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Results tagged with real-analysis
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user 32389
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
2
votes
How can we obtain the $-\frac{4\pi}3\mu(x)$ term?
This is just a further detalization of the Carlo Beenakker's answer. Let us write $$K_{ik}=\int\left (\frac{\partial^2}{\partial r_i \partial r_k}\frac{1}{r}\right )\mu(\vec{y})d\vec{y},$$ where $r_i= …
- 16.5k
14
votes
Accepted
On an example of an eventually oscillating function
In addition to Noam Elkies and David Speyer's answers. "Harder" explanation is made somewhat "softer" in http://www.maths.bris.ac.uk/~majpk/papers/37.pdf (Summability of alternating gap series, by J. …
- 16.5k
3
votes
Introductory texts to mathematics
I would recommend Mathematical Omnibus: Thirty Lectures on Classic Mathematics, by D. B. Fuks and Serge Tabachnikov: http://books.google.ru/books?id=bomkJMq2H9sC&source=gbs_similarbooks and also books …
22
votes
3
answers
2k
views
The origin of the Ramanujan's $\pi^4\approx 2143/22$ identity
What is the origin of the Ramanujan's approximate identity $$\pi^4\approx 2143/22,\;\;\tag 1$$ which is valid with $10^{-9}$ relative accuracy? For comparison, the relative accuracy of the well known …
- 16.5k
4
votes
Accepted
"Values" of divergent integrals
For an attempt of such a theory, see http://carlossicoli.free.fr/B/Burgin_M.-Hypernumbers_and_Extrafunctions__Extending_the_Classical_Calculus-Springer(2012).pdf
(Hypernumbers and Extrafunctions: Exte …
- 16.5k
4
votes
Numerically computing $\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$
How Gauss (supposedly) calculated this integral in terms of AGM (namely $M(1,\sqrt{2})$) is outlined in http://home.sandiego.edu/~langton/gaussagm.pdf (Gauss, recurrence relations, and the AGM, by Sta …
- 16.5k
21
votes
0
answers
651
views
A multiple integral
Let us consider the multiple integral
$$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots
\int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots
\cos {(s_{2n-1 …
- 16.5k
2
votes
Stokes theorem with corners
This book http://www.math.wustl.edu/~sk/books/root.pdf (Geometric Integration Theory, by S.G. Krantz and H.R. Parks) is a self-contained introduction to geometric measure theory. See also Hassler Whit …
- 16.5k
1
vote
Quantum Mechanics and bilinear optimal control theory
In addition to the literature indicated by Carlo Beenakker: in this 2015 review http://arxiv.org/abs/1508.00442 (Training Schrödinger's cat: quantum optimal control, by S.J. Glaser et al.) the authors …
- 16.5k