For questions about or related to trigonometric series.

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17
votes
0answers
316 views

Simplify the sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes $\le N^2$

I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum. I ...
9
votes
0answers
114 views

Which Fourier series are “legal”?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$ f(x)=\sum_{n\geq 0}...
6
votes
0answers
33 views

Certain symmetrized product of cosines - can it be transformed into more manageable form

I am interested in the following expression: $$ F_{k_1,\ldots,k_n}(t):=\sum_{\sigma\in S_n}\cos(\sigma(1)k_1t)\cos(\sigma(2)k_2t)\cdots\cos(\sigma(n)k_nt) $$ where $k_1, \ldots, k_n$ are natural ...
6
votes
0answers
101 views

On $\sum a^n \tan(n\theta)$

It is well known that $$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}$$ $$\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$ ...
6
votes
0answers
242 views

On the series $ \displaystyle \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) $.

Is the following series ‘summable’ in the sense that it may be divergent but we can attach a meaning to it? $$ \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) = (\text{reg}) \...
5
votes
0answers
137 views

Value of a Sine-Like Infinite Product

Does the following infinite product have a "nice" closed form? $$ P = \prod_{k=2}^{\infty} \left(\left(1 - \frac{1}{k^2}\right)^\dfrac{(-1)^k}{k}\right) $$ I know that without the power one could ...
4
votes
0answers
82 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
4
votes
0answers
78 views

find the closed form of $\sum_{k=1}^\infty\left(\frac{\sec{kz}}{k^2}\right)^2$

How to evaluate $\displaystyle\sum_{k=1}^\infty\left(\frac{\sec{(k\pi\sqrt{5})}}{k^2}\right)^2$? In general, how to find the closed form of infinite series $\displaystyle\sum_{k=1}^\infty\left(\frac{\...
4
votes
0answers
165 views

It is possible to get a closed-form for $\sum_{n=1}^{\infty}\frac{\sin(\frac{3\pi}{n})}{n^2}$?

I think that will not be useful to compute the Apéry's constant as $$\zeta(3)=\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{1}{n^2}\int_0^{\frac{\pi}{2n}}\sin^2(3x)dx+\frac{1}{3\pi}\left(\sum_{n=1}^{\infty}\...
4
votes
0answers
98 views

Has this function a name?

Has this function a name? $$f(x) = \prod_{i=2}^{\lceil \frac{x}{2} \rceil} \sin\left( \frac {\pi x}{i}\right)$$ (the product of $\sin( \frac {\pi x}{i})$ for $i=2,3,...,\lceil x/2 \rceil$)
3
votes
0answers
51 views

Lip $\alpha$ trigonometric series

Assume we have a trigonometric series $$ f(x)=\sum_{n=1}^{\infty} a_n\sin nx \in \text{Lip }\alpha, \, 0<\alpha <1. $$ Is there anything we can say about the series $$ g(x)=\sum_{n=1}^{\infty} |...
3
votes
0answers
61 views

Trigonometric sum evaluation

Let $q$ a prime number and $1 \leq a<q$ a positive integer. We know from Ramanujan identity that $$\underset{h=1,\left(h,q\right)=1}{\overset{q}{\sum}}e^{2\pi ih\frac{a}{q}}=\underset{h=1}{\overset{...
3
votes
0answers
112 views

Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq i}^n\frac{\sin(\theta_j-\theta_i)}{\left(1-\cos(\theta_j-\theta_i)\right)...
2
votes
0answers
48 views

Estimation of trignometric polynomial and Lipschitz estimation

Let $\mathcal{T}_n$ denote the linear space of trigonometric polynomials of degree up to $n$ and $$E_n(f)=\inf_{P\in\mathcal{T}_n} \|f-P\|_2=\|f-S_Nf\|_2=\left(\sum_{|k|>n}|\widehat{f}(k)|^2\right)^...
2
votes
0answers
112 views

Is there a relationship between a function's period and number of roots?

