# 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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### Polya-Laguerre Class and inversion formula

Is it possible to write the following statement: If (i) $E(s)=\frac{1}{\int_{0}^{+\infty}K(x) x^{s-1} dx}$ (ii)$E(s) \in E_{0}$ (i.e $E(s)= e^{bs} \prod_{k=1}^{\infty}(1-\frac{s}{a_{k}}) e^{s/a_{k}}$...
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### On integral relating logarithmic of absolute value of Zeta function:

Sorry for such a direct question: Consider the following integral: $$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da$$ How to find the nature of $I(t)$ as $t\rightarrow\infty$?
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### Riemann-Stieltjes integral of a distribution function

I recently learned the basics of Riemann-Stieltjes integral, and based on the sources I found, we can define the expectation of random variables quite naturally with the R-S integrals: if $X$ is a ...
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### Decide the order of of an integration involving the $\log$ function

Let $$A_n=\int_{n^{-\frac{1}{2}}}^{1}\frac{\log(nx)}{nx(\log\log(nx)-\log\log(1+x))}dx.$$ I want to discribe the order of $A_n$, by geting a progressive formula or a good lower bound for it. The order ...
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### Laplace transform calculation

Please can someone help me? I have tried to find the Laplace transform of the form: $$\int_{0}^{+\infty} (v+1)^{\nu}(2v+1)^{k}v^{\alpha} \exp(-pv), \mbox{ where }\alpha,\nu, k \mbox{ are integers }$$ ...
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### Generalizing contour integration to quaternions and bicomplex numbers

I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true ...
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### Question on an exercise from Terry Tao's blog

I've been reading Tao's An introduction to measure theory,a draft can be found here.An exercise from it is Exercise 30 (Rising sun inequality) Let ${f: {\bf R} \rightarrow {\bf R}}$ be an absolutely ...
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### Sources for multiple Stieltjes integral

I'd like to know which sources (books or papers) provide a detailed discussion of multiple Stieltjes integral or multiple Lebesgue-Stieltjes integral.
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### A tricky integral to evaluate

I came across this integral in some work. So, I would like to ask: QUESTION. Can you evaluate this integral with proofs? $$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
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### Integration of fractional function over Rice distribution

Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as $$\int_0^{\infty } {\frac{1}{{1 + {x^a}}}} f\left( {x|y} \right)\, dx$$...
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### integral of fractional function

Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as $$\int_b^{ + \infty } {\frac{x}{{1 + {x^a}}}}.$$ Here $b$ is a ...
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### Trying to evaluate an integral relating to $\zeta (3)$

So similarly to my search for $\zeta (3)$ over at the mathematics stack exchange, I have continued to attempt to work towards a closed-form for it. The following integral is related to a search of ...
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### Expected distance between two uniform points in distinct rectangles

Are there any good approximations (especially upper bounds) for the quantity $E(\|X_1-X_2\|$), where each $X_i$ is uniformly distributed in a rectangle $[a_i,b_i]\times[c_i,d_i]$? It does not appear ...
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### Length of isoline $x(1-x)y(1-y)=c$

For the integral appearing in this answer, it may be beneficial to derive the length $L(c)$ of the isoline: $$x(1-x)y(1-y) = c,$$ where $x,y$ are ranging in $[0,1]$, and constant $c\in [0,\frac1{16}]$....
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### Integrating multinomial likelihood over intersection of two simplexes

I am trying to calculate this integral, coming from a multinomial likelihood with an extra condition ($\sum_{i=1}^n p_i w_i = 1$). Essentially it can be seen as integrating a Dirichlet pdf over a ...
### Distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Cauchy
Suppose that $X$ and $Y$ are Cauchy-distributed with $\gamma=1$, i.e., with PDF $\frac 1 \pi \frac 1 {1+x^2}$. I tried to find the distribution of $R = \sqrt{X^2+Y^2}$. The PDF of $R$ should be given ...