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I have proved this equality by means of Cauchy’s Theorem applied to an adequate function. Since my solution is too long to post it here, I posted it in arXiv: Juan Arias de Reyna, Computation of a De …

First we put it in the notation of Mathematica $K(k)$ is $K(k^2)$. So our function will be $$f(k)= k K(k^2)\sinh\Bigl(\frac{\pi}{2}\frac{K(1-k^2)}{K(k^2)}\Bigr).$$ Now we change variables (W486, Whit …

I have posted in arXiv:1702.05442 the English translation of a paper about Fabius function that I published in Spanish in 1982 (we will refer to it as (A)). With the Theorems in this paper the ques …

I consider the function $$f(t):=\int_0^t\frac{dx}{x}\int_0^x\frac{dy}{y}\int_0^y\frac{dz}{z}\bigl\{ \sin x+\sin(x-y)-\sin(x-z)-\sin(x-y+z)\bigr\}.$$ It has an asymptotic expansion with main terms $$f( …

In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$. Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are $$ …

With the aid of Mathematica I found that $$g(\theta)=\textrm{EllipticK}(a^2)-\textrm{EllipticF}(t,a^2).$$ I get the first terms of the asymptotic expansion $$\frac{\sqrt{\pi}}{2\sqrt{k}}-\frac{\sqrt{ …

In Gabcke thesis, that you can download at http://hdl.handle.net/11858/00-1735-0000-0022-6013-8 page 84 you can find Theorem: The incomplete gamma function $$\Gamma(a,x)=\int_x^\infty e^{-v}v^{a-1 …

This is Theorem 3 (p. 10) in the Report: J. van de Lune, Some inequalities involving Riemann's zeta-function, CWI Report ZW 50/75, CentruM Wiskunde & Informatica, Amsterdam,1975, in which the proo …

The function $\sigma\mapsto f(\sigma+it)$ is periodic with period $1$, for any value of $t$. Therefore $$f(\sigma+it)=\sum_{n\in Z} c_n(t)e^{2\pi i n\sigma}.$$ It follows that $$c_n(t)=\int_0^1 f(x+i …

Not an answer but too large for a comment. In my paper with van de Lune On the exact location of the non-trivial zeros f Riemann's zeta function, it is proved. Theorem. If $f\colon\mathbf{R}\to\math …

Not exactly an answer, but too long for a comment. Assume there is such approximation to the identity. We may define a linear form $u\colon L^1(T)\to C$ by $$u(f)=\lim_{\varepsilon\to0} f*k_{\varep …

In the paper J. Rosenblatt, Convergence of Series of Translations, Math. Ann. 230 (1977) 245-272 it is proved a general Theorem (Theorem 2.5) that in particular proves the convergence a. e. . Als …

This is only a suggestion about how to get more terms of the asymptotic expansion. Numerical experiments suggest that the sequences $x_{2n}$ and $x_{2n+1}$ have different asymptotic expression of type …

We have the following characterization: The space $V:=\{\widehat f\colon f\in L^1(\textbf R)\}$ is equal to the space $W=\{h*g\colon h \text{ and } g\in L^2(\textbf R)\}$. That is the Fourier transf …