The integration tag has no usage guidance.

**2**

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224 views

### Coutour Integral of Gamma Functions

How do I solve the Integral
$$ \frac{1}{2\pi j} \oint
\frac{b^{ - s} \Gamma[2 + i - s] \Gamma[s] \Gamma[-1 - i + s]}{
(2 + i - s) \Gamma[3 + i - s]} \:\mathrm{d}s$$
This integral is an inverse ...

**3**

votes

**2**answers

661 views

### Stokes theorem for manifolds without orientation?

Hello!
In textbooks Stokes theorem is usually formulated for orientable manifolds (At least I couldn't find any version not using orientability). Is Stokes theorem: ...

**1**

vote

**2**answers

343 views

### Defining definite integral using indefinite integral.

Sometimes definite integral is defined using antiderivatives:
$$\int_{a}^b{f(t)dt}=F(b)-F(a)$$
where $F$ is any continuous function such that:
$$(\forall t\in[a,b]\setminus C)(F'(t) \space exists ...

**4**

votes

**1**answer

497 views

### Is there a closed form expression/series expansion for $\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$ ?

I've been trying to find a closed form expression/series expansion for the following integral without success:
$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...

**1**

vote

**1**answer

320 views

### Why is there a formula for symbolic differentiation (chain and product rules) but not for symbolic integration? [duplicate]

Possible Duplicate:
Why is differentiating mechanics and integration art?
There is a formula for the derivative of any product, composite or sum of functions, in terms of the derivatives of ...

**7**

votes

**3**answers

822 views

### Nontrivial trivial integrals

I posted this question to stackexchange and after 24 hours it's got five votes and no answers, so let's see if mathoverflow can say more than that.
Consider two propositions in geometry:
...

**3**

votes

**2**answers

519 views

### Sum involving binomial coefficients

I have the following sum
$\sum_{j=1}^K {K \choose j} (-1)^{j+1}/j$. Now I can write this as the integral $\int_{-1}^0 \frac{(1+x)^K - 1}{x} dx$. However, I wonder whether there is a closed form ...

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vote

**0**answers

174 views

### convergence of sets and limit of an integral

Let $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}$ be compact sets.
Let $f:X\times Y\rightarrow\mathbb{R}$ be a $C^{1}$ function.
Let $s:Y\rightarrow X$ be a function (not necessarily continuous).
...

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votes

**1**answer

372 views

### An infimum of integrals of a positive function.

Hi,
I have a question concerning integration theory I can't figure out, maybe someone can help:
Fix $\varepsilon>0$ and consider $\delta \colon [0,1] \to (0,\infty)$ measurable. Is it then true ...

**3**

votes

**1**answer

718 views

### definition of operator valued integral with spectral measure

I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011).
There, they work on a Hilbert space $H$ and on the ...

**1**

vote

**1**answer

602 views

### What are integration on fractal? [closed]

Who can explain the proof of the formula (2.12) given here: J. Phys. A: Math. Gen. 20 (1987) 3861-3875. Printed in the UK ...

**3**

votes

**1**answer

539 views

### Integral of Modified Bessel Function of the Second Type

Given the identity
$$ \int^\infty_0 K_v\left(\alpha\sqrt{x^2+z^2}\right) \frac{x^{2\mu+1}}{\left(\sqrt{x^2+z^2}\right)^v}\:\mathrm{d}x = \frac{2^\mu \Gamma(\mu+1)}{\alpha^{\mu+1}z^{v-\mu-1}} ...

**2**

votes

**1**answer

203 views

### hypervolume under the square of an n-simplex

I posted this question at math.stackexchange.com, reformulated and posted again both times without much luck. I also asked a math professor at my uni who suggested I post it here. Hopefully, it is ...

**1**

vote

**0**answers

91 views

### Integrating B-Spline composed with log

If $f$ is a real B-Spline and $a, b$ are real numbers, then is there a numerically stable way to evaluate the following expression?
$\int_a^b f (\log x) \mathrm{d}x$

**0**

votes

**0**answers

303 views

### Help with an irregular integral

I am looking for help with doing the following integral :
$$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in ...

