# 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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### 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. https://drive.google.com/file/d/...
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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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### 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 ...
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### 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 ...
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### Help with Proving : Period estimation for for concatenated sequences

Assuming I have two 8-bit random number sequences $s[n]$ and $d[n]$ which each have a period of $X$ and $Y$ respectively. Therefore: $$s[n+X] = s[n]\\ d[n+Y] = d[n]$$ If they were concatenated ...
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### Defining periodic functions?

Consider $f: (a,a+p] \rightarrow \mathbb{R}$. What is the "formula" for the p-periodic function $g$ which has the property that $g(x + np) = f(x)$ for all $x$ in $(a, a+p]$? I am well aware of how ...
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### Function with Multiple Periods

Basically I'm trying to fit some data with seasonal effects to a periodic function, and the problem I'm running into is that the local minima usually occur around April, and the local maxima usually ...
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### Solving differential equation with Fourier-series-inhomogenity

Let $\lambda$ be a real number , $(c_k)$ a complex sequence with $\mid c_k \mid \leq C(1+\mid k \mid)^{-2}$ for all k with a constant $C \geq 0$. Find all periodic, two times differentiable ...
Consider an arbitrary function, something like $f\left ( x \right )=\arccos \left ( \sin \left ( 4x \right ) \right )$. Its graph looks like this: I was greatly confused by the image below, because ...
### All solutions to functional equation $f(x+1)-f(x)=1$
I was thinking of the possibility of finding all solutions other than $f(x)=x$ for the functional equation: $f(x+1)-f(x)=1$ If there are other solutions, what will be some restrictions for the ...