All Questions
Tagged with integration real-analysis
352 questions
6
votes
3
answers
554
views
Computing the volume of a simplex-like object with constraints
For any $n \geq 2$, let
$$D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] =
\{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid
\sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \},$$
where $r \...
- 61
6
votes
1
answer
741
views
Is the following integral nonzero?
Recently I met an integral as follow:
$$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+...
- 1,997
6
votes
2
answers
635
views
Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?
PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim
If the integral
$$
\int_0^{2\pi} e^{i\...
- 405
6
votes
2
answers
681
views
integral depending on a parameter
Let $f$ be a real-valued continuous function on the interval $[0,1]$ and satisfy the following estimate
$$
\left|\int_0^1 f(t) e^{st}dt\right|\le Cs^{\frac12},\quad s>1,
$$
where the constant $C$ ...
- 375
6
votes
1
answer
193
views
Oscillatory integrals with a decaying factor in the integrand
Lemma 4.5 of Titchmarsh's book The Theory of the Riemann Zeta function says (slightly rephrased):
Let $F$ be a twice differentiable real function such that $ F''(x) \geq r > 0$ for all $x$ in $[a,b]...
- 63
6
votes
1
answer
433
views
Triangle inequality for Ito integral?
For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say
$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$
Now if ...
- 536
6
votes
1
answer
489
views
Henstock, Differentiation under the integral sign
Does anyone know, where I can find the proof of necessary and sufficient conditions for differentiating under the integral sign in case of Henstock integral? Here are the theorems but not all the ...
- 63
6
votes
0
answers
431
views
How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?
In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
- 2,836
6
votes
0
answers
2k
views
Interchange of integral and infimum
Can anyone please suggest how to justify widely used formula for interchange of integral and infimum:
$
\inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt,
$
where $ U\...
- 61
5
votes
2
answers
2k
views
Elementary calculus estimate or not?
Does there exist a constant $C>0$ such that for all $f \in H^3(\mathbb R)$
$$\int_{\mathbb R} \vert x f''(x) \vert^2 \ dx \le C \int_{\mathbb R} \vert f'''(x) \vert^2 + \vert x^3f(x) \vert^2 + \...
user69109
5
votes
2
answers
559
views
Compute $ \int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx $ [closed]
How can I compute this integral?
$$
\int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx
$$
- 71
5
votes
1
answer
2k
views
A rather curious equality: is this true?
I came across (coincidentally) two integral evaluations, which seem to agree according to numerical tests. It did not seem easy to convert one into the other.
QUESTION. Is this true?
$$\int_0^1\...
- 43.2k
5
votes
1
answer
2k
views
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 ...
- 95
5
votes
2
answers
301
views
Euler–Maclaurin formula in $\mathbb{Z}^d$
I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as
$$
\sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x)
$$
where $d\ge 2$ is an integer, $a,b \...
- 446
5
votes
1
answer
561
views
Upper bound an integral with exponential function
I am working on my research about approximation a function. I come up with the following integral. I run some simulations and saw that the integral would converge to zero as n goes to infinty. Here is ...
- 87
5
votes
1
answer
881
views
Two integral representations for $\zeta(3)$ from Zurab's integral and standard formulas for the gamma function
This morning I wrote with the help of a CAS, and integral representation for the Apéry's constant $\zeta(3)$ and some standard formulas two formulas involving this constant. I would like to know if ...
user75829
5
votes
1
answer
322
views
Proving an integral identity
Let $ f : \left[ 0, 1 \right] \rightarrow \mathbb{R} $ be a continuous function. Knowing that: $$ \int_0^1 (2x-1)f(x)dx = 0 $$ Show that $ \exists c \in \left(0,1\right)$ such that: $$ \int_0^c (x-c)...
- 151
5
votes
1
answer
355
views
Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\frac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} \, dx \, dy <+\infty$
I want to know whether or not
$$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}} \frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} \, dx \, dy <+\infty.$$
Here $D $ denotes the ...
- 61
5
votes
2
answers
647
views
Dominated convergence 2.1?
After this question : Dominated convergence 2.0?
