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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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        A complex integration formula

        I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula: $f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) e r f(b \cos \...
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        Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?

        Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
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        43 views

        integral of the product of an arbitrary function and a linear function [closed]

        In engineering mechanics, a classic approach for the calculation of displacements (virtual work method) requires the evaluation of the definite integral of the product of two continuous functions in $[...
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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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        207 views

        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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        249 views

        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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        167 views

        Legendre Polynomial Integral over half space

        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} = \...
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        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 $\operatorname{li} (x)$ ...
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        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-...
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        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 ...
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        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 ...
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        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 ...
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        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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