# 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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### Prove or disprove: If $f$ is periodic, then $|f|$ is also periodic.

I started with the definition of periodic function and absolute value function. And I do it with discussing different cases of $x$ and $p$. But I got stuck with when $-p\leq x <0$ , I want to ...
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### References about functions of the following form $f(t)=f(t+P(t))$

Where I can find any references about real functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(t)=f(t+P(t)),\forall t\in \mathbb{R}$ ?
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### Why are the points of a period module necessarily isolated?

Given a non-constant meromorphic function $f(z)$ then how can we say that the absence of a finite accumulation point of our module $M$ implies that $f$ is constant? I understand that an accumulation ...
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### Integrals of a nonnegative, symmetric, periodic, continuous function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be such that $f$ is nonnegative, symmetric, $\ell$-periodic in both variables, zero exactly on the diagonal of its domain, and continuous. Specifically, $f$ has the ...
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### When talking about periods of a function why is it common to introduce a factor of $2$?

In many books I have read, when talking about periodic functions, they tend to write the period as $2\omega$. Why do we need the $2$? I haven't come across any working where the factor of $2$ is ...
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### proving periodic function f is uniformly continuous [duplicate]

So I'm new to uniform continuity and this is just an exercise that was scribbled on the board in my real analysis class. But it's is a tricky question for me: if $f:R\to R$ is periodic with period P ...
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### Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.

Let $f(x) = |x|$ for $x \in [-1,1]$ and extend $f$ to all of $\mathbb R$ by requiring $f(x+2)=f(x)$. Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$. This is easy to observe on a graph but my ...
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### How can $e^x$ be restated for small $x$?

Suppose I have the following equation: $$1=\frac{S_0}{\gamma}(1-e^{-\gamma T_n})+\lambda\int^{t_{n+1}}_{t_n}e^{-\gamma(t_{n+1}-\tau)}g(\tau)d\tau.$$ If I make two ...
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### Periodic boundary conditions: image summation vs periodic solution

Let's say we have a linear PDE and its solution is the field $\sigma(x,y)$. I want to use this field in a simulation, and in order to avoid boundary effects, I want to use Periodic Boundary ...
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### Is there a natural, circular (toric) Gaussian density?

I am looking for density distributions over a circular set (think about $[-\pi, \pi[$ or $\Delta+[0, T[$ in general $\forall T \in \mathbb{R}^{+*}, \Delta \in \mathbb{R}$). Is there a density ...
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### Closed periodic curve

Let $\textbf{$\gamma$}: \mathbb{R} \rightarrow \mathbb{R}^n$ be a smooth curve and let $T \in \mathbb{R}$. We say that $\textbf{$\gamma$}$ is $T$-periodic if ...
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### A function that is zero for every integer multiple of $k$

I'm new to this kind of question so it may be a trivial one but I can't find a general solution: I'm searching for a function that has a value of $0$ for every integer multiple of some fixed real ...
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### Proof: sum, multiplication and division of two periodic function

Proof that the sum, multiplication and division of two periodic function with the same period, is again a periodic function Please help, I need these three proofs for my calculus homework
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### Fundamental matrix of $y''+\epsilon f(t)y=0$

I converted this ode into a linear matrix form like $y'=Ay$ and tried to solve this, but I couldn't find a fundamental solution which satisfies $\Phi (0)=I$, which is required in one of my assignment ...
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### secret formula for the “sin” wave with variable rising/falling edge

My math is pretty much forgotten. I was wondering if someone can take a look at this and share what's the formula for creating something like this. ...
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### Passage in a proof from Hofer-Zehnder

The proof I'm referring to is to the following theorem. Assume $S$ is a compact regular and strictly convex energy surface for the Hamiltonian field $X_H$ in $\mathbb{R}^{2n}$. Then $S$ carries a ...
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### How to calculate the period of a nonlinear Diff Eq?

I have this Diff Eq: $$2g''+g'^{2}+ag+a-b=0$$ We can manipulate this into an equivalent relation: $$g'^{2}=ce^{-g}-ag+b$$ ...which makes c a conserved quantity: ...
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### Why do the periods of a meromorphic fuction form a discrete module?

I've come across this statement: If $f$ is a nonconstant meromorphic function, the module $M$ containing all its periods cannot have an accumulation point, since otherwise $f$ would be a ...
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### The number of periodic solutions of Abel differential equations

I'm asked to proof the number of T-periodic solutions of ODE $$x'=p_0(t)x^3+p_1(t)x^2+p_2(t)x+p_3(t)$$ is at most three, where $$p_i(t)$$ is continuous functions with period T (i=0,1,2,3) and ...
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### Evaluating a definite integral of a Bessel-type function

I have an expression as follows: $\int_{0}^{2\pi} \sin{(x\sin{(\theta}) - n\theta)}\mathrm{d}\theta$ For real $x$ and $\theta$ and positive integer $n$. From plugging it into Mathematica with ...
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### Show that $f(x):=\cos(x^2)$ is not periodic.

How can I proof that the following function $f(x):=\cos(x^2)$ is not periodic? I think that I should find the zero points of the function but I don't know how to calculate it. Thank you very much ...
### Let $f\colon\Bbb R\to \Bbb R-\{3\}$ be a function such that there exist $T>0$ with $f(x+T)=\frac{f(x)-5}{f(x)-3}$ for every $x\in\Bbb R$.
Let $f\colon \Bbb R\to \Bbb R-\{3\}$ be a function with the property that there exist $T>0$ such that $f(x+T)=\frac{f(x)-5}{f(x)-3}$ for every $x\in \Bbb R$. Prove that $f(x)$ is periodic and find ...