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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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        Integration over two functions at different time points

        I need to integrate a function of the form: $H(t) = \int_0^t f(u)g(t-u) du$ The functions aren't simple and so I'm using numerical methods to calculate the value of $H$. However, I need to evaluate ...
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        150 views

        Uniformly Bounded

        If $a_1<1$, $a_1+a_2+a_3>1$, for $x,y,z>0,$ (1) define a fucntion $$H(x,y,z)=\frac{x^{\frac{1}{2}}\int_0^{\infty}\frac{1}{t^{a_1} (1+t)^{a_2+1} (1+t+z)^{a_3}}\exp\big\{-\frac{x}{1+t}-\...
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        38 views

        What's the moment of inertia of a swept n-sphere?

        I live in $n$ dimensions. I have a uniformly dense $n$-capsule $S$ of radius $r$ and semi-length $x$ - that is, I took an $n$-ball of radius $r$ from $[-x, 0, ..., 0]$ and swept it through to $[x, 0, ....
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        42 views

        Changing width of integration

        For $F: \mathbb{R} \to \mathbb{R}$ and $w \in \mathbb{R}$, define $g_w(a) = f(a + w) - f(a)$. Given $g_1$ (and not $F$), how can we compute $g_w$ for other $w$? We can assume $F$ is well behaved: $...
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        72 views

        Laplace transform of the tetration (integral or series)

        How to get some insight in the following integral: \begin{equation} \mathcal{I}(s)=\int_0^\infty x^{-x}e^{sx}\text{d} x \end{equation} where $s$ is real (and the lower integration bound may be set ...
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        109 views

        Closed form of integration of modified Bessel function composed with trigonometric function times a linear term

        Assuming one draws two points from a von Mises distribution on a circle, I am looking for the expected distance between two such points. Given the pdf of a centered von Mises distribution $$f_X(t \...
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        779 views

        Fubini without CH

        In Real and Complex Analysis, Rudin gives an example (due to Sierpinski) of a function $f:[0,1]^2\to[0,1]$ separately Lebesgue-measurable in each argument, such that $$ \int_0^1 dx\int_0^1f(x,y)\,dy \...
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        1answer
        51 views

        Domain transformation

        Let $\Omega$ be a 2D domain with boundary $\Gamma$. Suppose its boundary can be parametrised by a smooth function $\phi:I \to \mathbb{R}^2$, $I:=[0,1]$. I know the following transformation holds: $$ \...
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        115 views

        a question of definite integral [closed]

        1.$$\int_{0}^{1} \frac{1}{1+e^{-(x+\ln(u/(1-u)))/\tau}}\, du$$ 2.$$\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{+\infty}\frac{e^{-u^{2}/2}}{1+e^{-(x-u)/\tau}}\,du$$ please help me. I tried to use MATLAB but ...
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        Integrating over a hypercube, not a hypersphere

        Denote $\square_m=\{\pmb{x}=(x_1,\dots,x_m)\in\mathbb{R}^m: 0\leq x_i\leq1,\,\,\forall i\}$ be an $m$-dimensional cube. It is all too familiar that $\int_{\square_1}\frac{dx}{1+x^2}=\frac{\pi}4$. ...
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        1answer
        542 views

        An integral involving the argument of the Gamma function and the Riemann Hypothesis

        Evaluate $$I=\int_{0}^{\infty} \frac{t\arg \Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$ where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$ Note that $I$ converges ...
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        How to fix multi-valued function on contour?

        I am sorry to ask such an embarrassingly simple question here. My question is about contour integral of the multivalude function. I want to calculate the Fourier transformation of a muti-valued ...
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        33 views

        Integrability green function

        This is maybe a trivial question but i need some clarification to make it clearer in my mind. Consider the fundamental solution of the equation $ \partial_{t} u - \partial^{2}_{xx}u=0$ given by the ...
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        1answer
        79 views

        Multidimensional improper Riemann integrals with oscillatory kernels: Existence

        I have asked this question three weeks ago here https://math.stackexchange.com/questions/2998601/does-this-oscillatory-integral-exist/2998930#2998930 but received no relevant answers. Let $n\geq 2$ ...
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        49 views

        Existence of a function satisfying some integral conditions

        I need help to prove the existence of a real function $h(x) \in C^1$ with condition that near zero $h(x) \sim \ln(x)$ and near infinity $\lim h(x)_{x \to \infty} = \infty$ such that following ...

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