0
votes
1answer
28 views

Generalized Logarithmic Integral - reference request

This page at I&S forum defines the Generalized Logarithmic Integral as $$L\left[ \begin{matrix} a,b,c \\ d,e,f \end{matrix};z\right] =\int_0^z \frac{\log^a x \log^b(1-x)\log^c(1+x)}{x^d (1-x)^e ...
10
votes
2answers
242 views

What meaning did Riemann assign to $dx$?

Detlef Laugwitz wrote a monumental biography of Riemann. The book was translated into English by Shenitzer. Laugwitz discusses Riemann's fundamental essay Uber die Hypothesen, welche der Geometrie ...
5
votes
0answers
49 views

Integration of bundle-valued differential forms

The literature, at least textbooks, seems to be very scarce on the topic of integrating bundle-valued differential forms. So I wonder where can I read on the topic? I want to see usual theorems, like ...
2
votes
0answers
53 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
3
votes
1answer
62 views

Integral of products of cosines

Given $m+1$ integers $\alpha_0,\ldots,\alpha_m\geq 1$, I was trying to get a nice closed formula for the integral $$ \int_0^\pi\cos(\alpha_1\theta)\cdots\cos(\alpha_m\theta)d\theta. $$ More precisely, ...
3
votes
0answers
31 views

Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
0
votes
2answers
60 views

Resources for learning integral calculations

I am willing to learn about integrals . So i wonder is there any systematic book about the topic that goes progressively in difficulty and complexity . My current level is about knowing the basic ...
0
votes
0answers
32 views

Reference Request: Fubini's theorem for non-negative functions

I have never seen this (1st page) formulation of Fubini's theorem in the literature. Does anyone know where I can find it? In every calculus book (e.g. Apostol, Courant, etc.) I looked, the authors ...
2
votes
0answers
67 views

Simplifying a Vector Integral

While reading the book - Cercignani, Theory and Applications of Boltzmann Transport Equation (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \xi_l$ ...
11
votes
2answers
234 views

Why is Volume^2 at most product of the 3 projections?

Is there a simple proof for $$ \text{Vol}^2(P)\le \prod_{i=x,y,z} \text{Area}(\text{Proj}_i(P)), $$ where $P\subset \mathbb R^3$ and $\text{Proj}_z(P)$ denotes the projection of $P$ to the $z=0$ ...
0
votes
0answers
26 views

Reference request: substitution in Riemann–Stieltjes integrals

Suppose $h$ has bounded variation (or if you must, suppose $h$ is non-decreasing) $g$ is differentiable everywhere (for suitable values of "everywhere"), and $f$ is well-behaved enough for this to ...
2
votes
1answer
40 views

Is there a general algorithm to solve computable integral equation?

Hilbert's tenth problem ask for the general algorithm(finite number of operation) to solve of all Diophantine problems.Today, it is known that no such algorithm exists in the general case. What ...
1
vote
0answers
34 views

The Fourier sine transform of $f(x)/\sin x$

Is the following result $$\lim_{\lambda \to \infty} \frac{2}{\pi} \int_0^\infty \frac{f(x)}{\sin x} \sin(\lambda x) \, dx = f(0) + 2\sum_{k = 1}^\infty f(k\pi),$$ where $\lambda$ is an odd integer, ...
1
vote
0answers
44 views

Repository of functions which do not have elementary integrals [duplicate]

If there is some function and I suspect that the primitive function cannot be expressed using elementary functions, I would like to have some argument that there indeed is no such expression. One ...
1
vote
3answers
125 views

Does anyone know which textbook this question is from? $\displaystyle \int_0^x \frac{\sin t}{t+1}dt > 0$ for all $x>0$

So I'm currently doing integration problems preparing for final exam, and I would like to practice problems like the following: Prove that $\displaystyle \int_0^x \frac{\sin t}{t+1}dt > 0 \ \ $ ...
0
votes
0answers
40 views

Reference for theorem? Inequality of integrals of increasing function over two distributions

