Articles of integration

Integral related to a geometry problem

In the question Geometry problem involving infinite number of circles I showed that the answer could be obtained by the sum $$ \sum_{k=0}^{\infty}\int_{B_{k}} {4 \over \,\left\vert\,1 + \left(\,x + y{\rm i}\,\right)\,\right\vert^{\,4}\,} \,{\rm d}x\,{\rm d}y\,, $$ where $ B_k = \left\{\, z \in \mathbb{C}:\ \left\vert\,z – \left[\frac{1}{2} + \left(k + {1 \over 2}\right) {\rm i}\right]\,\right\vert\ \leq\ […]

Do we have $\lim_{h\to 0^+}\int_0^\infty|f(t+h)-f(t)|dt=0$?

Let $f:\mathbb{R^+}\to\mathbb{R}$ be an integrable function ($f\in L^1(\mathbb{R}^+,\mathbb{R})$). Do we have $$\lim_{h\to 0^+}\int_0^\infty|f(t+h)-f(t)|dt=0$$ ? How can we prove it ?

Endpoint-average inequality for a line segment in a normed space

Let $X$ be a normed vector space over $\mathbb R$. What is the smallest universal constant $C>0$ such that the inequality $$\|x\|\le C\int_0^1 \|x+tv\|\,dt\tag{1}$$ holds for all $x,v\in X$? Geometrically, (1) means that the norm of an endpoint of a line segment can be majorized by the average of the norm over said line segment. […]

Simple Integral Involving the Square of the Elliptic Integral

I have, $$ \int uE^{2}\left(u\right)du $$ where $E$ is the complete elliptic integral of the second kind: $$ E\left(k\right)=\int_{0}^{\frac{\pi}{2}}d\theta\sqrt{1-k^{2}\sin^{2}\left(\theta\right)} $$ I’ve tried integrating this by looking for something that differentiates into $E^2(u)$, though can’t seem to find anything. Any help?

A closed form for the integral $\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy$

Yesterday, during reviewing my old lecture notes on advanced quantum mechanics, i stumbeled over the following integral identity, which seems, on a first glance, too nice to be true $$ I_{A,B}=\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy=\sqrt{\frac{i\pi}{B }}e^{i(\sqrt{A}+\sqrt{B})^2} $$ with $A,B>0$ After working on it for a few hours i came up with a solution, which i think is […]

Topology on the space of universally integrable functions

Let $X$ be a compact space. Let us call a function $f:X\to {\mathbb C}$ universally integrable if it is integrable with respect to each regular Borel measure $\mu$ on $X$ (i.e. a positive functional on ${\mathcal C}(X)$, according to Riesz–Markov–Kakutani representation theorem). Let us denote by ${\mathcal U}(X)$ the space of all universally integrable functions […]

Definite integral of a product of normal pdf and cdf

Denote the pdf of the standard normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int_{-\infty}^y \phi(x)\Phi(\frac{x−b}{a})dx$? Notice that this question is similar to an existing one, the only difference being that I’m computing the integral over $(-\infty, y)$ for some real $y$, rather than over the entire real […]

Integral equation solution: $y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$

Integral equation $$y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$$ has: a unique solution for $\lambda \neq \frac{4}{\pi +2}$; a unique solution for $\lambda \neq \frac{4}{\pi -2}$; no solution for $\lambda \neq \frac{4}{\pi +2}$, but the corresponding homogeneous equation has a non-trivial solution; or no solution for $\lambda \neq \frac{4}{\pi -2}$, but the corresponding homogeneous equation […]

Integral of Dirac delta function/distribution $\delta(x)$ with upper boundary equal to zero

I would like to find the value of $$\int_{a<0}^0 \delta(x) dx$$ In particular, I would like to know if I can break down the integral $$\int_a^b \delta(x)f(x) dx=\int_a^0 \delta(x)f(x) dx + \int_0^b \delta(x)f(x) dx $$ with $a<0$ and $b>0$ and $f(x)$ a well-behaved function. Is it wrong to break down the integral like this, doing […]

Variable in Feynman Integration

Evaluate: $$I=\int_{0}^{\frac{\pi}{2}} \ln(2468^{2} \cos^2x+990^2 \sin^2x) .dx$$ Now, a friend suggested that to evaluate the integral $I$, I rewrite the integral as $$f(y)=\int_{0}^{\frac{\pi}{2}} \ln(y \cos^2x+ \sin^2x) .dx$$ where $y=\dfrac{2468^2}{990^2}$. $$$$But, how can we do this? $$$$Could somebody please explain how this parametrization was done? Many thanks! $$$$The suggested solution:$$$$ $$f(y) = \int_{0}^{\pi/2} ln( y^{2}cos^{2}x + sin^{2}x)$$ […]