Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics.

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38
votes
8answers
4k views

Does the formal power series solution to $f(f(x))= \sin( x) $ converge?

I have spent some time using gp-pari. There is, of course, a formal power series solution to $ f(f(x)) = \sin x.$ It is displayed, below, identified by the symbol $g$ because I am not entirely sure ...
54
votes
2answers
16k views

Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows: All numbers are divided into two classes: those ...
9
votes
3answers
929 views

Limit cycles as closed geodesics(in negatively curved space)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$: \begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation} This equation defines a foliation on $\...
74
votes
9answers
16k views

How to solve $f(f(x)) = \cos(x)$?

I found the following interesting equation on some web page I cannot remember: $f(f(x))=\cos(x)$ Out of curiosity I tried to solve it, but realized that I do not have a clue how to approach such an ...
9
votes
1answer
552 views

Solution of linear ODE

Let $A=A(t)$ be a smooth one parameter family of $n\times n$-matrices, $n\ge 2$. It seems that the solution of linear ODE $$\dot x= Ax$$ can not be written in a closed form using $\int$, $A$, $x(0)$ ...
102
votes
6answers
8k views

Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference: $$ \frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2}, $$ where the ...
30
votes
1answer
3k views

Geometric interpretation of the half-derivative?

For $f(x)=x$, the half-derivative of $f$ is $$\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}} x = 2 \sqrt{\frac{x}{\pi}} \;.$$ Is there some geometric interpretation of (Q1) this specific derivative, and, (...
21
votes
8answers
9k views

Does the exponential function have a square root?

(asked by Nathaniel Hellerstein on the Q&A board at JMM) Is there a "half-exponential" function $h(x)$ such that $h(h(x))=e^x$? Is it unique? Is it analytic? Related question: Is there an ...
4
votes
0answers
3k views

Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

This problem seems like a nightmare to me. I tried to expand $K_{\omega}^f(t)$, but I am clueless of getting some kind of a closed form or some kernel like structure. If I try to take derivative, ...
10
votes
0answers
536 views

Multiple Integral (American Mathematical Monthly problem 11621 and its generalization)

AMM problem 11621 asks to calculate the integral $$I_2=\int\limits_{-\infty}^{\infty}ds_1\int\limits_{-\infty}^{s_1}ds_2 \int\limits_{-\infty}^{s_2}ds_3\int\limits_{-\infty}^{s_3}ds_4 \;\cos{(s_1^2-...
2
votes
2answers
560 views

Convergence of Newton series for sin ax

Let's define half discrete-analytic function as a function whose Newton series converges to that function for each $x>0$: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)=\sum_{m=0}^{\...
15
votes
2answers
777 views

Asymptotic approximation of $x^\alpha$ by entire functions

Given a non-integral real $\alpha$, is there an entire (see http://en.wikipedia.org/wiki/Entire_function) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$ for $x\rightarrow+\infty$ (with $...
3
votes
0answers
106 views

Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii). In a joint paper that I am ...
123
votes
16answers
18k views

How do I make the conceptual transition from multivariable calculus to differential forms?

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...
73
votes
4answers
12k views

Does the inverse function theorem hold for everywhere differentiable maps?

(This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.) Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each ...
29
votes
2answers
4k views

Is there a category structure one can place on measure spaces so that category-theoretic products exist?

The usual category of measure spaces consists of objects $(X, \mathcal{B}_X, \mu_X)$, where $X$ is a space, $\mathcal{B}_X$ is a $\sigma$-algebra on $X$, and $\mu_X$ is a measure on $X$, and measure ...
21
votes
24answers
27k views

Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
16
votes
5answers
11k views

Criteria to determine whether a real-coefficient polynomial has real root?

