Questions tagged [trigonometric-series]

For questions about or related to trigonometric series, i.e. series of the form $a_0 + \sum_{n = 1}^{\infty} (a_n \cos{nx} + b_n \sin{nx})$ or $\sum_n c_n e^{inx}$.

Filter by
Sorted by
Tagged with
0 votes
0 answers
35 views

Central Limit Theorem for irrational rotation

If $ X=\mathbb{R}/\mathbb{Z}$ be the unit circle with $\mu$ Lebesgue measure, and suppose $\alpha\in X$ irrational, and $T:X\rightarrow X$ is defined by $T(x)=x+\alpha$. we want to find a function $f\...
user avatar
  • 1
0 votes
0 answers
20 views

A complicated sum of cosines involving sums inside the cosine arguments

Let $n$ be an integer greater than $1$. How do I show that $$\sum_{L_{1}=0}^{n-2}\sum_{L_{2}=0}^{L_{1}}\left(-1\right)^{L_{1}+L_{2}}\left(1+\left(-1\right)^{L_{1}+L_{2}}\cos\left(\pi\sum_{k=L_{2}+1}^{...
user avatar
1 vote
1 answer
91 views

Need advice on approximating the sum of a trigonomitric series which (I think) has no analytical solution

I've encountered a maths problem in a programming project I'm working on. I've tried a lot of things already, and I'm feeling very swamped with maths that is way above my head. I need to find: $$f_N(x)...
user avatar
  • 21
0 votes
0 answers
64 views

$ \sum_{k=1}^{n} \arctan(k)=? $ Using argument of complex number

Question: \begin{align} \sum_{k=1}^{n} \arctan(k)=? \end{align} My approach: \begin{align} \prod_{k=1}^{n} (1+ik)=a+ib\qquad(1)\end{align} \begin{align} \Rightarrow arg \Bigg(\prod_{k=1}^{n} (1+ik)\...
user avatar
12 votes
3 answers
208 views

Proving $\left|\sin x\right|\leq\frac{(2m+1)!}{2^{4m}(m!)^2}\left[\binom{2m}{m} - \sum_{k=1}^{m} \frac{2}{4k^2-1} \binom{2m}{m+k} \cos(2kx) \right]$

Update. Based on @Exodd's answer, it turns out that the upper bound is equal to $$ T_{2m}(\cos x) = \sum_{k=0}^{m} (-1)^k \binom{1/2}{k}\cos^{2k} x, $$ where $T_{2m}(x)$ is the degree $2m$ Taylor ...
user avatar
0 votes
0 answers
21 views

Notation for Expresion of Tangent of Summation of Angles

I am proving, assuming this expression leads to something correct, that the tangent of a summation of angles is an expression involving sums of products of the tangents of each angle, like this: $$ \...
user avatar
  • 2,700
1 vote
0 answers
29 views

Bound on amount of minima of finite Fourier sum

Consider a finite Fourier sum of the form $$f(\theta) = \sum_{i=1}^n r_i \cos(m_i \theta) \,,$$ where $n \geq 1$ is an integer, the $r_i$ are positive real numbers and the $m_i$ are integers. Is there ...
user avatar
  • 109
0 votes
1 answer
17 views

Showing that $R= \frac{1}{n}\sum_{k=1}^n\cos(\theta_k - \overline{\theta})$ given a system of equations

Let angles $\theta_1,\dots,\theta_n$ be given and define $C = \frac{1}{n}\sum_{k=1}^n\cos(\theta_k)$ and $S = \frac{1}{n}\sum_{k=1}^n\sin(\theta_k)$. Then, there exists a value $\overline{\theta}$ ...
user avatar
0 votes
0 answers
22 views

Relating complex logarithm power series to the arctan power series

I've been trying to directly convert the expected complex logarithm power series to the arctan series without success. I expected some interplay between the terms so that imaginary terms would cancel, ...
user avatar
0 votes
0 answers
40 views

Is it possible to express $\sin(\frac{k\pi}{4})$ as something of the form $(-1)^{f(k)}$?

