page for the discrete Fourier transform, the first sentence says: In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a ...
0
votes
2answers
242 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
1
vote
0answers
210 views

How can I find the compact trigonometric Fourier series from these signals?

I've been stuck on this for a while, but how exactly would I go about calculating the compact trigonometric Fourier series for both of these signals? I have a general formula down for it, but I just ...
1
vote
2answers
389 views

Proving that two periodic functions are orthogonal

Suppose we have a periodic function $f_K(\vec x)$. We want to show that $\int {f_K^*(\vec x) f_{K'}(\vec x) d\vec x} = \delta_{KK'}$, where the integration is over the period of $f(x)$. I know this ...
1
vote
1answer
78 views

Alternative complete bases for Fourier Series.

Knowing that $$\left\{ \sin\left(kx\right)\right\} _{k\in\mathbb{N}}$$ and $$\left\{ \cos\left(kx\right)\right\} _{k\in\mathbb{N\cup}\left\{ 0\right\} }$$ are complete systems in $L^2(0,\pi)$. How ...
1
vote
2answers
271 views

Trigonometric Identities and Fourier Series

I have the series: $$2+\sum_{m=1}^n 4(-1)^m\cos(m\pi x)$$ Here, $x\in (-1,1)$. I need to show that this equals some fraction with only cosine terms and $n$ (no $m$). Just looking for some ...
1
vote
1answer
575 views

Odd harmonics only in Fourier transform

If a function is even or odd, it implies that there are respectively only cos and sine terms in its Fourier expansion. But is there a condition for a function to have an expansion with only odd or ...
4
votes
1answer
98 views

Total variation of a Fourier series

Let $f(x) = f(x+2\pi)$ be a bounded real function given by the Fourier series of the form $$ f(x) = \sum_{k=1}^N a_k \sin(kx + \phi_k). $$ What is the total variation $V(f)$ of this function over one ...
9
votes
7answers
464 views

Why does this Fourier series have a finite number of terms?

I am learning about Fourier series in class and the basic form of a Fourier Series is $$a_{0}+\sum_{n=1}^{\infty} [a_{n}\cos(nx)+b_{n}\sin(nx)]$$ so a fourier series should have an infinity number ...
3
votes
2answers
94 views

Need help with this integral using trig identities

I am trying to integrate the function $$\int_{-\pi/2}^0 \sin(2x)\cos(nx) \, \mathrm{d} x.$$ My professor has an answer of $$\frac{-2\cos(\frac{n \pi}{2})+1}{n^{2}-4}.$$ When I do this problem, I ...
5
votes
1answer
123 views

A trigonometric identity

If one sees the simplification done in equation $5.3$ (bottom of page 29) of this paper it seems that a trigonometric identity has been invoked of the kind, $$\ln(2) + \sum _ {n=1} ^{\infty} ...
3
votes
2answers
130 views

Sum the series: $ S = \frac{1}{2} \cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4} \sin{2\alpha} + \cdots \ \text{ad inf}$

How do I sum the following series? $$ S = \frac{1}{2} \cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4} \sin{2\alpha} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \sin{3\alpha} + \cdots \ \text{ad ...
1
vote
2answers
470 views

Please check my answer to $\sum_{i=1}^n \frac{\sin{(ix)}}{i} < 2\sqrt{\pi}$

$$\sum_{i=1}^n \frac{\sin{(ix)}}{i} < 2\sqrt{\pi}$$ I have this answer, please let me know if there is a more beautiful proof. My answer: at first, we prove two inequalities: If $x\in ...
1
vote
1answer
244 views

Sum of sinusoidals (frequency/phase of acoustic beats)

I have a function that's the sum of two sinusoidals: $ A \cos(\Theta_1 + \omega_1 t) + B \cos(\Theta_2 + \omega_2 t) $. It basically forms an acoustic beat pattern. I need to find the frequency of ...
1
vote
1answer
287 views

expression for Dirichlet's kernel like sum

It is given in the book that the Dirichlet's kernel $D_n(t) = 1/2 + \sum\limits_{k=1}^{n} \cos(kt)$ is given as $\frac{\sin(n+1/2)t}{2\sin(t/2)}$. I'd like to know if there is any such expression for ...
1
vote
1answer
492 views

Relation between “harmonic form” and fourier series?

I am currently prepping for uni having been a few years out of the studying loop (programming as it happens). Anyway, I've been re-reading my A-level notes/exercises and looking through OpenCourseWare ...
5
votes
5answers
2k views

Example of a trigonometric series that is not fourier series?

My textbook doesn't give any example of this kind of series. Could you provide some? Trigonometric series is defined in wikipedia as : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...