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Results tagged with real-analysis
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user 113397
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
4
votes
1
answer
104
views
Dividing a spherical cap into three equal wedges
Background: Optimal ways to cut an orange.
In this problem, we have a spherical orange, and we do not wish to eat its central column which is modelled as a cylinder. Part of the procedure involves an …
- 1,459
3
votes
1
answer
240
views
Dividing a spherical cap into $n$ equal wedges
This is a follow-up of the question Dividing a spherical cap into three equal wedges where the $n=3$ case was shown.
Motivation: Optimal ways to cut an orange.
In this problem, we have a spherical ora …
- 1,459
5
votes
Accepted
Analytic expression for $\int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\p...
We seek \begin{align}I&=\int_0^\infty\frac{e^{-st}\sin t}{t(e^{at}+1)}\,dt\\&=\sum_{n\ge0}(-1)^n\int_0^\infty\frac{e^{-(s+a(n+1))t}\sin t}t\,dt=-\sum_{n\ge1}(-1)^n\arctan\frac1{s+an}\end{align} which …
- 1,459
8
votes
0
answers
291
views
Is there a real-analytic approach to evaluate a definite integral (with an elementary integr...
I have never seen a real-analytic approach to evaluate integrals of the form below
$$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary f …
- 1,459
4
votes
1
answer
183
views
Maximal increasing behaviour of $\sqrt{-\log x}\operatorname{Li}_{1/2}(x)$
This is an extension of a problem in mathematical biology. It appears that
For any $\varepsilon>0$, there exists an interval $I\subset(0,1)$ on which $\sqrt{-\log x}^{1+\varepsilon}\operatorname{Li}_ …
- 1,459