robjohn

West Hills, CA

none

Age: 55

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

aka Rob Johnson

$\LaTeX$ support for Chat

10h
revised Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $
fix typo
11h
comment Prove that an analytic function, real-valued on radii $[0, 1)$ and $[0, e^{i\pi\sqrt 2})$, is constant on the open unit disk
You might want to mention why it is important for this question that $\sqrt2$ is irrational; why the result is not true if $\sqrt2$ is replaced by a rational number.
11h
revised Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $
add an alternate result
16h
answered Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $
16h
revised Prove that $\int_0^\infty\frac{x^n}{1+e^{x-t}}\mathrm{d}x = \frac{t^{n+1}}{n+1} + o(t^n)$, when $t \to \infty,\,n\in\Bbb{R}^+$
add some local color
16h
answered Prove that $\int_0^\infty\frac{x^n}{1+e^{x-t}}\mathrm{d}x = \frac{t^{n+1}}{n+1} + o(t^n)$, when $t \to \infty,\,n\in\Bbb{R}^+$
21h
comment Evaluate $\int_1^\infty \frac {dx}{x^3+1}$
@Ant: I've added a small circle about $1$. Since $\log(x)$ is locally integrable, integrating up to the singularity can usually be ignored, but there's nothing wrong with being a bit more careful.
22h
revised Evaluate $\int_1^\infty \frac {dx}{x^3+1}$
add a fourth contour
22h
revised Evaluate $\int_1^\infty \frac {dx}{x^3+1}$
note the points for the residues
23h
comment Should chat have TeX support?
@Singh: Method 1: drag the link from the installation page to the bookmark bar. Method 2: right-click on the link on the installation page and choose "Bookmark This Link".
1d
revised Compute the fourier coefficients, and series for $\log(\sin(x))$
actually answer the question
1d
answered Compute the fourier coefficients, and series for $\log(\sin(x))$
1d
comment Should chat have TeX support?
@Singh: If you install the "start ChatJax" bookmark, either in a menu or on the bookmark bar, you should be able to execute it (select it from the menu or click on it in the bookmark bar) while the chat window is active and then the LaTeX should continuously render until the page is refreshed (then you need to execute the bookmark again).
1d
answered Evaluate $\int_1^\infty \frac {dx}{x^3+1}$
1d
comment Using numerical methods to calculate integral
@CollegeConfidential: Either Jack's formula $(3)$ or the formula in my deleted answer $$\frac1{10}\sum_{k=0}^{28}\frac{(-1)^k\,3^{2k+1}}{(2k+1)\,k!}$$ should give the result to $5$ decimals
1d
comment Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{\sqrt{n}}$ converge?
@mike: I have checked. The computation of the the asymptotic expansion was done using a Mathematica program that I wrote a long time ago and which has proven to be accurate. Have you checked numerically?
1d
revised Calculate $\lim_{x \to 0} (e^x-1)/x$ without using L'Hôpital's rule
add condition to make reciprocals act nicely
1d
comment Using numerical methods to calculate integral
@CollegeConfidential: Notice that that estimate is going to be divided by $10$, so an error of $10^{-4}$ when divided by $10$ is an error of $10^{-5}$.
1d
comment Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{\sqrt{n}}$ converge?
@mike: that is as good as any form. The terms for the lower limit (those with $m$) are usually collected into a single constant.
1d
answered Calculate $\lim_{x \to 0} (e^x-1)/x$ without using L'Hôpital's rule
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