I want to plot x against y in this relation:

# Tag: integral

## Limit of integral function with parameter

I’m having troubles finding a way to solve these kind of problems, Should I just use L’Hopital’s rule and move on from there?

$ \lim_{x \to 1+} {\int_1^x \mathrm{e}^{\mathrm{-t}^\mathrm{2}}\, dt\over \mathrm{(x-1)}^\alpha} -{1\over 2}$

Thank you a lot for your help.

## Integral operator

Assume that $ A: L^1([0,1]) ]\to L^1[0,1]$ is an integral operator with a real Kernel G(x,y). Then its adjoint operator $ A^*:L^\infty([0,1]) ]\to L^\infty [0,1]$ is an integral operator with a real Kernel G(y,x).

My question is. Whether the norms of $ A$ and $ A^*$ coincide?

## Formula for exponential integral over a cone

While reading ‘Computing the Volume, Counting Integral points, and Exponential Sums’ by A. Barvinok (1993), I came across the following:

“Moreover, let $ K$ be the conic hull of linearly independent vectors $ u_{1}, …, u_{n}$ so $ K = co(u_{1}, …, u_{n})$ . Denote by $ K^{*} = \{x\in \mathbb{R}^{n}: <x, y> \le 0 \text{ for all } y\in K\}$ the polar of K. Then for all $ c \in \text{Int} K^{*}$ we have

\begin{equation} \int_{K}exp(<c, x>)dx = |u_{1} \land … \land u_{n}|\prod_{i=1}^{n}<-c_{i}, u_{i}>^{-1} \end{equation}

To obtain the last formula we have to apply a suitable linear transformation to the previous integral. “

I have tried proving this but I can’t find relevant links to help me. Also, I’m unsure what $ |u_{1} \land … \land u_{n}|$ is supposed to mean. Would greatly appreciate if someone could point me to a relevant resource or provide proof. Thanks!

## Density of integral values of a rational function

Let $ \mathbf{x} = (x_1, \cdots, x_n)$ , and consider a rational function $ F : \mathbb{R}^n \rightarrow \mathbb{R}$ be given by

$ $ \displaystyle F(\mathbf{x}) = \sum_{i = 1}^m \frac{Q_i(x_1, \cdots, x_{n-1})}{R_i(x_1, \cdots, x_{n-1})} x_n^i,$ $

where $ Q_i, R_i$ are non-constant polynomials with integer coefficients. Moreover we may assume that $ R_i > 0$ for all $ \mathbf{x} \in \mathbb{R}^n$ , so $ F$ is well-defined everywhere.

I am trying to understand the counting function

$ $ N_F(X) = \# \{\mathbf{x} \in \mathbb{Z}^n : \lVert \mathbf{x} \rVert_\infty \leq X, F(\mathbf{x}) \in \mathbb{Z} \}.$ $

In particular, given $ x_1, \cdots, x_{n-1}$ one can always find $ x_n \in \mathbb{Z}$ such that $ F(\mathbf{x}) \in \mathbb{Z}$ , but the smallest possible choice of such $ x_n$ could very well be extremely large. Thus it is perhaps best to consider all of the $ x_i$ ‘s as varying at once.

The above observation also gives a somewhat straightforward upper bound for $ N_F(X)$ . In particular, having chosen $ x_1, \cdots, x_{n-1}$ the resulting function $ f(x) = F(x_1, \cdots, x_{n-1}, x)$ is then a rational polynomial in a single variable, and we can clear its denominator to obtain $ g(x)$ say, and then the question is equivalent asking for the density of $ x \in [-X,X]$ such that $ g(x)$ satisfies a certain congruence. However the modulus, equal to $ \text{LCM}_{1 \leq i \leq m} R_i(x_1, \cdots, x_{n-1})$ is typically much larger than $ X$ , since the $ R_i$ ‘s are assumed to be positive definite and in particular not linear. Therefore usually there is at most one root of $ g(x)$ in $ [-X,X]$ . It thus follows that $ N_F(X) = O\left(X^{n-1}\right)$ . I am looking for a bound that beats this, and perhaps close to what one might expect to be the exact asymptotic order.

## Double integral differentiation

Could somebody tell me how do I get the right side from the left side – which rule of differentiation of the double integral produces this?

\begin{align*}\frac{\mathrm{d}}{\mathrm{d}x} \int_{v=x}^\infty &\int_{u=0}^x f(v, u)\cdot g (v)\cdot g(u)\, \mathrm{d} u\, \mathrm{d} v =\ &=\int_0^\infty g(v)\;\mathrm{d}v\cdot \frac{\mathrm{d}}{\mathrm{d}x} \int_0^x f (v, u)\; g(u)\; \mathrm{d}u + \int_x^\infty \;g(u)\;\mathrm{d}u\cdot \frac{\mathrm{d}}{\mathrm{d}x} \int_0^x f (v, u)\; g(v)\; \mathrm{d}v\end{align*}

My ignorance is probably quite profound, and this is something really simple, but I cannot see. It’s been quite a while since I last dabbled in double integrals.

When I type

`d/dx(integral from v=x to infinity integral from u=0 to x f(u,v)g(u)g(v) du dv)`

into WolframAlpha, I get

\begin{align*}\frac{\mathrm{d}}{\mathrm{d}x} \Bigg(\int_{v=x}^\infty \int_{u=0}^x &f(v, u)\, g (v)\, g(u)\, \mathrm{d} u\, \mathrm{d} v\Bigg) =\ &=\int_x^\infty g(v) g(x) f(x,v) \mathrm{d}v – \int_0^x g(u) g(x) f(u,x) \mathrm{d}u,\end{align*}

which is correct, but I first need to understand how to arrive at the first formula, in order to apply the Leibnitz rule.

If someone knows how to align these equations here beautifully, do tell.

## Distribution of a linear pure-birth process’s integral

I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process:

$ $ Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k \bigg] $ $ where $ (Y_t)_t$ is a Yule process of intensity $ \lambda$ and $ k$ an arbitrary positive integer.

Are there any results regarding the $ Z_t$ ‘s distribution?

## Is there a name for this family of integral?

This one: $ \int_{0}^{\bar{x}}e^{-x^{a}}x^{b}(1-x)^{c}dx,a,b,c\ge0$ . When $ a=1,c=0,\bar{x}=\infty$ it is the gamma function.

## exact form of the integral of x^(-x) between 0 and infinity

I have searched the internet for an exact answer and although I have found many decimal approximations, https://www.wolframalpha.com/input/?i=integrate+x%5E(-x)+from+0+to+infinity I have not been able to find an exact form.

## Homology of linear groups over integral domains and their field of fractions

Let $ A$ be a noetherian integral domain of finite Krull dimension with the field of fractions of $ F$ . Consider the natural injections $ i_n:GL_n(A)\hookrightarrow GL_{n+1}(A)$ , $ j_n:GL_n(F)\hookrightarrow GL_{n+1}(F)$ , $ \varphi_n:GL_n(A)\hookrightarrow GL_n(F)$ and similarly $ \varphi_{n+1}$ . Let $ x\in H_k(BGL_{n+1}(A))$ and assume that $ \varphi_{*(n+1)}(x)$ lies in the image of $ j_{*n}$ i.e. for some $ y\in H_k(BGL_n(F))$ , $ j_{*n}(y)=\varphi_{*(n+1)}(x)$ . Does that imply that $ y$ is in the image of $ \varphi_{*n}$ ?