For example $\cos(k\pi)=(-1)^k$ similarily, $\sin(\frac{k\pi}{2})=(-1)^\frac{k-1}{2}$ when k is odd, and 0 otherwise. Is there a similar representation for $\sin(\frac{k\pi}{4})$
user avatar
1 vote
0 answers
23 views

How to prove that $\sum_{ k= 1}^{N-1} \sin(\frac{k}{N} \pi)^{-2} = (N^2 - 1)/3$? [duplicate]

I find the equality \begin{equation} \sum_{k = 1}^{N-1} \sin(\frac{k}{N}\pi)^{-2} = \frac{N^2 - 1}{3}, \end{equation} during study. Just wonder how can I prove it? Short matlab code to do the ...
user avatar
  • 73
0 votes
0 answers
11 views

Fundamental period of sum of two sinusoidal signal and period

i would like to understand why is used LCM to find period of two sinusoidal signal and GCD for finding fundamental period of the same signal, let us consider following example: $x(t)=30*sin(2*\pi*...
user avatar
0 votes
2 answers
39 views

How to show $\displaystyle\sum_{j=1}^n \cos\left((k+l)\frac{\pi (j-1/2)}{n}\right) = 0$

This result seems trivial, how would I show $$\sum_{j=1}^n \cos\left((k+l)\frac{\pi (j-1/2)}{n}\right) = 0$$ Where $0 \le k,l \le n-1$ And $k \neq l \neq 0$. I tried to use the exponential definition ...
user avatar
  • 403
1 vote
1 answer
64 views

$\sin\left(\frac{\pi}{2020}\right)+\sin\left(\frac{3\pi}{2020}\right)+...+\sin\left(\frac{2019\pi}{2020}\right)=\csc \left(\frac{\pi}{2020}\right)$ [duplicate]

How to prove that the below equality holds true? $$\sin\left(\frac{\pi}{2020}\right)+\sin\left(\frac{3\pi}{2020}\right)+...+\sin\left(\frac{2019\pi}{2020}\right)=\csc \left(\frac{\pi}{2020}\right)$$ ...
user avatar
  • 1,776
0 votes
1 answer
33 views

Weighted sum of cos(kx) (weighted Lagrange identities?)

I am currently stuck with the following sum: $$f(x)=\sum\limits_{k=1}^{N-1}a_k\cos(\frac{kx\pi}{N})$$ Here $x\in[0,N)$. The coefficients $a_k$ are decreasing ($a_1\geq a_2\geq...\geq a_{N-1}$). I am ...
user avatar
0 votes
1 answer
70 views

Proof That $\sin{(bt)} = b\sin{(t)}$?

Note the Taylor expansion for $\sin{x}$: $$ \sin{x} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots $$ Now consider the Taylor expansion for $\sin(bt)$: $$ \begin{align} \sin{(bt)} ...
user avatar
1 vote
0 answers
24 views

Why is the term $(2n+1)$ in the solution angles of sine and also in its taylor sum? Is this coincidence?

Given $ \sin (x) = a $ the solutions are given by (with $ A = \arcsin a $) $$ x = A + 2 \pi n$$ $$ x = (\pi - A) + 2 \pi n = - A + (\pi + 2 \pi n) = - A + (2n + 1) \pi $$ The taylor sum is given by: $...
user avatar
0 votes
0 answers
13 views

Show that $g(x)=\sum_{n \ge 1} \varepsilon_n b^{-n}\cos(b^n x)$ is in the Zygmund class $\lambda_*$ if $\varepsilon_n \rightarrow 0$

In his book Trigonometric Series, Zygmund says : Theorem : Let $b>1$, $\varepsilon_n \rightarrow 0$ and $$g(x)=\sum_{n \ge 1} \varepsilon_n b^{-n}\cos(b^n x)$$ then $g \in \lambda_*$ Which means ...
user avatar
0 votes
0 answers
50 views

How do you find the roots of a sine series?

