A Short Proof of a Concrete Sum [PDF]
arXiv, 2016 We give an alternative proof of a formula that generalizes Hermite's identity. Instead involving modular arithmetic, our short proof relies on the Fourier-type expansion for the floor function and on a trigonometric formula.
arxiv
Some implications of a new definition of the exponential function on time scales
, 2010 We present a new approach to exponential functions on time scales and to timescale analogues of ordinary differential equations. We describe in detail the Cayley-exponential function and associated trigonometric and hyperbolic functions. We show that the
Cieśliński, Jan L.
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A pair of optimal inequalities related to the error function [PDF]
arXiv, 1997 The Error Function \begin{eqnarray} V(x) & \equiv & \sqrt{\pi} e^{x^2} [1 - \hbox{erf}(x)] \\ & = & \int_0^\infty \frac{ e^{-u} }{\sqrt{x^2 + u}} du = 2 e^{x^2}\int_x^\infty e^{-t^2} dt \nonumber \end{eqnarray} arises in many contexts, from probability to mathematical physics.
arxiv
arXiv, 2004 This is a translation into English from the original Latin of Leonhard Euler's Exercitatio analytica, Nova Acta Academiae Scientarum Imperialis Petropolitinae 8 (1794), 69-72; E664 in the Enestrom index. In it Euler uses the infinite product identity for sin to prove some properties of the series for the Riemann zeta function for even n.
arxiv
The integrals in Gradshteyn and Rhyzik. Part 1: A family of logarithmic integrals [PDF]
Scientia, Series A: Mathematical Sciences, Vol. 14 (2007), 1-6, 2007 We present the evaluation of a family of logarithmic integrals. This provides a unified proof of several formulas in the classical table of integrals by I. S. Gradshteyn and I. M. Rhyzik.
arxiv
APP accumulates with presynaptic proteins around amyloid plaques: A role for presynaptic mechanisms in Alzheimer's disease? [PDF]
Alzheimers Dement, 2022 Jordà-Siquier T +8 more
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The integrals in Gradshteyn and Ryzhik. Part 8: Combinations of powers, exponentials and logarithms [PDF]
arXiv, 2007 We present the evaluation of some definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik where the integrand is a combination of powers, exponentials and logarithms.
arxiv
The integrals in Gradshteyn and Ryzhik. Part9: Combinations of logarithms, rational and trigonometric functions [PDF]
arXiv, 2007 The classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik contains many definite integrals where the integrand is the product of a rational function times the logarithm of another rational function. We begin the systematic evaluation of these integrals.
arxiv
On some determinants involving the tangent function
, 2019 Let $p$ be an odd prime and let $a,b\in\mathbb Z$ with $p\nmid ab$. In this paper we mainly evaluate $$T_p^{(\delta)}(a,b):=\det\left[\tan\pi\frac{aj^2+bk^2}p\right]_{\delta\le j,k\le (p-1)/2}\ \ (\delta=0,1).$$ For example, in the case $p\equiv3\pmod4 ...
Sun, Zhi-Wei
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Coefficient estimates for some classes of functions associated with \(q\)-function theory
, 2017 In this paper, for every $q\in(0,1)$, we obtain the Herglotz representation theorem and discuss the Bieberbach type problem for the class of $q$-convex functions of order $\alpha, 0\le ...
Agrawal, Sarita
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