## how can I plot two variables (inside a definite integral) against each other

I want to plot x against y in this relation:

## 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 $$K^{*} = \{x\in \mathbb{R}^{n}: \le 0 \text{ for all } y\in K\}$$ the polar of K. Then for all $$c \in \text{Int} K^{*}$$ we have

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

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 $$\varphi_{*n}$$?