I would like to know if there is an analytical way to find the roots of series of sine functions given by $$ \sum_{n=0}^N \sin \left( \omega_n t \right) = 0 $$ Here, $\omega_n = \frac{1}{h}\textrm{...
user avatar
  • 191
0 votes
0 answers
113 views

Fitting a sum of sinusoids to known data

I have been trying for several months to solve what looks (on the surface) like a simple problem. I have 9 data points (t,y) and it is known that these points represent the turning points of a ...
user avatar
1 vote
1 answer
100 views

Baby Rudin's proof of Riemann Localization Theorem (8.14)

Question How the boundedness of $g(t)\cos\left(\frac{t}{2}\right)$ and $g(t)\sin\left(\frac{t}{2}\right)$ on $[-\pi,\pi]$ shows their Riemann-integrability in Rudin's proof? This is needed in the end ...
user avatar
0 votes
0 answers
30 views

Finding a periodic function with this fundamental cycle

I'd like to write a smooth ($C^{\infty}$), periodic function $f(x)$ with a fundamental cycle that looks like this: I include no scale because I'm only interested in the signs of $f(x)$, $f'(x)$, and $...
user avatar
  • 30.3k
1 vote
1 answer
60 views

Sinusoids as solutions to differential equations

It is well known that the function $$t \mapsto a \cos(\omega t) + b \sin(\omega t)$$ is the solution to the differential equation: $$x''(t) = -\omega^2 x(t)$$ with the initial conditions $x(0) = a$ ...
user avatar
0 votes
0 answers
37 views

Is my assumption in levy bending plates right?

I have a coding project where I have to solve an equation that could be simplified as: $$ \sum_{i=1}^{m} \sin(i*x)*f(C_i)=0, \forall x \in [0,\pi], $$ where I have to find all the $C_i$ (not $x$), $...
user avatar
4 votes
1 answer
153 views

What is known about sums of the form $\sum_{n=-\infty}^{\infty} \operatorname{sinc} (n^{p})$?

Recently, I've become fascinated with the whole 'sum = integral' concept. The sinc function harbours some great examples. For instance, the authors R. Bailie, D. Borwein and J. Borwein described in ...
user avatar
  • 5,027
1 vote
1 answer
59 views

Trigonometric terms for floor function $Q_k(n)$

I'm working on some problems of number theory and somehow I could manage to find a more general formula for some problems. However, I needed to define a function $$Q_k(n) = \text{floor}(\frac n{k}) \...
user avatar
1 vote
0 answers
87 views

How to calculate sums like $\sum_{n=2}^{\infty} \frac{ sinc (-4 \cdot \pi (n-3)(n-4) ) }{n(n-1)} $ with the Residue Theorem?

I'm trying to compute sums like $$S:= \sum_{n=2}^{\infty} \frac{ \operatorname{sinc} (-4\pi (n-3)(n-4) ) }{n(n-1)} $$ by means of the Residue Theorem, which states that $$\lim_{k \to +\infty} \sum_{k=-...
user avatar
  • 5,027
0 votes
1 answer
135 views

$\sum _{k=1}^n\:\left(\cos\left(\frac{2\cdot k\cdot \pi }{n}\right)-2\:+\:i\cdot \sin\left(\frac{2\cdot k\cdot \pi }{n}\right)\right)$

$\sum _{k=1}^n\:\left(\cos\left(\frac{2\cdot k\cdot \pi }{n}\right)-2\:+\:i\cdot \sin\left(\frac{2\cdot k\cdot \pi }{n}\right)\right)$ Normally the general factor is $a(n)=\cos\left(\frac{2k\pi }{n}\...
user avatar
0 votes
0 answers
54 views

Proof of series expansion of $\operatorname{arctanh}(e^{is})$ for $s\in(0,\pi)$.

I would to know how to prove that the series expansion of $\operatorname{arctanh}(e^{is})$ for $s\in(0,\pi)$ is: $$\sum_{m=1}^{+\infty}\dfrac{e^{is(2m-1)}}{2m-1}.$$ We know by Taylor series that $$\...
user avatar
  • 43
2 votes
3 answers
96 views

How to calculate the inverse of $f(t) = \frac{(k/2)(\sin(t) + t) - t}{2π (k/2 - 1)}$?

