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22h
comment How to find $\nabla$ in spherical coordinates
I'd have thought that $x=r\sin \theta \cos\phi, y= r \sin \theta \sin \phi, z=r \cos \theta$ and the chain rule for partial derivatives is enough (though messy).
1d
comment Integral $I=\int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx$
@Integrals Substitute $t=u^2$, and use $$B(x,y)=\int_0^\infty \frac{t^{x-1}}{(1+t)^{x+y}}dx,$$ where $B(x,y)= \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}$ is the beta function.
1d
comment How to convert a integral into the another?
@Victor An excellent explanation is here.
1d
comment Closed form for $\int_0^1 \frac{x^a dx}{(1+x^b)^c}$.
This is essentially what the other answers do, the equivalence between their forms and your idea basically comes from the definition of the hypergeometric function.
1d
comment Which $n$th order differential equations have $n$ linearly independent solutions?
Thanks. However, you seem to have just proven the case for $n=2$, not general $n$ (or, indeed, for other types of differential equation).
1d
comment Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.
@AlexB. I've encountered a few things that need $p$-adic numbers, but never a particularly good introduction on them. I know this is not really to do with the original question, but do you know of a good introduction? Regardless, thanks for the speedy response.
2d
comment Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.
@AlexB. What theory is necessary to understand Tate's proof?
2d
comment Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$
@RandomVariable Oh, and only take the limit after differentiating?
2d
comment Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$
@RandomVariable Is there a rigorous way one could justify that that would make the integral converge (I'm really bad with this, despite trying to learn some analysis)?
2d
comment Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$
I don't know how to justify playing with divergent integrals.
2d
comment Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$
@Integrals $\frac{d}{d \alpha}I(\alpha)= \int_0^1 \frac{\ln(x) x^\alpha}{x^n-1}dx$.
Apr
15
comment What would be an example such that $aH=bH$ but $Ha \neq Hb$?
math.stackexchange.com/questions/532191/…
Apr
13
comment Lang's proof of Cauchy's Theorem
Thank you. I don't know why this has taken me so long to grok.
Apr
13
comment Lang's proof of Cauchy's Theorem
In 'if $G$ has exponent $n$ then the order of $G$ divides some power of $n$', does he mean the smallest $n$?
Apr
3
comment Is there a slowest rate of divergence of a series?
@user139964 This is interesting. Can one construct an infinite sequence of series, each term diverging slower than the last, using the Ackermann function?
Apr
2
comment What is $\sum_{n=0}^{\infty}|a_nz^n|^2=\frac{1}{2 \pi}\int_{-\pi}^{\pi}|f(ze^{it})|^2dt$ for?
This is related, in case you didn't already know.
Apr
2
comment Closed form or asymptotic expansion for $\int_0^m \frac{n^x}{\Gamma(x+1)}dx$?
Yes, I thought of it after making my request more precise to Antonio.
Apr
2
comment Closed form or asymptotic expansion for $\int_0^m \frac{n^x}{\Gamma(x+1)}dx$?
@AntonioVargas Sorry, I meant $n \to \infty$ with constant $m$.
Apr
2
comment Closed form or asymptotic expansion for $\int_0^m \frac{n^x}{\Gamma(x+1)}dx$?
@AntonioVargas I was referring to asymptotic expansions (as described here), I'll edit the question accordingly. I grant that $\sum_{i=0}^{m-1}\frac{n^i}{i!}$ may not be the first term of an asymptotic expansion for the integral, it was just an idea.
Apr
2
comment Closed form or asymptotic expansion for $\int_0^m \frac{n^x}{\Gamma(x+1)}dx$?
@AntonioVargas Ideally for all $m$.