**2**

votes

**1**answer

1k views

### Why can't I interchange integration and differentiation here?

I think my questions relates to this other: "counterexamples to differentiation under integral sign"
In fact, it provides a counterexample
Consider $f(x,y)=y^3e^{-y^2x}$ and define $F(y) ...

**7**

votes

**0**answers

388 views

### Minkowski's Inequality for Integrals in Orlicz spaces

EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.
Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, ...

**4**

votes

**1**answer

562 views

### Extended integral in Spivak’s Calculus on Manifolds

On page 48 of Calculus on Manifolds Spivak defines (Riemann) integration over rectangles $[a_{1},b_{1}]\times\cdots\times[a_{n},b_{n}]\subset\mathbb{R}^{n}$. Then on page 55 he extends this integral ...

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votes

**5**answers

1k views

### An algebra of “integrals”

When discussing divergent integrals with people, I got curious about the following:
Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace)
$$\int_0^{\infty}: ...

**9**

votes

**5**answers

809 views

### $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral
$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$,
where $s$, $r$ and $m$>1 are positive integers.
My question is whether a closed form ...

**1**

vote

**2**answers

91 views

### Finding Kuramoto Model coupling strength with limits?

The following is an equation describing the coupled phases of N oscillators according to the Kuramoto model:
$$
1 = K \int_{-\pi/2}^{\pi/2}\cos^{2}\left(\theta\right)\,
{\rm ...

**4**

votes

**1**answer

189 views

### Practical way to check for geometric convergence

Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution.
When I measure the convergence rate ...

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votes

**4**answers

399 views

### Reference for integral of functions taking values in a topological vector space.

(Note that I am interested in the Gelfand-Pettis integral specifically, as opposed to, for example, the Bochner integral.) I have tried Googling things like "integral topological vector space", ...

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votes

**3**answers

5k views

### Proof without words for surface area of a sphere [closed]

I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the ...

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votes

**1**answer

281 views

### A particular kind of Cauchy Principal Value integral

I am sorry to bother the community with such a narrow question, it may perhaps be a little specific. As I study Random Matrix Theory, I often have to solve integrals of the form
$$\mathcal{P} ...

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votes

**1**answer

348 views

### Modern version of an inequality of R. M. Gabriel for contour integrals

I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm:
Functional Analysis and Operators by Brigitte Vallée, which cites a 1928 article by R. M. Gabriel for the following ...

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votes

**1**answer

100 views

### Dertivative of a Special Function with respect to Order

The marcum Q-function is defined by
$$ Q_m(a,b) = \int^\infty_b x \left(\frac{x}{a}\right)^{m-1} \exp\left(-\frac{x^2+a^2}{2}\right)
I_m\left(a x\right)
\:\mathrm{d} x,$$
where $m\in\mathbb{N}$ , ...

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votes

**1**answer

557 views

### integrate of functions involving floor

Is there any exact formula or at least exact inequalities for the following intehral
$$
\int_{2}^{x}{{\rm d}t \over \left\lfloor\vphantom{\large h}%
\log(x)/\log(t)\right\rfloor
\log\left(t\right)}
...

**2**

votes

**1**answer

379 views

### A multiple definite integral.

I'm unable to find an easy way to compute the following multiple definite integral.
Apologies if it is trivial.
Let $C$ be a $N \times 1$ real vector.
Let $M$ be a $N \times n$ real matrix.
Let ...

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votes

**2**answers

3k views

### About the definition of Borel and Radon measures

I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature.
More precisely, I have a doubt about the very definition of ...

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votes

**1**answer

456 views

### Certain compact subset of $L_1$

Let $(\Omega,\Sigma, \mu)$ be a probability measure and $X$ a Banach space. I am interested in subsets $F\subseteq L_\infty (\mu,X)$ that satisfy these two compactness conditions:
$F$ is a ...

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votes

**4**answers

3k views

### Is $x \, \tan(x)$ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $x \, \tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in ...

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vote

**2**answers

585 views

### Gauss Legendre Method for Implicit Integration

Methods that are usually adopted for time integration in transport phenomena problems are either:
Euler (explicit, first-order accurate)
$\frac{dY}{dt}=f(t,Y)$
$Y^{n+1}=Y^n+\Delta t f(t,Y^n)$
...

