Reputation
25,986
Next tag badge:
90/100 score
48/20 answers
Badges
1 26 77
Newest
 Nice Answer
Impact
~308k people reached

1h
comment How to find $\int \frac{\ln(x)}{x^2}dx$
As an aside, for $n>-1$ we have $~\displaystyle\int_1^\infty\frac{\ln^n(x)}{x^2}~dx~=~n!$
3h
awarded  Nice Answer
14h
comment a function defined as an integral can be continued analytically
In general, $~\displaystyle\int_0^\infty\frac{t^n}{e^t+1}~dt~=~n!~\eta(n+1),~$ and $~\displaystyle\int_0^\infty\frac{t^n}{e^t-1}~dt~=~n!~\zeta(n+1),~$ see the Dirichlet $\eta$ and Riemann $\zeta$ functions for more information, both of which possess well-known analytic continuations.
14h
revised compute $\lim_{n\rightarrow\infty}\sum_{k=1}^n \sin(\pi \sqrt{k/n}) (1/\sqrt{kn})$
added 1 character in body
14h
answered compute $\lim_{n\rightarrow\infty}\sum_{k=1}^n \sin(\pi \sqrt{k/n}) (1/\sqrt{kn})$
14h
comment compute $\lim_{n\rightarrow\infty}\sum_{k=1}^n \sin(\pi \sqrt{k/n}) (1/\sqrt{kn})$
I believe you forgot a factor.
16h
comment Find the sub-area of a circle cut by chords
Hint: Divide the region into a triangle and a slice.
20h
comment Prove the Maclaurin-series representation of $ \sqrt{1 + x} $.
See binomial series.
20h
comment $\sum\limits _{n=1}^{\infty}\frac{1}{n^s}=50$ Riemann-Zeta
@robjohn: other that perhaps pulling out a CAS - Precisely. There is no closed form for it. Numerical algorithms are the only way out.
21h
answered $\sum\limits _{n=1}^{\infty}\frac{1}{n^s}=50$ Riemann-Zeta
21h
comment Integral involving Bessel functions of the first kind
@Chappers: That book is well above my pay-grade. :-$)$ I was hoping for an easier approach, one that even the likes of me could understand. :-$)$
23h
comment Is $f(z)=\sum_{k=1}^{\infty}\frac{1}{(k+\frac{1}{k})^{z}}$ somehow related to Riemann's zeta function?
$g(x)=x$ and $h(x)=x+\dfrac1x$ are two wholly different geometric shapes. The former is a straight line, the latter a hyperbola.
23h
comment Is $f(z)=\sum_{k=1}^{\infty}\frac{1}{(k+\frac{1}{k})^{z}}$ somehow related to Riemann's zeta function?
No, there is no meaningful connection between the two, other than their superficial similarity in both definition and numerical value. If anything, I would search for a connection to the polygamma function (a generalization of harmonic numbers), as well as trigonometric and hyperbolic functions.
23h
comment Integral involving Bessel functions of the first kind
I posted a question asking for a proof of the above identity weeks ago, but found no satisfactory answer.
1d
comment Integral involving Bessel functions of the first kind
$I~=~\dfrac2{3\pi}.$
1d
answered Lambert W-Function
1d
comment Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$.
Hint: Since $n!~=~1\cdot2\cdot3\cdots(n-1)~n$, an obvious approach would be rewriting $n^2$ as $n~(n-1)+n$.
1d
comment solving the integral of $e^{x^2}$
See error function, Liouville's theorem, and the Risch algorithm.
1d
comment holomorphic function writen as a serie
$f_a(z)=(1+z)^a.~$ See binomial series.
2d
answered Evaluate $-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$ in $\gamma$.