Given a polynomial equation $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, where $n$ is even and all the coefficients $a_i$ are real, what is the best way to determine whether it has a real root or not? I ...
15
votes
2answers
2k views

Ideals of the ring of smooth functions

The ring $C^\infty(M)$ of smooth functions on a smooth manifold $M$ is a topological ring with respect to the Whitney topology and the usual ring operations. Is it possible to describe, maybe under ...
11
votes
5answers
1k views

Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same. Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
7
votes
1answer
2k views

Existence/Uniqueness of solutions to quasi-Lipschitz ODEs

Would the Picard–Lindelöf theorem still be true if the requirement that f be Lipschitz continuous in y was replaced with the requirement that f be almost Lipschitz in y? If not, are there any moduli ...
10
votes
2answers
3k views

Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb R^k\...
18
votes
1answer
497 views

Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...
13
votes
1answer
1k views

Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?

Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$ runs over the integers? The existence of the limes inferior follows from Dirichlet's approximation theorem, but the ...
15
votes
3answers
1k views

Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for $C^k$-...
10
votes
1answer
321 views

On the independence of lower and upper asymptotic and Banach densities

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \...
9
votes
1answer
943 views

Has anyone seen this series?

I come across the following infinite series. $$ \sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}. $$ In particular, I am interested in the case where $a=1/4$. ...
4
votes
1answer
739 views

Why are all these families of polynomials finally log-concave?

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$. (Notation: in the following, the $a_k$ ...
10
votes
1answer
491 views

A tricky tractrix question about vertical tangents

This is raised by a recent question occurring in combinatorial geometry. It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ (...
3
votes
2answers
197 views

functions with orthogonal Jacobian

I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = \...
2
votes
2answers
1k views

How to compute \prod_{n=1}^{\infty} (1-p^{-n})

We know it converges for any prime p. I just want to know how to compute its exact value: $$\prod_{n=1}^{\infty} (1-p^{-n})$$
1
vote
1answer
440 views

New differintegral formula: how is it related to other differintegral formulas?

Lets define new differintegral formula as $$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ or, equivalently, $$\mathbb{D}^s_xf(x)= \lim_{t\to s} \...
107
votes
26answers
34k views

What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
89
votes
17answers
20k views

Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by parts/...
62
votes
21answers
9k views

Probabilistic Proofs of Analytic Facts

What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...
45
votes
22answers
13k views

Interesting Calculus Questions/Exercises

I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...
52
votes
9answers
9k views

Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)

Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
58
votes
15answers
9k views

What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

Before you close for "homework problem", please note the tags. Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...
78
votes
8answers
6k views

Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?

Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$? (I think it is.) If so, how would one prove this? (To confirm: This is the power series for $e^x$, except with the ...
33
votes
11answers
30k views

Why is the gradient normal?

This is a somewhat long discussion so bear with me. There is a theorem that I have always been curious about from an intuitive standpoint. This is an issue that has been glossed over in most ...
33
votes
15answers
11k views

What is the Implicit Function Theorem good for?

What are some applications of the Implicit Function Theorem in calculus? The only applications I can think of are: the result that the solution space of a non-degenerate system of equations ...
59
votes
1answer
4k views

A hard integral identity on MATH.SE

The following identity on MATH.SE $$\int_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{dx}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8}$$ seems to be ...
41
votes
16answers
6k views

How helpful is non-standard analysis?

So, I can understand how non-standard analysis is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta ...
30
votes
6answers
3k views

Taylor's theorem and the symmetric group

Anytime I see an $n!$ in some formula, my instinct is to look for the symmetric group on $n$ letters coming in somewhere. I have never done this seriously with the $n!$ in Taylor's theorem. Question: ...
12
votes
4answers
9k views

Visualization of Riemann–Stieltjes Integrals

The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating Riemann–...
30
votes
4answers
2k views

Which differential equations allow for a variational formulation?

Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation $$ \frac{d}{dt}\frac{\...
34
votes
4answers
3k views

Which functions of one variable are derivatives ?

This is motivated by this recent MO question. Is there a complete characterization of those functions $f:(a,b)\rightarrow\mathbb R$ that are pointwise derivative of some everywhere differentiable ...
25
votes
2answers
8k views

Irrationality of $ \pi e, \pi^{\pi}$ and $e^{\pi^2}$

What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?
31
votes
7answers
2k views

How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?) Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
36
votes
5answers
1k views

Undergraduate ODE textbook following Rota

I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do ...