This function arises from a specific problem in optical engineering. I am modeling a light source whose amplitude is being modulated sinusoidally. I have derived this function which describes the ...
user avatar
2 votes
1 answer
120 views

$\cos(\cos(\cos(\cos(\cos(\cos(\cos(....(\theta)))))))))$ approaches a constant as the number of cosines increases [duplicate]

I was experimenting with the software geogebra, and playing with a couple of unusual trigonometric functions and I encountered a quite strange phenomena when I entered this input - $$f(\theta)=\cos(\...
user avatar
0 votes
0 answers
53 views

Apply the Abel summation formula to $\sum_{i=1}^k \operatorname{sinc} \bigl( \pi (x-i p) \bigr)$

As part of an exploration of the Abel Summation formula (see here), I am looking at an impulse train $T$ made up of $k$ $\operatorname{sinc}$ pulses at intervals $p$ along the $x$ axis: $$T(x):=\sum_{...
user avatar
28 votes
1 answer
706 views

Is $\sum_{a=0}^m\sum_{b=0}^n\cos(abx)$ always positive?

Fix integers $m,n\geq0$. Do we have the inequality $\displaystyle\sum_{a=0}^m\sum_{b=0}^n\cos(abx)>0$ for all $x\in\mathbb{R}$? We can also write this function as \begin{align*} \sum_{a=0}^m\sum_{...
user avatar
1 vote
1 answer
73 views

Simplify infinitely recursive function $f(a,b) = a+b\sin( a + b\sin( \cdots ( a + b\sin(a) ) \cdots ) )$

Is there a mathematical method to simplify this infinitely recursive function? I have tried to approximate it, but for work with the ranges I am considering, precision becomes an important factor. the ...
user avatar
0 votes
1 answer
56 views

Solve definite integral of infinite series of a complex function

I need to calculate the following integral. $$\int _{\frac{-2 \pi}{L}}^{\frac{2\pi}{L}} \left( \sum _{n=0}^{\infty} \frac{\left(j \chi \rho \cos\left(\xi - \theta\right) \right)^n}{n!}\right) d\xi$$ ...
user avatar
3 votes
2 answers
75 views

Is there a positive integer $N$ for which $\sum_{n=1}^N \cos\left(n^2x\right)=0$ for every $x\in[0,2\pi]$?

I'd like to know whether or not the trigonometric sum $$\sum_{n=1}^N \cos\left(n^2x\right)$$ is ever identically $0$ on $[0,2\pi]$. Plotting the graph of $\sum_{n=1}^N \cos\left(n^2x\right)$ for a few ...
user avatar
  • 4,305
0 votes
1 answer
49 views

Pade approximation of $1-\frac{(1-x^2)\sin^2(\theta)\sin^2(\theta-y)}{(1-(1-x^2)^{1/2}\cos(\theta)\cos(\theta-y))^2}$ upto $2^{nd}$ order

I am very new to Pade' approximation concept, so some detailed derivation for the approximate result of the following function would be very helpful. The function that I wish to approximate in the ...
user avatar
  • 69
1 vote
1 answer
53 views

Sum $\sum_{r=1}^n \cos(2.(\frac {3^rx}{3})).\csc (3^rx)$

Prove that $$\sum_{r=1}^n \cos(2.(\frac {3^rx}{3})).\csc (3^rx) = \frac{1}{2\sin x}-\frac{1}{2\sin (3^nx)}$$ My attempt: $$\Sigma \frac{\cos(\frac {2.3^rx}{3})}{\sin(3^rx)} = \Sigma \frac{1-2\sin^2(\...
user avatar
  • 120
1 vote
0 answers
51 views

Challenging series with cosine, sine and a variable to be expressed as a polynomial

In my research work, the following two series come out describing the 1st and 2nd order of a physical phenomenon: $$\mathfrak{f}_1(\alpha )=2\cdot\sum_{k=1}^{+\infty} \ \begin{cases} \...
user avatar
1 vote
1 answer
76 views

Challenging series of a fraction with a cosine and a variable

Is it possible to express the following series explicitly (e.g. as a polynomial in $\alpha$): $$f(\alpha )=\sum_{k=1}^{\infty} \frac{\cos(\alpha2\pi k)}{(2\pi k)^2((2\alpha k)^2-1)^2} , $$ where $0\...
user avatar
1 vote
1 answer
132 views

Uniform convergence of a series of functions with cos(x)

I have to solve the following problem about series of functions: Study the uniform convergence of the following series : $$\sum_{n=1}^\infty \frac{1}{\sqrt{n}}\left(1-\cos\left(\frac{x}{\sqrt{n}}\...
user avatar
  • 99
0 votes
1 answer
72 views

How to write a square of a trigonometric polynomial cosine?

