# Tagged Questions

Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

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### Comparing log functions of CDFs and PDFs (related to order statistics) with non-log functions of the same

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Consider any natural $n\geq3$ and any real $a$ and $c$ such that $a\leq c$, and $\rho\geq0$. We want ...
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### Regularity of measures in the theorem of Riesz

There are two concurrent theories of measure/integration on a locally compact topological spaces: either as positive linear forms on the space of continuous functions with compact support, or as Borel ...
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### Example of progressively measurable process that is not predictable

Is there an example of progressively measurable process that is not predictable? This question is motivated by Revuz-Yor, Continuous Martingales and Brownian Motion http://www.springer.com/gb/book/...
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### Legendre Polynomial Integral

How can I evaluate $$\int_{-1}^1 P_n(x)P_l(x)x^k dx$$ when $k$ is even? Or what might be a source where I could find integrals like this?
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### Integrability property of polynomials in several variables

This might be very trivial, or not. Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there ...
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### Generalizations of the Euler-Maclaurin Summation Formula

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely Specifically, I would like to ...
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### Prove this function is increasing

I'm stuck in showing that the following function is increasing over the domain $\left[0,\hat{b}\right]$: \begin{eqnarray} \Pi\left(z\right) & = & \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}...
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### interpretation of a singular integral

There is a post on MSE about a principal value integral in this paper. It has not received much attention even with a bounty, and since it concerns a published paper, I believe this is a better forum ...
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### Summation of series involving $\sinh$ of a square root

Consider the following series: $$S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})}$$ From the physical ...
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### How can we obtain the $-\frac{4\pi}3\mu(x)$ term?

Given the expression $$K_{ik} := \frac{\partial}{\partial x_k} \int_{\mathscr X} \frac{y_i-x_i}{|y-x|^3} \mu(y) dy,$$ where $\mathscr X=\mathbb R^3$, how does one derive the expression \begin{align} ...
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Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\... 0answers 80 views ### The integral of$\exp(-|x-a|)$over an even dimensional sphere I'm after a reference for an integral. For$m$a positive integer and$R>0$let$S^{2m}_R\subset \mathbb{R}^{2m+1}$denote the radius$R$sphere of dimension$2m$. Suppose that$a$lies inside ... 3answers 3k views ### What is the standard notation for a multiplicative integral? If$f: [a,b] \to V$is a (nice) function taking values in a vector space, one can define the definite integral$\int_a^b f(t)\ dt \in V$as the limit of Riemann sums$\sum_{i=1}^n f(t_i^*) dt_i$, or ... 0answers 78 views ### Integration and Inverse Function Theorem Apologies if this sounds too silly for advanced math people here. It's long since I moved from mathematics to medicine and this problem appears in my research. For an$f^{-1}\in C^{1}([a,b])$, ... 0answers 34 views ### Generalizing Integration by parts for general bounded continous measure Consider a probability measure$d\mu = w(t) dt$with$w(t)\in L^1(I)$,$I =\left[ 0,1\right]$. What are the minimal assumption I can take on two functions$f,g:I\ \to \mathbb{R}$so that an ... 1answer 187 views ### Extension of a function from almost everywhere to everywhere The informal general question is: let$f$be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend$f$to the "remaining" points? Example: Let$f(x)=\...
Let $F_1(\mathbf{x}), F_2(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous polynomial. For the system of equations $$F_1(\mathbf{x})= F_2(\mathbf{x}) =0,$$ we have the ...