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### Integral $\int_0^{\frac{-\pi}{2}} \frac{(2\pi i + \epsilon e^{i\theta})^4i\epsilon e^{i\theta} d \theta} {e^{2 \pi i + \epsilon e^{i \theta}} - 1}$

I'm not sure to see how the integral $$\int_0^{\frac{-\pi}{2}} \frac{(2\pi i + \epsilon e^{i\theta})^4i\epsilon e^{i\theta} d \theta} {e^{2 \pi i + \epsilon e^{i \theta}} - 1}$$ yields a finite value ...
89 views
+100

### How can I solve to a closed form of this sequence?

I was studying some things and I’m stumped with the sequence $$c_n=-\frac{1}{n}\sum_{k=0}^{n-1}c_{n-k-1}a_{k+1}, c_0=1$$ Does it have a closed form? I know it needs to have, because we can see by ...
3 views

### Solve $\dfrac{x-1}{2}\geq \dfrac{2}{3} \geq 1-\dfrac{x}{6}$

Solve $\dfrac{x-1}{2}\geq \dfrac{2}{3} \geq 1-\dfrac{x}{6}$ Multiply everything by $6$: $3x-3 \geq 4 \geq 6-x$ Evaluate separately: $3x-3 \geq 4 \Rightarrow 3x \geq 7 \Rightarrow x \geq \dfrac{7}{3}$ ...
1 vote
827 views

50 views

### Inverse Fourier Transform - convolution of exponential and rectangular window

I'm trying to get the response in the time domain of the convolution between the exponential $u(t)e^{-at}$ and the rectangular window ($u(t+1)-u(t-1)$). I had already obtained its result by ...
27 views

### Weird error behavior when implementing Euler's method for solving ODEs in Python

I am solving this particular ODE: $$\begin{cases} u'(t)=-2\,tu^2/20, \quad t \in [0,\sqrt{20}]\\ u(0) = 1 \end{cases}$$ which the analytical solution is given: $$u(t) = \frac{1}{1 + t^2/20}.$$ ...
1 vote
3k views

### Bootstrap estimation of the 95% confidence intervals for the 95% quantile for gamma distribution

I cant find any where information or algorithm how to apply in steps the bootstrap procedure to estimate the 95% confidence intervals for the 95% quantile from a random sample. Does anyone knows how ...
1 vote
96 views

### Probability of n Periodic Events per Attempt, Skipping m = 20 Attempts in a row

The problem: There is a system with a clock ticking at a constant rate measured every second. Within this system there are n loops that are ongoing. The period of the loop is between 80 and 120 ...
12 views

1 vote
152 views

### A question based on intermediate fields and separability

This problem was asked in my Abstract algebra assignment and I was unable to solve it so I am asking for help here. Let E be an intermediate field . (a) If u $\in$ F is separable over K, then u is ...
154 views

### Closed form for $\prod_{n=1}^{\infty} \frac{e^{\frac{x}{2^n}}-1}{\frac{x}{2^n}}$

I would like to know if there is a closed-form for \begin{align*} \prod_{n=1}^{\infty} \frac{e^{\frac{x}{2^n}}-1}{\frac{x}{2^n}}, \end{align*} where $x\in\mathbb{R}$. The closest results I could ...
27 views

### Fourier analysis of sine: magnitude result clarification

I want to calculate the Fourier transform of $sine(2 \pi 60 t)$ and I have difficulty understanding/calculating the magnitude result. It is already known that the only frequency on this sine function ...
4 views

### One dimensional regularity obstacle problem ($H^2$ regularity)

I am reading the proof of the $H^2$ regularity of the obstacle problem in the book Numerical Methods for Nonlinear Partial Differential Equation by Sören Bartels. I understand everything he does in ...
267 views