# Tagged Questions

Questions on periodic functions, functions $f(x)$ that satisfy the identity $f(x+c)=f(x)$, for some nonzero $c$.

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### What is the period of $f(x)=\sin x\cos x$?

Problem We need to find the period of the following: $f(x)=(\sin(x))(\cos(x))$ using basic trigonometric identities which is as follows: My steps disclaimer! I know the steps but I will pin point ...
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### How to find f(23) given part of the graph and that f(x)=f(x+6)? [closed]

I am very confused, how can $f(x+6) = f(x)$ for all $x$ since $f(x+6)$ would be a shift version of $f(x)$, unless it is a straight line? is the answer D? $6+6+6+6 =24$ $\Rightarrow$ $f(23) = f(5) = 4$...
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### periodic period for Markov Chain

I don't understand why the only state with period > 1 is 1 Let's take state 2 for example, what's the period for state 2? Another question is, does an absorbing state(state 4 in this example) only ...
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### Number of zeros in cosine for one period

Stupid I know, but for $\cos(kx+\phi)$ is the number of zeros in the first period always two? Just unsure of myself.
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### Nonlinear first-order differential equation with periodic bounded solution

Let $f:\mathbb R\times \mathbb R\longrightarrow\mathbb R$ be a $C^1$ function, periodic in the first variable, and such that $f(t,1)\leq > 0\leq f(t,0)$ for all $t$. Consider the ...
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### Riemann integrability of trigonometric functions clarifying

Consider a function which is periodic with period $2\pi$, and is Riemann integrable on the closed interval $[-2\pi,2\pi]$. Now, can we say that the function is Riemann integrable on every bounded ...
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### Projection of periodic trajectories

Let $(\bar x(t),\bar u(t)),\, t\in [0,1]$ solution of $$\left \{ \begin{array}{l} \dot x_1 (t) = u(t)\, f(x(t)) \\ \dot x_2 (t) = u(t)\\ x_1(0) = 0, x_1(1)=1 \\ x_2(0) = x_2(1) \end{array} \right.$$ ...
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### Is this non-constant function periodic for every definable number?

Given the set $\mathbb{D}$ which contains all definable real numbers. The definition must not be infinite long. E.g. it contains $12$, $-3$, $\frac{1}{12}$, $\sqrt{2}$, $\pi^2$, $i+e$, Chaitin's ...
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### Integration of periodic function $f \in L^1([0, 2\pi])$

While studying trigonometric series and $L^p$ spaces I was wondering the following: Let's say we have a $2\pi$-periodic function $f \in L^1([0, 2\pi])$ satisfying $\int_{0}^{2\pi}f(x) \, dx = 0$. Is ...
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### Singularity of $f(z)=f(z+a)=f(z+b),\ a,b\in\mathbb C$ does not exist

Let $f(z)=f(z+a)=f(z+b),\ a,b\in\mathbb C$ be a not constant meromorphic function, which is periodic and let $a,b$ be lineare independent. Show that $f$ has no zeros or singularities on the boundary ...
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### Bound on Rayleigh quotient on $H^{1}_{\text{per}}[0,1]^{n}$

Let $Q=[0,1]^{n}$. Define a $Q$-periodic function as follows: A function $f:\mathbb{R}^{n}\to\mathbb{C}$ is said to be $Q$-periodic if for any $x\in\mathbb{R}^{n}$ and any $z\in\mathbb{Z}^{n}$, it ...
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### What is the period of $f(x) = \cos (x) \cos(2x) \cos(3x)$? [closed]

What is the period of $f(x) = \cos(x) \cos(2x) \cos(3x)$? Please tell me the method plus the logic behind solving these kind of problems .. Plus is there any property for even functions like even ...
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### What is the period of $(2007)^{\sin x}$?

What is the period of $(2007)^{\sin x}$? Please explain how to proceed and what's the technique to generally solve these kind of problems.
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### For what value of $v_0$ is the solution periodic?

A solution of the second-order differential equation $$x''+x-x^3=0$$ satisfies the initial condition $x(0)=0$ and $x'(0)=v_0$. For what value of $v_0$ is the solution periodic? I have tried ...
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### Decomposition of periodic functions

Suppose that $f$ is a periodic function defined on the integers with period $mn$, with $m$ and $n$ coprime integers. Does there necessarily exist a function $g$ with period $m$ such that $f-g$ is ...
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### Floquet Boundedness for Floquet multiplier $|\lambda_i|=1$

The statement: Consider the system $x'=A(t)x$, where $A(t)$ is a periodic matrix with period T. If $|\lambda_i|=1$ then the corresponding Jordan block to $e^{TR}$ is diagonal. The constant matrix R ...
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### If $f(x+1)= -f(x-1)$, prove that the function f is a periodic function and its period is $4$

I sure do know that I should arrive to the point where $f(x)= f(x+T)$. I've tried replacing $x+1=a$ but the problem seems to get more and more complicated.
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### Differential equations that have non-sinusoidal periodic solutions

Examples help but I mostly just want to know what the criteria is for an equation to give non-sinusoidal periodic functions as solutions. https://en.wikipedia.org/wiki/Periodic_function the first ...