# 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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### number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
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### Finding period of a function? Am I doing something wrong?

I need to find the period of the function $$\large{\frac{\sin (\sin {nx})}{\tan{\frac{x}{n}}}}$$. According to me, the period of $\sin (\sin (nx))$ should be $\large{\frac{2\pi}{n}}$ and the period ...
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### Is there exact formula that returns minimal period of a periodic function?

Is there exact formula that returns minimal period of a periodic analytic function? For constant it should return 0, for non-periodic functions - infinity. I only came to the following but it ...
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### Differential Equation has a unique solution periodic

Let $A(t)$ continuous and periodic of period $S$ in $\mathbb{R}$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that ...
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### Integration of nontrivial trigonometric functions

First an example which I know how to solve. If we have the following integral $$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)}dt$$ there is a very practical way to evaluate it by interpreting it as some ...
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### Estimating pseudo-periodicity of signals

I have pressure data which are measured at a given point in a standing wave. These data(signals) are 'almost' sinusoidal in nature. Each cycle may slightly vary from the original signal i.e the ...
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### Cayley on “trivial transformations”

In his 1854 paper, "Deuxième mémoire sur les fonctions doublement périodiques" ("Second memoir on doubly periodic functions"), Cayley discusses (what we would today describe as) a certain class of ...
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### Does weak convergence (of a periodic function) imply weak-* convergence for the derivative?

Lets assume we have $b(x,t) \in L^\infty$ periodic in x and $\frac{db}{dt} \in L^{p}, p\in\mathbb{N}$. $b(x,t)$ converges weakly to an $f(t)=\int b(x,t)dx \in L^q$ for all $1<q<\infty$. Also,...
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### Does Every Periodic Function Have An Associated Differential Equation?

My question is the opposite to proving the existence of periodic solutions to ODE's. Assume that $\ f(z)$ is a periodic function over the $\mathbb{R}$ , or doubly periodic over some lattice $\Lambda$ ...
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### Why is $\int\limits_{u}^{u+\omega_j} f'(z)/f(z) dz \in 2 \pi i \mathbb{Z}$?

Because $f(u) = f(u+\omega_j)$ for $j \in \{1,2\}$ it applies $$\int\limits_{u}^{u+\omega_j}\frac{f'(z)}{f(z)}dz \quad \in 2\pi i \mathbb{Z} \quad \text{for} \quad j=1,2,$$ Hello, I write my thesis ...
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### period of the sum and the product of sin(ax) and cos(bx)

Take the function f(x)=sin(ax)cos(bx) , with a,b>0 . We know that the fundamental period of sin(ax) is p= 2π/a and the fundamental period of cos(bx) is q=2π/b. Suppose there are positive integer n ...
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### Ode with Piecewise function

We can write this $$12x"+36x'+48x=f(t)$$ my main problem is how to solve this non-homogeneous ODE I know how to do this as 2 different ode unfortunately its not in a syllabus which doesn't use ...
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So the task is to integrate $${\frac{1}T}\int_0^{2\pi}tf(t)\,dt$$ where $T=2\pi$ and $$f(t) = \left\{ \begin{array}{ll} -t^2 & \quad -\pi < t < 0, \\ t^2 &... 0answers 55 views ### Period of hypocycloid Hypocycloid is defined by the following parametric equations:$$x(\phi) = (a - b) \cos(\phi) + b \cos\big(\dfrac {a - b} b \phi\big)\\ y(\phi) = (a - b) \sin(\phi) - b \sin\big(\dfrac {a - b} b \...
Suppose $\alpha, L>0.$ Under what conditions (between $\alpha, L$) the Riccati equation $d\Phi/dz=2i[\Phi(z)^2+\alpha Cos(2\pi z/L)\Phi(z)+1]$ can have a periodic solution with period $L$ (under ...