robjohn

West Hills, CA

none

Age: 56

the mean square The Mean Square
(with one standard deviation and several unusual ones)

aka Rob Johnson

$\LaTeX$ support for Chat

22h
comment Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$
@Aldon: I have expanded the explanation of the Complex Analysis Approach. Note the explanation of $(8)$.
22h
revised Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$
expand the explanation
23h
comment Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$
@Aldon: you missed the next line, which is a continuation.
23h
comment Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$
@user1952009: indeed, there are many ways to compute the integral.
1d
comment Determining the angle degree of an arc in ellipse?
Between $\theta=0$ and $\theta=2\pi$, $b\,\text{EllipticE}\left(\theta,\frac{b^2-a^2}{b^2}\right)$ gives the entire circumference. Perhaps I am missing something.
1d
awarded Revival
1d
answered How does computing the determinant of a matrix with unit vectors give you the Cross Product?
1d
revised Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$
complete the real approach since the answer was given in the complex approach
1d
revised Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$
add an image of $\gamma$
1d
revised Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$
add a complex analysis approach
1d
answered Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$
1d
revised Prove that $\int_{0}^\pi x^{2k} \cos(h x) dx\geq 0$.
generalize and simplify the proof
1d
comment Prove that $\int_{0}^\pi x^{2k} \cos(h x) dx\geq 0$.
Inequality $(5)$ was discussed in chat a little over a year ago.
1d
answered Prove that $\int_{0}^\pi x^{2k} \cos(h x) dx\geq 0$.
1d
revised Prove $\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}dx$ exists.
improve exposition a bit
2d
comment Prove $\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}dx$ exists.
@RRL: sorry, I fixed the typos, but forgot to address your question about CMVT. Two applications of CMVT give $$\begin{align} \frac{\cos(h)-1}{h^2} &=\frac{-\sin(h_{1a})}{2h_{1a}}\\ &=\frac{-\cos(h_1)}{2} \end{align}$$ where $0\lt h_1\lt h_{1a}\lt h$ and $$\begin{align} \frac{h-\sin(h)}{h^2} &=\frac{1-\cos(h_{2a})}{2h_{2a}}\\ &=\frac{\sin(h_2)}{2} \end{align}$$ where $0\lt h_2\lt h_{2a}\lt h$
2d
comment Prove $\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}dx$ exists.
Would the downvoter care to comment?
2d
answered application of L'Hopital's rule?
2d
comment Prove $\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}dx$ exists.
@RRL: Thanks for noticing. I have fixed a few +/- typos.
2d
revised Prove $\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}dx$ exists.
fix typos
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