# 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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### Constant functions periodic?

I dont understand the meaning of this line in my book - " $\sin^2x + \cos^2x$ is periodic but the fundamental period is not defined. " Why is the period not defined? $F(x)$ is $1$ here so it is a ...
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### Integral $\int_{-\infty}^{\infty}e^{-\frac{u^2}{2 t}}\,\cosh\left(u\right)\,K_{1/2+i\,u}\left(\frac{1}{4x}\right)\,du,\hspace{0.5cm}x,t>0$ is positive

I would like to prove that this function $$F_t(x)=\int_{-\infty}^{\infty}e^{-\frac{u^2}{2 t}}\,\cosh\left(u\right)\,K_{1/2+i\,u}\left(\frac{1}{4x}\right)\,du,\hspace{0.5cm}x,t>0$$ is positive, ...
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### In deriving $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$, why doesn't this imply $f$ is constant?

I was wondering if anyone can help me with this, if f(x) is a periodic function with period T then it satisfies $$\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)\;dx$$ for all $a \in \Bbb R$. It is clear that ...
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### An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.

I want to prove: For an integrable function $f(x)$ and periodic with period $T$, for every $a \in \mathbb{R}$, $$\int_{0}^{T}f(x)\;dx=\int_{a}^{a+T}f(x)\;dx.$$ I tried to change the values and ...
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### Integral of periodic function over the length of the period is the same everywhere

I am stuck on a question that involves the intergral of a periodic function. The question is phrased as follows: Definition. A function is periodic with period $a$ if $f(x)=f(x+a)$ for all $x$. ...
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### Definite integral of a periodic funtion, offset by the period, equals the original definite integral

Suppose $f: \mathbb R \to \mathbb R$ is Riemann integrable on every finite interval and periodic with period $T>0$. Then for every interval $[a,b]$: $$\int_a^b f = \int_c^d f,$$ where $c = a+T$ ...
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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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### 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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### 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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### Is $(\sin{x})(\sin{\pi x})$ periodic?

I'm wondering, is the function $f=(\sin{x})(\sin{\pi x})$ is periodic? My first inclination would be two assume that if the periods of the individual sine expressions, $p_1 \text{and}\space p_2$ have ...
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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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### Integrate a periodic absolute value function

$$\int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor$$ I ...
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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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### Sum of two periodic functions is periodic?

I have following paragraph taken from the Stanford's study material. Question: Is the sum of two periodic functions periodic? Answer: I guess the answer is no if you are Mathematician, yes ...
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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 ...
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Before this question is removed for being a duplicate, let me specify what I'm looking for. The book should have a treatment of the foundations of periodic/trigonometric functions. Students are often ...