I want to know, what about the case when $h\in L^1([0,1])$.
The completed question :
Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging ...
- 4,074
5
votes
1
answer
588
views
On the Riemannian integrability of the bounded derivative
Let $f:[a,b]\to\mathbb R$ be a differentiable function with $f'$ bounded. According to this post, $f'$ is not necessarily Riemann integrable on $[a,b]$, see also Volterra's function.
I wonder, if $f'$...
- 299
5
votes
1
answer
155
views
Which averages of products of a function give a norm?
Let $f: [0,1] \rightarrow \mathbb{R}$ be a bounded measurable function. For some real non-negative numbers $a_1, a_2, b_1, b_2$ with $a_1+b_1=a_2+b_2=1$ consider the quantity
$$N(f)=\int_{[0,1]} \int_{...
- 2,288
5
votes
1
answer
426
views
When is the Radon-Nikodym derivative locally essentially bounded
Let $\mu\lll\nu$ be $\sigma$-finite Borel measures, which are not finite, on a topological space $X$. Under what conditions is $0<\operatorname{ess-supp}(\frac{d\mu}{d\nu}I_K)<\infty$ for every ...
- 5,405
5
votes
1
answer
432
views
Functions that are Khinchin integrable but not Henstock-Kurzweil integrable
I posed this question on Mathematics SE recently, though by the total lack of attention it has gotten, I do not anticipate an answer and bring it here.
What are some Khinchin integrable $f$ which ...
5
votes
1
answer
914
views
Extension of a function from almost everywhere to everywhere
The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points?
Example: Let $f(x)=\...
- 7,182
5
votes
2
answers
288
views
Inverse Mellin transform of 3 gamma functions product
I want to calculate the inverse Mellin transform of products of 3 gamma functions.
$$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$
Above contour integral has ...
- 59
5
votes
1
answer
226
views
Multidimensional integrals that diverge by oscillation
It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq 0$...
- 19.7k
5
votes
0
answers
652
views
Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
- 782
5
votes
0
answers
266
views
Hadamard lemma without integration
Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero.
By the product ...
- 10.5k
4
votes
2
answers
2k
views
Does a function exist which is not Riemann integrable and satisfies the given condition:
I am looking for a function $f:[0,1]\rightarrow \mathbb{R}$ which is not Riemann integrable such that
$$\sum_{k=0}^n |f(x_k)-f(x_{k-1})|^2 <1$$ for every choice of $0=x_0\le x_1 \le \cdots \le x_n =...
- 151
4
votes
2
answers
303
views
Express $\int_0^{\pi/2}\{ \operatorname{gd}^{-1}(x)\}dx$ as series of special functions, with $\operatorname{gd}^{-1}(z)$ the inverse Gudermannian
I know that in the literature there are a lot of integrals involving the fractional part (and other floor and ceiling functions). For some of these integrals is provide its evaluation as a series or ...
4
votes
2
answers
592
views
From Zurab's integral representation for the Apéry's constant to almost impossible integrals
I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...
4
votes
1
answer
1k
views
Does the Lebesgue Differentiation Theorem hold for regular polytopes?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
- 4,597
4
votes
3
answers
598
views
Meaning of divergent integrals
In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. ...
4
votes
2
answers
872
views
What are the criteria for an elementary function to be infinitely integrable in elementary functions?
What features of elementary functions define a class of functions whose consecutive indefinite integration also gives an elementary function?
Is there a way to check whether a given elementary ...
- 10.1k
4
votes
1
answer
548
views
Two definitions of $L^p$ spaces that are not always equivalent
There are two definitions of $L^p(S, \Sigma,\mu)$ in the literature. (Here $S$ is a set, $\Sigma$ is a $\sigma$-algebra of subsets of $S$ and $\mu$ is a positive measure.) The two definitions are ...
- 41
4
votes
2
answers
718
views
Integrate $1/(x_1x_2\cdots x_n)^k$ for $1\le x_i \le a$, where product of coordinates satisfies $ b\le x_1\cdots x_n\le c$
I need to integrate
$$
\int\limits_{[1,a]^n} \frac {\chi(\{ b \le x_1 \cdots x_n \le c \})} {( x_1 x_2 \cdots x_n)^k} \,dx_1 \cdots dx_n,
$$ where $\chi(E)$ is the characteristic function of a set $E$....