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votes

**2**answers

225 views

### Integrating a product

By trying to find a marginal distribution I came accross integration of the product series. For the sake of generality, lets assume the integral is of following form:
$$\int \prod_{k=1}^{n}\left ( ...

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votes

**2**answers

1k views

### Borel sets preserved under open maps?

Given open map f: $R^n$ to $R^n$ such that each open set $U\in R^n$, $f(U)$ is also open. Are Borel sets in $R^n$ preserved under f?
Motivation: Pre-image of Borel sets under continuous map is a ...

**3**

votes

**2**answers

345 views

### Integration in several variables and elementary applications

This fall I'm teaching the "second half" of the standard entry-level undergraduate multivariable calculus course: the focus is on double and triple integrals, path integrals, Green's theorem, Stokes' ...

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votes

**3**answers

504 views

### Defining the integral of a function using the product measure

Imagine that we're trying to define the expression
$$\int_U f(x)dx$$
in a rigorous way.
Assume that $f:X \rightarrow \mathbb{R}^{\geq 0}$ where $(X,\mu)$ is a measure space, and suppose that $U$ is a ...

**12**

votes

**3**answers

2k views

### Counterexamples to differentiation under integral sign?

I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...

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votes

**3**answers

327 views

### Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?

What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$?
Or any good reference for tools to tackle this question?
...

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**0**answers

135 views

### Boundedness of Riemann-like sums on unbounded interval

Hi
I am trying to find suitable conditions (integrability, growth...) on a function $f:\mathbb{R}\to \mathbb{R}$ such that:
\begin{equation}
\sum_{k\in\mathbb{Z}}f(kh)h= \mathcal{O}(1),\qquad h\to ...

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votes

**2**answers

500 views

### Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...

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votes

**1**answer

950 views

### Integration on high dimensional sphere

Hi, I need to integrate a function on an n-dimensional sphere surface. One way is to use the triangle function like: http://en.wikipedia.org/wiki/N-sphere#Spherical_volume_element, however, it is too ...

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137 views

### convergence of multiple integral

I am searching for some theorems and books about convergence of multiple integrals of the form:
$$
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\;f(x,y)\;\mathrm{d}x\,\mathrm{d}y.
$$
In particular, ...

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votes

**3**answers

7k views

### Integration of the product of pdf & cdf of normal distribution [closed]

Denote the pdf of normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int \phi(x) \Phi(\frac{x -b}{a}) dx$? Notice that when $a = 1$ and $b = 0$ the answer is ...

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102 views

### Variation of a Function

Let $g$ be a function of finite $q$ Variation and $f$ be a function of finite $p$ Variation, and $\frac{1}{q}+\frac{1}{p}>1$.
What can be said about the variation of $H$ with ...

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votes

**1**answer

403 views

### Fourier coefficients of a rational function

Any ideas how to compute or to approximate integral
$$\int_{0}^{1}\frac{(x+a)^{2q}(x+b)^{2q}}{(x-1)^{4q}+x^{4q}}\exp({-2\pi i x y})dx$$
where $q \in \mathbb{N}$ and $a,b =-2,-1,0,1$, $y \in (0,1)$

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vote

**0**answers

202 views

### Evaluating the integral $\int_{a}^{+\infty} \frac{\exp(-bx)}{x+c} Ei(x) dx$

I'm trying to evaluate or simplify this integral:
$$I_{a,b,c} = \int_{a}^{+\infty} \frac{\exp(-bx)}{x+c} Ei(x) dx $$
with $a,b,c \in \mathbb{R}_+^*$.
and $ Ei(x) =\int_{-\infty}^{x} ...

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vote

**0**answers

394 views

### Question about Riemann integral and total variation [closed]

Let $g$ be Riemann integrable on $[a,b]$, $f(x)=\int_a^xg(t)dt$ for $x∈[a,b]$.
How to show that the total variation of $f$ is equal to $∫_a^b|g(x)|dx$?

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votes

**2**answers

308 views

### Is there a good comparative study of the Banach integral?

The Banach integral is elegant in its definition, and I am intrigued as to why it is so rarely seen. Is it in practice difficult to calculate from the definition? And are there any other problems with ...

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votes

**2**answers

447 views

### One-sided Cauchy principal value

What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely,
$ PV \int_a^b f(t) dt = ? $,
where the integral is convergent in the upper limit, but ...