I'm struggling with a Fourier serie. I need to find the Fourier series of the following function.
That's the function under study:
$f(t)=\left[\sqrt{1-k^2\sin^2t}\,\right]$.
The function is even and $\pi$-periodic.
The Fourier series should be in this form:
$f(t)=\frac{a_0}2+\s...
I want to find all critical points of the following nonlinear system:
$$\def\b{\begin{pmatrix}}\def\e{\end{pmatrix}}$$
$$\b y_1' \\ y_2'\e = \b 5y_2 -15 \\y_2^2 - y_1 ^2\e$$
Then use linearisation to find the type and stability of the critical points.
So first of all, finding the critical po...
First, let $-\log(\sin(x))=y$ that yields
$$\underbrace{\int_0^1 \log(y) \log(1-e^{-2 y}) \ dy}_{\displaystyle \pi^2 \log(A)-\frac{1}{12}\pi^2 \log(A)}-\frac{1}{2}\underbrace{\int_0^1 \log(\pi^2+y^2) \log(1-e^{-2 y}) \ dy}_{\displaystyle\sum_{k=1}^{\infty} \frac{\operatorname{Ci}(2k \pi)}{k^2} -...
First, let $-\log(\sin(\theta))=y$ that yields
$$\underbrace{\int_0^{\infty} \log(y) \log(1-e^{-2 y}) \ dy}_{\displaystyle \pi^2 \log(A)-\frac{1}{12}\pi^2 \log(\pi)}-\frac{1}{2}\underbrace{\int_0^{\infty} \log(\pi^2+y^2) \log(1-e^{-2 y}) \ dy}_{\displaystyle\sum_{k=1}^{\infty} \frac{\operatornam...
We have the following representation of pi:
$$\pi=\lim_{n\to\infty}2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\dotsb+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}}}}}_{n\text{ square root signs}}$$
which can be proven using the identity $\sin\left(\dfrac\pi{2^{n+1}}\right)=\dfrac12\underbrac...