How to write a square of a polynomial of the form $$\left(1 + 2\sum_{k=1}^n a_k \cos k \theta\right)^2$$ with an explicit formula for just the coefficient of $$\cos k\theta$$ in terms of $k$ and the ...
user avatar
  • 11
0 votes
1 answer
51 views

Simplify infinite cosine series

I'd like to simplify the following expression $$\sum_{n=1}^\infty \cos\left(\frac{n\pi x}{L}\right)e^{-a\left(\frac{n\pi}{L}\right)^2}-(-1)^{-bn\left(\frac{n\pi}{L}\right)^2}\cos\left(\frac{n\pi x}{L}\...
user avatar
  • 75
1 vote
0 answers
126 views

Evaluating $\mathrm{\int\limits_0^1 D(x) dx=I_2(1)-1-\sum\limits_{n=2}^\infty \frac{n\cos(n!)}{n!^2}}$. D(x) is the Darboux fractal function.

This question takes inspiration from Evaluating $$\mathrm{\int_0^1 ?(x)dx}$$ and other fractal questions. Let me introduce the Darboux function which has fractal properties due to being a continuos ...
user avatar
  • 5,219
1 vote
1 answer
76 views

Is there an easy way to solve this real-world circle/congruent triangles/trigonometry problem?

This is a real-world problem I've got to deal with, so help appreciated. The diagram looks like this: The diagram shows 4 congruent sectors of a circle radius r, each of angle 2A, cut into triangles ...
user avatar
  • 1,021
5 votes
1 answer
283 views

If $\sin(n!\, x)\to 0$ as $n\to +\infty$, is then $x$ inevitably a rational multiple of $\pi$?

If $x$ is a rational multiple of $\pi$, for a natural number $N$ big enough $\sin(n!\,x) = 0$ for all $n\geqslant N$ and then $\sin(n!\,x)\to 0$ as $n\to +\infty$. However, I'm not so sure about the ...
user avatar
4 votes
1 answer
392 views

On $\mathrm{\sum\limits_{n=0}^\infty \left(C(n)-\frac{\sqrt\pi}{2\sqrt2}\right)+ \sum\limits_{n=0}^\infty \left(S(n)-\frac{\sqrt\pi}{2\sqrt2}\right)}$

This question will take inspiration from Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$? and On $\mathrm{\sum\limits_{x=1}^\infty Ci(x)}=\frac{\ln(2)+\ln(\...
user avatar
  • 5,219
4 votes
1 answer
297 views

On $\mathrm{\sum\limits_{x=1}^\infty Ci(x)}$.

I have held out on asking this question as it seems a bit simple, but I have also asked some similar summation questions. This brought about the idea of adding it to the collection. The problem uses ...
user avatar
  • 5,219
0 votes
1 answer
27 views

Can the coefficients of a trigonometric series bounded by its infinite norm?

Suppose $$f(x) = \sum_{k=1}^n c_k\cos(kx)$$ satisfies $$|f(x)|\le 1,\quad \forall x\in[0,2\pi]$$ then does there exist a constant $A$ which does not rely on $n$ and the inequality $$\sum_{k=1}^n |c_k|&...
user avatar
  • 329
5 votes
1 answer
136 views

Showing $ n=\sum_{k=1}^{(n+1)/2}\sin{\frac{2\pi k}{n+2}}\sin{\frac{\pi(n-2(k-1))}{n+2}}\sec^2{\frac{\pi(n-2(k-1))}{2n+4}}$ for natural $n$

Good afternoon, I am a little confused and intrigued by this finite summation formula I came up with. If $n$ is a natural number then $$ n=\sum_{k=1}^{(n+1)/2}\sin{\frac{2\pi k}{n+2}}\sin{\frac{\pi(n-...
user avatar
  • 524

1
2 3 4 5
17