Let: $f(x,a,l)=\prod _{k=a}^l\sin \left(\frac{\pi x}{k}\right)$ and $f(x,k)=\sin \left(\frac{\pi x}{k}\right)$ I came up with this equation to find the period $T(f(x,a,l))=2\,{{\pi }^{l-a+1} \left(...
2
votes
0answers
152 views

Proving that sin(x)/x=(1-x^2/pi^2)(1-x^2/4pi^2)(…)

I am faced with explaining to a bunch of people who have taken a course in real analysis, but no course in complex analysis, why $\sin(x)=x\prod_{n\geq1}(1-x^2/\pi^2n^2)$. I vaguely remember being ...
1
vote
0answers
25 views

Explaining a Series Expansion for the Divisor Function

The "sum-of-divisors" function, defined as $\sigma(n)=\sum_{d\,\mid\,n}d$, can be expressed as $$\sigma(n)=\sum_{i=1}^n\sum_{j=1}^i\cos\Big(2\pi n \frac{j}{i}\Big)$$This expansion makes logical sense. ...
1
vote
0answers
33 views

A geometric proof for the “small angle approximation” for the sine, cosine and tangent

How can be deduced the so called "small angle approximations" for sine, cosine and tangent, namely $\sin \theta \approx \theta$ $\tan\theta \approx \theta$ $\cos\theta \approx 1-\frac{\theta^2}{2}$ ...
1
vote
0answers
50 views

Laurent series for $\cot(\pi z)/z^2$

I'm looking for the series for $\frac{cot(\pi z)}{z^2}$ using the residue theory, where the function denotes a circle about the origin with a radius of $k+\frac{1}{2}$ I found that the residues of $\...
1
vote
0answers
84 views

Value of $\tan(1^{\circ})+\tan(2^{\circ})+\tan(3^{\circ})+…+\tan(89^{\circ})$

Is there any method to find value of $$\tan(1^{\circ})+\tan(2^{\circ})+\tan(3^{\circ})+\dots+\tan(89^{\circ})$$ without using calculator. To find the same sum for sine and cosine , I used De-Movier's ...
1
vote
0answers
33 views

Integral of sums of basic trigonometric polynomials

My textbook (RCA, Rudin) asserts (in the proof of theorem 5.15) that $$ \lim_{n\to\infty}\lVert{\sum_{k=-n}^n}e^{ikt}\rVert_1=\infty. $$ Why is this true? I tried using Euler's formula to reduce the ...
1
vote
0answers
38 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
1
vote
0answers
61 views

The derivative of the trigonometric polynomial

For $n\in\mathbb{N}$, let $\varphi:\mathbb{R}\to\mathbb{R}$ be a $4n-$ periodic function s.t. $$\varphi(t)=n-|n-t|,\quad -n\leq t\leq 3n$$ For each $j\in\mathbb{Z}$ define $$c_j=\int_0^1\varphi(4n t)...
1
vote
0answers
44 views

How to simplify summation containing trignometric function?

Prove that $$2 \sum\limits_{n=1}^\infty \left(\frac{x}{b}\right)^n \frac{\sin A}{n} = \tan^{-1}\left(\frac{2bx \cos A}{b^2-x^2}\right)$$ I have tried to do by changing this into imaginary part of ...
1
vote
0answers
38 views

Definite integral of recursive functions

Let $F_0(x)=x, F_1(x)=4x(1-x), F_{n+1}(x)=F_1(F_n(x))$. Prove that for $n\in\mathbb{Z_0}^+$: $\int^1_0 F_n(x)\,dx = \frac{2^{2n-1}}{2^{2n}-1}$ Here is progress that I have: I used the substitution $x=...
1
vote
0answers
140 views

Fourier series of half of $\sin(\pi x)$

So my question is: Find the Fourier series (using integrals) for the half wave rectified sine function: $$f(x)= \begin{cases}0&-1<x<0\\ \sin(\pi x)& 0<x<1\end{cases}$$ ...
1
vote
0answers
67 views

Summation of trigonometric series

The first part of the question requires me to show that the sum of $$cos(2n - 1)x = \frac{sin(2Nx)}{2sin(x)}$$ from $n = 1$ to $N$. This I have done by considering the real part of the geometric ...
1
vote
0answers
72 views

Is $\sum_{k=2}^{\infty}{\sin(k!)}/{(k\log(k))}$ convergent or divergent? How can I prove it?

