For questions about or related to trigonometric series.

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0
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3answers
333 views

How to find the sum of a geometric progression involving cos using complex numbers?

Use $ 2\cos{n\theta} = z^n + z^{-n} $ to express $\cos\theta + \cos3\theta + \cos5\theta + ... + \cos(2n-1)\theta $ as a geometric progression in terms of $z$. Hence find the sum of this progression ...
1
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2answers
69 views

Solving for linear velocity

A wheel is rotating at 3 radians per second and the wheel has an 80 inch diameter To the nearest foot per minute, what is the linear velocity?
2
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1answer
24 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
22 views

How do I find the direction angle of a vector?

v= -3i-10j I found the dot product to be 109 and the magnitude to be the square root of 11881. I divided it out and it came out to be one. I don't know what I'm doing wrong. Please help.
4
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0answers
91 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 $i=2,3,...,\lceil x/2 \rceil$)
4
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0answers
153 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}) ...
3
votes
0answers
63 views

Proof of a series containing normal as well as hyperbolic functions

Show that $$\sum _{n=0}^{\infty}\frac{\sinh(\pi n \sqrt 2)- \sin(\pi n \sqrt 2)}{n^3{\cosh(\pi n \sqrt 2})- \cos( \pi n \sqrt 2)}=\frac{ \pi^3}{18 \sqrt 2}$$ I have no hint as to how to even start.
1
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0answers
61 views

L1 norm of a trigonometric polynomial

For a real $x$, $f(x) = \sum_{k=-T}^{T}e^{ikx}$ is the well known Dirichlet kernel. It is also known that $\|f\|_{L_1}=\int|f(x)|dx \le C_1\log T + C_2$ for some $C_1,C_2$ independent of $T$. ...
1
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0answers
70 views

Extensions of Ramanujan's Ramanujan Cos/Cosh Identity

The Ramanujan Cos/Cosh Identity is as stated here . Then there is a line . Equating coefficients of $\theta^0$, $\theta^4$, and $\theta^8$ gives some amazing identities for the hyperbolic secant. ...
1
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0answers
25 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
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0answers
31 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 ...
0
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0answers
15 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 ...
0
votes
0answers
7 views

Is there a specific name for Fourier cosine series divided by its input?

I am thinking about a univariate model $$ y=a_{0}x^{-1}+\sum_{k=1}^{n}a_{k}\cos(kx)x^{-1}. $$ It seems that this form looks like a Fourier cosine series w.r.t. $x$ divided by $x$. Could you tell me ...
0
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0answers
35 views

Proving a Certain Inequality that Involves the Sinc Function

Could someone kindly show me how to rigorously prove that there exists a constant $ C > 0 $ such that $$ \forall N \in \mathbb{N}: \quad \sup_{x \in \mathbb{R}} \sum_{\substack{k \in \mathbb{Z} \\ ...
0
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0answers
122 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 ...
0
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0answers
138 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 $K$ ...