I have a monotone increasing function $H(x)$ and two distributions with CDFs $F_1$ and $F_2$, where $F_1(x) \leq F_2(x)$ everywhere. The domain is $[0,\infty)$. This seems like it must be true: $$ ...
3
votes
0answers
65 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
0
votes
0answers
16 views

Weak stochastic integral

I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prevot/Rockner [PR07]: $ \int_0^T { \langle \Psi dW(t), \Phi(t)\rangle }$ A few useful ...
1
vote
0answers
49 views

Books explaining differentiation under the integral sign

I've heard that this is a great tool to have in you math toolkit, but I cannot comprehend this method just from the wiki entry and 2 page pdf files. I'm looking for a book which has problems ...
3
votes
1answer
65 views

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is 'supposed' to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ ...
0
votes
2answers
49 views

What is list of common integral that have no closed form?

What is list of common integral that have no closed form? It's diffucult for me to google it for some reason.
3
votes
2answers
314 views

List of functions not integrable in elementary terms

When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the ...
3
votes
1answer
62 views

Intuition for chains and cochains

I'd like to get some "geometric," "physical," (or other form of) intuition for chains, cochains, and their relationship to integration on manifolds at an elementary level. In particular, it would be ...
1
vote
1answer
98 views

How can I find who discovered this integral?

I need to find the first paper/author to document this integral $$\int\log^nx\;\mathrm dx=(-1)^n\;\Gamma(n+1,-\log x)\quad n\in\Bbb N_0$$ To prevent this in the future, is there a service in which I ...
0
votes
0answers
41 views

Reference needed for integration on boundary of Lipschitz domain

I need a reference for a definition of an integral of a function $f:\partial\Omega \to \mathbb{R}$ over the boundary of a Lipschitz open domain $\Omega \subset \mathbb{R}^n$ (the usual domain in ...
10
votes
3answers
285 views

Examples of adding a constant to integration by parts.

The formula for integration by parts is given by $$ \int uv'=uv-\int u'v $$ As most of you know. The result is invariant if we use$v=v+c$, instead of $v$ where $c$ is some arbitary constant. $$ ...
2
votes
1answer
175 views

Continuity of parameter dependent integral (source needed)

I am looking for a reference from a book for the result of continuity of an integral (found in http://www.encyclopediaofmath.org/index.php/Parameter-dependent_integral): Let $D \subset ...
4
votes
0answers
70 views

The Leibniz rule in Euler's works

Does anyone know if the Leibniz rule (the method of differentiation under the integral sign), or a variation thereof, has ever appeared in any of Euler's papers? Any references would be appreciated. ...
8
votes
1answer
115 views

Integrals 3.384 from Gradshteyn and Ryzhik

I'm interested in understanding the computation of $$\int_{-\infty}^\infty\frac{e^{-ip x}}{(1 + ix)^{2u}(1-ix)^{2v}}\mathrm{d}x,$$ which is evaluated in 3.384.9 of Gradshteyn and Rhysik for ...
2
votes
1answer
65 views

How to do integration with substitution

Let's suppose we have the following integral and substitution for $u=x^2$: $$\int x^2\;\mathrm dx=\int u\;\mathrm du\tag1$$ The solution for the following integral is: $$\int x\;\mathrm ...
0
votes
1answer
84 views

Prove the Jordan lemma i.e. $\int e^{-R\sin{\theta}}< \pi/R$

In complex variables my instructor wrote on the board "Jordan's Lemma", and then, somewhat imprecisely, $$\int e^{-R\sin{\theta}}< \pi/R \;\;\;\; \text{ e.g. } \int \frac{s \sin{x}}{x^2 + 2x + ...
4
votes
1answer
63 views

How to choose contour in $\mathbb{C}$ to do Residue Integration.

I'm almost sure that there's not any simple way to answer this question, but I'll try. I'm studying complex variables and the method of calculating improper integrals with residues but I'm struggling ...
3
votes
3answers
1k views

How to master integration and derivation?