How can I prove that $~\displaystyle\sum_{k=2}^{\infty}\frac{\sin(k!)}{k\log(k)}~$ converges ? Both Leibniz's criterion and Dirichlet's test seem rather inadequate for handling this particular task.
1
vote
0answers
53 views

Approximation of functions by trigonometric polynomials?

I'm seemingly not understanding the fact that a continuous function $f$ that is periodic on [0, 1) can be approximated by a trigonometric polynomial of the form $$g(x) = \sum_{n = 0}^k c_n e^{2\pi inx}...
1
vote
0answers
73 views

Harnack's curve theorem for zero sets of real trig polynomials in two variables?

Is there a result like Harnack's curve theorem for real trig polynomials which gives bounds for the number of connected components the zero set can have in terms of the degree? More specifically, let ...
1
vote
0answers
341 views

Fourier Series; odd and even half-range expansion

I have some standard Fourier series questions which I cannot solve. My fourier series is defined like this: $$s(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos (nx) + b_n \sin (nx))$$ For $f(t) = \...
1
vote
0answers
30 views

integrate trigonometric function

I have to determine the value of $\int_{-\pi }^{\pi } \left(\cos (3\cdot x)\cdot \sin (x)^2\right)^2 \, dx$. What I did was rewriting the integrand as $$\frac{1}{16}(6\cdot \sin (x)-\sin (3\cdot x)-3\...
1
vote
0answers
45 views

Understanding the indices in a Fourier series

Sometimes the truncated Fourier series of a function with Fourier coefficients $\hat{u}_k$ is written $$\sum_{k=-N}^N\hat{u}_ke^{ikx}$$ which is a linear combination of $\cos(nx) +i\sin(nx)$ for ...
1
vote
0answers
157 views

Cosine series as a Fourier series.

Theorem: Let $a_{n}\downarrow 0$ and suppose that $\left(a_{n}\right)$ is a quasi-convex. Then $\displaystyle \frac{a_{0}}{2}+\sum a_{n}\cos nx$ is the Fourier series of the $L^{1}$ function $\...
1
vote
0answers
218 views

Trigonometric series as a Fourier series of essentially bounded function.

A trigonometric series $ \displaystyle \frac{a_0}{2}+\sum_{n=1}^{\infty}(a_{n}\cos nx +b_{n}\sin nx)$ is a Fourier series of a essentially bounded function if and only if there exists a constant $K$ ...
0
votes
0answers
49 views

Expression for $\arccos\frac2\pi$

Are there any 'nice' expressions for $$\arccos\frac2\pi$$ By 'nice', I mean involving rationals, square roots, powers of $\pi$, etc... I'm hoping there is something that doesn't involve sum or ...
0
votes
0answers
24 views

What is support and spectrum of this nonnegative trigonometric function (or Finite Fourier Sum )?

This is a follow up of another question. The zeros of the following cosine sum shows the prime distribution, and the gap between the zeros can help to study the gap between prime numbers. $$ P(p_i,x)=\...
0
votes
0answers
23 views

On a certain series of cosines

For natural numbers $p$ and $q$, compute the value of $$\displaystyle \sum_{k=0}^{q-1} \cos^{p} \left(\dfrac{2\pi k}{q}\right).$$ I got the answer $$\dfrac{q}{2^p} \sum_{l=1}^{p} \binom{p}{l} \...
0
votes
0answers
31 views

For a real number $r>0$ for the function $\cot(z)$ has a Laurent series where 0<|z|<r. What is the largest possible r?

