# Questions tagged [periodic-functions]

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

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### If a function $f$ is $L$-periodic then $f'$ has $2$ zeros in $[0,L)$?

Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a differentiable and odd function. If $f$ is periodic and the (minimal) period $L>0$, then $f'$ has $2$ zeros in $[0,L)$? For example, this ...
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### Sine Parametric function exercise

Find the biggest negative value of $a$ , for which the maximum of $f(x) =sin(24x+\frac{πa}{100})$ is at $x_0=π$ The answer is $a=-150$, but I don't understand the solving way. I would appreciate if ...
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### How to find fundamental period?

Suppose we have a function $h(x) = f(x) + g(x)$ where $f$ and $g$ are periodic functions with fundamental period $T_1$ and $T_2$ respectively. We know that the least common integral multiple of $T_1$ ...
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### Period of trig functions

How to prove the periodicity and then find the primitive period of summation, products and compositions of trig functions? Is it possible to prove that the primitive period of sine function is $2\pi$ ...
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### Finding the function $f(x)$ through the given set of conditions

I got terribly stuck while solving this question recently Let $f(x)$ be a real valued function such that $$f(0)=\frac{1}{2}; \quad f(x+y) =f(x) f(a-y) +f(y) f(a-x).$$ I do not want the rigorous ...
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### Period of $\sin^{6}(x)+\cos^{4}(x)$

I am trying to find the fundamental period of $\sin^{6}(x)+\cos^{4}(x)$. Now the period of $\sin^{6}(x)$ is $\pi$ and that of $\cos^{4}(x)$ is also $\pi$, so expectedly the period of the expression ...
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### Every period of function is multiple of fundamental period

Suppose $f : \mathbb{R} \to \mathbb{R}$ is periodic with $T$. Is it necessarily true that $T$ is a multiple of fundamental period $T_0$? Obviously every multiple of $T_0$ is a period. Will the other ...
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### Why does this functional equation follow from periodicity?

I'm reading a proof where they say that given $$\phi(x) = \Gamma(x)\Gamma(1-x)\sin \pi x$$ and $$g(x) = [\log \phi(x)]''$$ then, since $g$ is periodic with period 1, it satisfies the functional ...
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### Trying to see if there is correlation between the Corona-Virus transmission rate per day in NY city (COVID-19) and the humidity that day

There should be a correlation between relative humidity and the transmission rate of COVID19. However, since there are no data available for transmission rate per day, I used the # of deaths as a ...
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### Periodic, Infinitely Differentiable Function Dense in $L^2$

I have encountered an interesting question, which seems to have a simple solution. Consider $E$ as the set of $2\pi$ periodic, complex-valued, infinitely differentiable functions s.t. $\forall f\in E$...
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### Prove Existence Periodic Solution Nonlinear Pendulum with Torque

Suppose we have the following system $$\theta'=v$$ $$v'= -bv - \sin\theta + k.$$ I need to prove that there exists a periodic solution in the region where $k>1$. The hint that was given is to ...
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### How to get the periodic solution of the nonlinear PDE?

How to get the periodic solution of the nonlinear PDE? i.e. the equation $iq_{t} +q_{xx} = i(|q|^{2}q)_{x}$ has the priodic solution $q = ke^{ia[x-(a-k^{2})t]}$, where $a$ and $k$ are ...
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### Generalization of periodicity

We know that a periodic function (e.g. a trigonometric function) has the property $$f(x+n\Lambda)=f(x) \qquad n\in\mathbb Z$$ A Bessel function is not exactly periodic, because the value of the ...
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### Functional equation $f(x+1)=af(x)+b$

Functional equation $f(x+1)=af(x)+b$ There was a question I solved a few days back that asked for a closed form of an equation for a given system. The function came down to this equation which I ...
Find the only periodic solution for $y'+y=b(x)$ with $b:\Bbb{R}\to\Bbb{R}$ has a period of $2T$ and is $1$ for $x (0,T)$ and $-1$ for $x (-T,0)$. The ODE is easy to solve: $y(x) = \exp(-x)\cdot c+1$ ...