We have learnt in school about derivation and integration, however I find my knowledge fairly poor. I mean I have problems with taking the derivative/integrating even simple functions. So I would like ...
0
votes
1answer
32 views

Integrals via Infinite Series

On the bottom of page 24 & top of page 25 of this pdf an integral is beautifully computed by breaking it up into an infinite series. Is there any reference where I could get practice in working ...
3
votes
2answers
129 views

Reference for integration

Does anyone have a good reference for a book that already assumes knowledge of calculus/analysis and whose main focus is computing more difficult integrals? I'm looking for something which will have a ...
3
votes
2answers
195 views

Good books to learn Riemann integration

I am looking for a good text book to learn Riemann integration. Please suggest books with theories and proofs comprehensively explained.
2
votes
0answers
216 views

Cauchy's theorem for integral homotopic closed curve in $G\subset\mathbb{C}^n$.

Recall Cauchy's theorem (third version in the Conway's book "Function of one complex variable", thm 6.7. page 90 in the second edition): Let $f$ be an analytic function on $F\subset\mathbb{C}$ and ...
3
votes
2answers
169 views

Derivative of Integral (in Fourier transform)

I've taken some analysis, but somehow Fourier transforms were never brought up until they were assumed to be familiar. Fun. Anyway, in a class example (showing the integral of a Gaussian is again a ...
6
votes
1answer
140 views

Integrals of matrix functions

I've stumbled across some math I've never really encountered before, and I would love it if someone could provide me with some useful references and texts on it. I'm dealing with integration over the ...
4
votes
1answer
91 views

Matrix-product-integrals?

Whereas the conventional "sum integral" is $$ \lim_{\Delta x\to 0} \sum_i f(x_i)\,\Delta x, $$ a "product integral" is $$ \lim_{\Delta x\to 0} \prod_i f(x_i)^{\Delta x}. $$ Now you're thinking: just ...
7
votes
1answer
156 views

Does this inner product on $L^1([0,1])$ have a name?

Math people: For $f, g \in L^1([0,1])$, define $$\langle f,g \rangle = \int_0^1 \int_0^1 f(t)g(t')\exp(-|t-t'|)dt'\,dt.$$ Although we don't normally think of $L^1([0,1])$ as an inner product space, ...
0
votes
1answer
71 views

Integrating trig functions with $R(\frac {z+1/z} {2}, \frac {z - 1/z} {2i} )$

Someone told me that there is a method for integrating rational functions $R(\cos{\theta}, \sin { \theta})$ by doing contour integration of the complex function $$\frac {R \left( \frac {z + \frac1z} ...
2
votes
3answers
420 views

What texts do you recommend to study calculus?

I've studied calculus 2 years from Arabic text . It was great text , which is supported with huge amount of examples and exercises , Now , I find it's a good step to study the material in English as ...
0
votes
1answer
130 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
4
votes
3answers
152 views

Notes about evaluating double and triple integrals

I'm searching notes and exercises about multiple integrals to calculate volume of functions, but the information I find in internet is very bad. Can someone recommend me a book, pdf, videos, ...
5
votes
0answers
145 views

Algorithm to calculate multiple integral.

One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
0
votes
1answer
44 views

Outer measure defined by a continuous and bijective function

This problem is from K.T. Smith's Primer of Modern Analysis: Let $\psi: \mathbb{R}^d \to \mathbb{R}^d$ be continuous and one-to-one on an open set $\Omega \subset \mathbb{R}^d$ and define $$\nu(A) ...
17
votes
1answer
315 views

References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$

While extending my calculation techniques, with aid of Mathematica, I found that \begin{align*} \int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right)^{3} \, dx &= -6 \zeta '(-1) ...
2
votes
3answers
111 views

Reference request for practicing integration

I'm looking for a reference -- of any kind, website or book -- for practicing integration techniques. The only book I have on hand is Stewart's calculus book, but that's not quite what I'm looking ...
6
votes
3answers
249 views

An “Itzykson-Zuber”-like integral

I been told that there exists an integration formula, which states (or something of this sort) $$ \int_{U(N)} dU \det[(\mathbb I+XUYU^{-1})^{-r}\propto ...