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# Questions tagged [integration]

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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### Marginalize over a vector of functions

I'd like to compute the following integral: $$\int\exp\bigg[-\sum_{i=1}^{3}\frac{1}{2\sigma^2_i}\big(\mathbf{y}_{t}^i-f_i(\mathbf{x}_{t})\big)^2\bigg]P(\mathbf{f})\mathrm{d}\mathbf{f}$$ where we ...
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### Integrating multiplication of two piecewise functions

Problem I'm having difficulty with computing analytical solution to convolution of two piecewise functions: The form above can be simplified to following (where in place of f1,f2, consider ...
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### Function orthogonal to powers of $1/\left(1+x^2\right)$

Does there exist any continuous function $f:\mathbb{R}\to\mathbb{R}$, $f(x)/(1+x^2)\in L^1(\mathbb R)$, such that $f(0)=1$ and $$\int_{-\infty}^{\infty}\frac{f(x)}{\left(1+x^2\right)^p}dx=0$$ for ...
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### Simplify Wasserstein distance between Gaussians with binary cost function

Let $\mu_1$ and $\mu_2$ be 1D gaussian distributions with means $m_1$ and $m_2$ respectively and common variance $\sigma$. Let $\Omega$ be a closed subset of $\mathbb R^2$, and consider the cost ...
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### On $\zeta(7)$ as the integration of the product of an indefinite integral due to Lobachevskii by a power of the inverse Gudermannian function

In this post I invoke certain function from a post of this site MathOverflow it is [1] (please see further references from the post, authors from the Springer link of the cited literature and answers ...
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I need to compute the following integral $$I_{n,m} := \int_0^1 P_n(x) P_m(x) \; \mathrm{d}x$$ where $P_n$ is the Legendre polynomial. For an even sum $n+m=2l$ it is easy to show that $$I_{n,m} = \... 0answers 65 views ### Calculating \int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx, where \operatorname{ali}(x) is the inverse function of the logarithmic integral It is well-known that we can compute the closed-form of the integrals$$\int_1^{\infty}\frac{\log x}{x^2}dx$$and$$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$where \operatorname{li} (x) ... 0answers 73 views ### Express \zeta(n), in particular the Apéry's constant, in terms of series involving Gregory coefficients or the Schröder's integral The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function \zeta(n), in particular \zeta(3), in terms of series involving the so-... 0answers 138 views ### Change of variables for p-adic integral Say p is an odd prime. Suppose I have a measure \mu on \mathbf{Z}_p. As in II.4.3 in Colmez - Fonctions d'une variable p-adique, I can restrict \mu to 1+p\mathbf Z_p, and there is a ... 1answer 150 views ### Integration with values in a topological vector space Is there a general theory of integration of functions with values in a topological vector space (not necessarily locally convex)? Browsing through mathoverflow posts, I came across a discussion ... 1answer 190 views ### Help in proving, that \int_{0}^{\infty} \frac{1}{\Gamma(x)} d x=e+\int_{0}^{\infty} \frac{e^{-x}}{\pi^{2}+(\ln x)^{2}} d x using real methods only I hope this does not seem like a too easy question for Overflow. I would like to find an easier method than mine to prove the above statement for the Fransén-Robinson Constant. My first method was to ... 1answer 188 views ### Limits of a family of integrals Assume \lambda_1+\lambda_2=1 and both \lambda_1 and \lambda_2 are positive reals. QUESTION. What is the value of this limit? It seems to exist.$$\lim_{n\rightarrow\infty}\int_0^1\frac{(\...

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