- 91
4
votes
1
answer
1k
views
Indefinite integral of squared hypergeometric function
I am trying to compute the indefinite integral
$$
\int_0^u {}_2F_1\left(\frac{1}{4},\frac{5}{4},2,1-v^2\right)^2 dv
$$
for $0<u<1$. Using Clausen's formula for the square of the hypergeometric ...
- 165
4
votes
1
answer
424
views
An exercise on log-concave random variable on the real line
Let $X$ be a real random variable with log-concave density $f$. Assume that $E(X) =0$ and $E(X^2)=1$.
Show that there is a universal (independent of $X$) constant $c>0$ such that:
$$P(X\in[-1/2;0])\...
- 245
4
votes
2
answers
305
views
Generalization of van der Corput's estimate on oscillatory integrals
Question: Given exponents $0<\alpha<\beta$ and an interval
$[a,b]\subset(0,\infty)$ are there constants $C,d>0$ such that for any
$\lambda_1,\lambda_2\in\mathbb{R}$,
$$\left|\int_a^be(\...
- 1,701
4
votes
1
answer
191
views
Scaling of double convolution
I am interested in the scaling of
$$F(x_1,x_4)=\int_{\mathbb R^2} e^{-\vert x_1 -x_2 \vert -\varepsilon \vert x_2 -x_3 \vert- \vert x_3 -x_4 \vert } \ dx_2 dx_3 $$
In particular, I suspect that
$$F(...
- 192
4
votes
1
answer
461
views
How to get an expression for this integral (Numerically/Analytically)
I have the following problem:
I need to evaluate the integral $$\int_{\cos(\alpha)}^1 P_l(t)P_{l'}(t) \, dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the $l$-th ...
user37929
4
votes
1
answer
351
views
Asymptotic behaviour of function using Fox $H$-function representation
In equation (9) of this paper, it is claimed that the limiting behaviour
$$
\int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk
\sim
\frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}...
- 85
4
votes
1
answer
166
views
Estimate an improper integral
Suppose that $f$ satisfies $a$-Hölder condition on $[0,1]$ ($0<a<1$). Fix $0<b<a$. For any $x\in [0,1]$ and sufficiently small $\delta>0$, is the following inequality established?
$$ \...
- 419
4
votes
1
answer
282
views
How to estimate the order of this integral with parameter
Some introduction:
Given a homogeneous structure called "dilation" in $R^n$: For $t\geq 0$
$$D_t: R^n\rightarrow R^n$$
$$D_t(x)=(t^{a_1}x_1,...,t^{a_n}x_n)$$
where $1=a_1\leq...\leq a_n$, ...
- 561
4
votes
1
answer
143
views
Mean value of a function associated with continued fractions
Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let
$$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$
What is the mean value of $d$?
- 3,443
4
votes
1
answer
4k
views
Why can't I interchange integration and differentiation here?
I think my questions relates to this other: "counterexamples to differentiation under integral sign"
In fact, it provides a counterexample
Consider $f(x,y)=y^3e^{-y^2x}$ and define $F(y) =\...
- 53
4
votes
1
answer
114
views
Antiderivative of $f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$
I need to know the primitive function (Antiderivative) of this function:
$$f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$$
where
$K$ is an integration constant,
$d=-\frac{1}{2p}$ with $p<...
- 51
4
votes
1
answer
347
views
Two similar integrals
Let $n$ be a given even positive integer. We have the following integral
\begin{align}
\int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{i=1}^n\prod\limits_{j=1}^n(...
- 1,997
4
votes
1
answer
414
views
Convergence of the Double Integral of a Polynomial Reciprocal
Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:
(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;
(ii) $f$ is non-degenerate, in the sense that there isn't a ...
- 3,142
4
votes
1
answer
860
views
Lebesgue's integrability condition in several variables
The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
- 21.8k