For questions about or related to trigonometric series.

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6
votes
1answer
144 views

Is there any identity for $\sum_{k=0}^{n-1}\tan(x+ka) $??

I found this series $$ \sum_{k=0}^{n-1}\tan\left(\theta+\frac{k\pi}{n}\right)=−n\cot\left(\frac{n\pi}{2}+n\theta\right) $$ but it's not what I need.
5
votes
1answer
61 views

query about the cosine of an irrational multiple of an angle?

de Moivre's identity $$ (\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta $$ only applies as written when $n \in \mathbb{Z}$. if the exponent is a fraction $\frac{m}{n}$ then there will ...
3
votes
1answer
236 views

Extremal points of a sum of trigonometric functions

Show that the sum of trigonometric functions $$ f(x,y,z)=\cos(x+y+\alpha_1)+\cos(x-y+\alpha_2)+\cos(y+z+\alpha_3)\\+\cos(y-z+\alpha_4)+\cos(x+z+\alpha_5)+ \cos(x-z+\alpha_6) $$ where the $\alpha_i$ ...
2
votes
1answer
52 views

Solving $\operatorname{ctg} x=x/b$

I have no problems finding first solution (both: $b \to 0$ and $b \to \infty$). My solutions on photos. I got stuck trying to find solution when $x \to \infty$. As I think, solution for $x$ will have ...
1
vote
1answer
31 views

Find the limit of $\lim_{j\rightarrow \infty}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)$

Can anyone give a hint on how to see if the following has a limit? $f$ stands for frequency. $$\lim_{j\rightarrow \infty}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)$$ I've tried a few ...
1
vote
1answer
35 views

Trigonometry system - complex conjugate

I have the following function $$e_n(t) = e^{2\pi int}, t \in R, n \in Z$$ Could anyone explain how one can go from this: $$e_m(t) \bar e_n(t) $$ to $$e^{-2\pi i(m - n)t}$$ Shouldn't it be $e^{2\pi ...
1
vote
1answer
91 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
0
votes
1answer
33 views

When does $-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$ where $z$ is a complex variable?

Let $z$ be a complex variable. Is there someone who can show me when does :$$-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$$ Note: I have tried using trigonometric formulas but it didn't work. Maybe I ...
0
votes
1answer
44 views

Can you raise trigonometric functions to a non-integer power?

I don't inmediately see any reason why you could not yet I have never come across it. For any answer given reasoning would also be appreciated! Thank you
0
votes
1answer
59 views

Sum of trigonometric series $\sum_{m=1}^{N-1} \frac{\sin(4\pi mk/N)}{\sin ^2 (\pi m/N) }$

Anybody has some ideas to prove the following identity? \begin{equation} \sum_{m=1}^{N-1} \frac{\sin(4\pi mk/N)}{\sin ^2 (\pi m/N) }= 0 \end{equation} where $N$ is an integer greater than $1$, $k$ ...
8
votes
0answers
104 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 ...
6
votes
0answers
67 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 ...
6
votes
0answers
85 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
234 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
118 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
148 views
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
47 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
59 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 ...
3
votes
0answers
104 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 ...
2
votes
0answers
36 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 ...
2
votes
0answers
47 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} ...
2
votes
0answers
110 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} ...
2
votes
0answers
134 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
20 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
32 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
48 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 ...
1
vote
0answers
43 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 ...
1
vote
0answers
107 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
61 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
67 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
44 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 ...
1
vote
0answers
63 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
314 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 ...
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
154 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
201 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
41 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
30 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
12 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} ...
0
votes
0answers
33 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
74 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
votes
0answers
24 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
24 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 ...
0
votes
0answers
30 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
57 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 ...
0
votes
0answers
77 views

Find a short formula for $\sin x+\sin (x+y)+\sin (x+2y)+. . .+\sin (x+(n-1)y)$

The answer is : $$\sin(\frac {x+x+(n-1)y}{2}) \dfrac {\sin \frac{ny}{2}}{\sin \frac {y}{2}}$$ I could've written the question as: Show that..., but then people would try induction. What I did: ...
0
votes
0answers
50 views

Binomial series for $2^{n-1}\cos^n\vartheta$ and $2^{n-1}(-1)^{\frac{n}{2}}\sin^n\vartheta$

Can somebody confirm for me whether the following series are correct? $$2^{n-1}\cos^n\vartheta=\cos ...