Related papers: Evaluation of an Integral

Through an inversion approach, we suggest a possible estimation for the absolute value of Mertens function $\vert M(x) \vert$ that $ \left\vert M(x) \right\vert \sim \left[\frac{1}{\pi \sqrt{\varepsilon}(x+\varepsilon)}\right]\sqrt{x}$…

Real and imaginary part of the limit 2N->infinity of the integral int_{x=1..2N} exp(i*pi*x)*x^(1/x) dx are evaluated to 20 digits with brute force methods after multiple partial integration, or combining a standard Simpson integration over…

In this paper we compute the 2-adic valuations of some polynomials associated with the definite integral $\int_{0}^{\infty} \frac{dx}{(x^4+2*a*x^2+1)^(m+1)}$

A class of log-trigonometric integrals are evaluated in terms of elliptic functions. From this, by using the elliptic integral singular values, one can obtain closed form evaluations of integrals such as \[…

In the world of mathematical analysis, many counterintuitive answers arise from the manipulation of seemingly unrelated concepts, ideas, or functions. For example, Euler showed that $e^{i\pi} + 1 = 0$, whereas Gauss proved that the area…

We provide additional methods for the evaluation of the integral \begin{eqnarray} N_{0,4}(a;m) & := & \int_{0}^{\infty} \frac{dx} {\left( x^{4} + 2ax^{2} + 1 \right)^{m+1}} \end{eqnarray} where $m \in {\mathbb{N}}$ and $a \in (-1, \infty)$…

It is argued that for certain meromorphic functions $u:\cal{R}\rightarrow\cal{R}$ and analytic function $ A_1$ and for any integrable function $F$, as long as it converges as a Cauchy Principal Value,, $$\int_{-\infty}^{\infty}A_1(x)F[u(x)]…

All integrals available in literature and books, that are related to Sinc(=sin x/x) function, are special cases of the general form of the integral given in the title. The evaluation of the integral is divided into two cases (i) $a$ and $b$…

Logarithmic integrals revisited. We consider integrals of the form $\int_0^1 \ln{\ln{(\frac{1}{x})}}R{(x)}{\rm d}x$ again, where $R{(x)}$ is a rational function, and we will explain a way to obtain their values.

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…

For formal multivariate power series $\varphi(x)$ an inversion formula of the form $$ \varphi^{-1}(x)=x +\sum_{m=1}^{\infty}\sum_{k=0}^m (-1)^k(m k)\varphi^{\circ k}(x) is offered$$.

The definite integrals $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^3\D x$, $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^2P_{\nu}(-x)\D x$, $\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^3\D x $ and…

Let M(f) denote the Midpoint Rule and T(f) the Trapezoidal Rule for estimating integral_a^b f(x) dx. Then Simpson's Rule = tM(f) + (1-t)T(f), where t = 2/3. We generalize Simpson's Rule to multiple integrals as follows. Let D be some…

We utilize a combination of integral transforms, including the Laplace transform, with some classical results in analytic number theory concerning the Riemann $\xi$-function, to obtain a new integral equation. We also provide a new proof of…

Let $\varphi(\tau)=\eta((\tau+1)/2)^2/\sqrt{2\pi}e^\frac{\pi i}{4}\eta(\tau+1)$ where $\eta(\tau)$ is the Dedekind eta-function. We show that if $\tau_0$ is an imaginary quadratic number with $\mathrm{Im}(\tau_0)>0$ and $m$ is an odd…

We prove that the backward shift operator on $H^4$ has norm equal to $\sqrt[4]{\varphi}$, with $\varphi = \frac{1 + \sqrt{5}}{2}$. Furthermore, we characterize all extremal functions; they are precisely the functions of the form \[ f(z) =…

In his third notebook, Ramanujan claims that $$ \int_0^\infty \frac{\cos(nx)}{x^2+1} \log x \,\mathrm{d} x + \frac{\pi}{2} \int_0^\infty \frac{\sin(nx)}{x^2+1} \mathrm{d} x = 0. $$ In a following cryptic line, which only became visible in a…

In this paper, we study the multiple integral $ \displaystyle I= \int_0^1 \int_0^1 \dots \int_0^1 f(x_1+x_2 + \dots +x_n) \, dx_1 \, dx_2 \, \dots \, dx_n$. A general formula of $I$ is presented. As an application, the integral $I$ with…

We study several integrals that contain the infinite product ${\displaystyle\prod_{n=0}^\infty}\left[1+\left(\frac{x}{b+n}\right)^3\right]$ in the denominator of their integrand. These considerations lead to closed form evaluation…

Based on the total integrability we first define an integral of a real valued function f as an interval function associated to its antiderivative F. By introducing the concept of the residue of a function into the real analysis, the…