Okay so I know that $\cot(z)$ expands as: $$ \cot(z) = \frac{1}{z} - \frac{z}{3} - \frac{z^3}{45} - \ldots$$ But I'm unsure how to find the largest $r$. I'm assuming this is the point where it ...
0
votes
0answers
26 views

Defining trigonometric functions for obtuse angle

I realized that following is apparently true: we had definitions of sine, cosine, etc. for angles of right triangle. Then, one suddenly draws X and Y axes, incorporates negative numbers, and sees that ...
0
votes
0answers
43 views

How can I change a summation of cosines to a product of cosines for higher degree functions?

I was wondering if I can get an alternate form of a sum over cosines $$\sum_{n=1}^m\cos{f(n)}$$ and I found that I can. We must however make a modification to the upper limit with $m\rightarrow2^m$. ...
0
votes
0answers
32 views

Sum of treble trigonometric Series

I am very much thankful to you if you can help me to find summation of following series. $$\sum_{i=1}^{N-2} \sum_{j=i+1}^{N-1} \sum_{k=j+1}^{N} [ \sin (ix-jx) + \sin (jx-kx) + \sin (kx-ix)]^2$$
0
votes
0answers
20 views

Find convergent majorant series

I have a series $\sum_{n=1}^\infty \frac{1}{n^3} (\cos(nt) - 2 \sin(nt))$ and I want to find a convergent majorant series. I know that I have to find $k_n$ such that $|f_n(t)| = |\frac{1}{n^3} (\cos(...
0
votes
0answers
39 views

$\sum_{n=0}^x \sin(n)$?

I saw an article somewhere asking about the question $\sin 1 + \sin 2 + \sin 3 + ... +\sin 90$, and the answer the article gave was ~$57.76..$ But a quick graph in Desmos reveals that the titular sum ...
0
votes
0answers
88 views

Express ∑sin(kx)/k as the sum of a function and the integral of another function

I wish to find functions $g_n$ and $h_n$ defined on $(0,π)$such that 1. $\displaystyle\sum_{k=1}^n \frac{\sin(kx)}{k}= g_n(x)-\int_x^π h_n(t) \, dt$ 2. $\displaystyle h_n(x)≥0$ For all $x \in (0,π)$...
0
votes
0answers
26 views

Can I use a change of variable to simplify this series of Sines?

I have a function that looks something like: A sin(x)+B sin(2x)+C sin(3x)+D sin(4x)+...+M sin(Nx) Is there some change of variable I can use to turn this into a polynomial? Or more generally - are ...
0
votes
0answers
31 views

simplify the summation of fraction of two sinusoidal functions

How can I simplify the expression bellow $$\sum_{r=0}^{N-1}\frac{\sin^2(\pi\epsilon)}{\sin^2\bigg(\dfrac{\pi(r-n+\epsilon)}{N}\bigg)}$$ where $n$ and $N$ are integers? Does this summation equal $1$...
0
votes
0answers
34 views

Least period of sum of simple sines and cosines.

Given a finite summation of sines and cosines of the form $$\sum\limits_{i=1}^{n} \left(A_i\sin(\omega_ix)\right)+\sum_{j=1}^p \left(B_j\cos(\sigma_jx)\right)$$ where $A_i,\omega_i\in\mathbb{Z}$ $\...
0
votes
0answers
66 views

Trigonometry - word problem - functions cosθ = adjacent/hypotenuse

I'm not sure what trigonometric equation I should use for this problem: At a certain instant, a ship was 5km south of a lighthouse. The ship was travelling westward and after 30 minutes it's bearing ...
0
votes
0answers
41 views

Fourier series question - represent $x$ as a series of $\cos$

I was asked to represent $f(x)=x$ in $(0,\pi)$ as a sum of $\cos$ functions, using fourier series. I couldn't solve it on my own, but here is what the teacher did, and I don't fully understand why ...