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Questions tagged [ca.classical-analysis-and-odes]

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

3448 questions
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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 function $g:(a,b)\rightarrow\mathbb R$ ? Of course,…
Denis Serre
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44
votes
1 answer

Microwaving Cubes

First a little background. Microwaves do not heat uniformly. To help overcome this, your food is rotated, however this is not usually sufficient to produce totally uniform heating. Informally, this is the question: Is there a way of moving our food…
Mark Bell
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37
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3 answers

Do these properties characterize differentiation?

Let $L: C^\infty(\mathbb{R}) \to C^\infty(\mathbb{R})$ be a linear operator which satisfies: $L(1) = 0$ $L(x) = 1$ $L(f \cdot g) = f \cdot L(g) + g \cdot L(f)$ Is $L$ necessarily the derivative? Maybe if I throw in some kind of continuity…
Steven Gubkin
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28
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8 answers

Is there any use for $\sin(\sin x)$?

The convention that $\sin^2 x = (\sin x)^2$, while in general $f^2(x) = f(f(x))$, is often called illogical, but it does not lead to conflicts because nobody uses $\sin(\sin x)$. But is this really true? Or is there a real-world application in which…
user22882
27
votes
2 answers

Zeros of Gradient of Positive Polynomials.

It was asked in the Putnam exam of 1969, to list all sets which can be the range of polynomials in two variables with real coefficients. Surprisingly, the set $(0,\infty )$ can be the range of such polynomials. These don't attain their global…
Anonymous
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23
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5 answers

Existence of a smooth function with nowhere converging Taylor series at every point

By Borel's theorem, for any sequence of real numbers $a_n,$ there is a $C^{\infty}$-function $f:\mathbb{R}\to\mathbb{R}$ whose Taylor series at 0 is $\sum a_nx^n.$ In particular, there are $C^{\infty}$-functions whose Taylor series at a point has 0…
zamanjan
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18
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An intriguing calculus question

Let $f:{\bf R}^n\to {\bf R}$ ($n\geq 2$) be a $C^1$ function. Is it true that $$\sup_{x\in {\bf R}^n}f(x)=\sup_{x\in {\bf R}^n}f(x+\nabla f(x))\hskip 3pt ?$$
Biagio Ricceri
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18
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4 answers

Is the following function decreasing on $(0,1)$?

Hi, I asked some time ago the following question on math.stackexchange, but I ask it here too since it remains unanswered. The question concerns a function I encountered during research : $$f(k):= k K(k) \sinh \left(\frac{\pi}{2}…
Malik Younsi
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17
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4 answers

If $f$ is smooth and even then there exists a smooth function $g$ such that $f(x)=g(x^2)$

Assume that a function $f: R \rightarrow R$ is smooth and even. Does there exist a smooth function $g:R \rightarrow R$ such that $f(x)=g(x^2)$ ?
rts
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Higher arcsin wanted

There is a known proof of the identity $$\sum_{k=0}^\infty \frac1{(2k+1)^2}=\frac{\pi^2}8\,\,\,(*)$$ (equivalent to $\sum \frac1{n^2}=\frac{\pi^2}6$) by expanding $\arcsin x$ as a power series $$\arcsin x=\sum_{k=0}^\infty…
Fedor Petrov
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15
votes
2 answers

Justifying the definition of arclength

Background One of my friends told me the following story: A child must walk from his home at point A = (1,0) to his school at point B = (0,1). The laws in his country state that you can only walk parallel to the horizontal and vertical axis. No…
Steven Gubkin
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15
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0 answers

Surprising approximate identity

While answering this MO question Connection between Bernoulli numbers and Riemann-Siegel theta function? Dan Romik found the following surprising approximate identity: $$\ln{8\pi}\approx \pi\left[…
Zurab Silagadze
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Simultaneous zero set of two equations in $\mathbb R^3$

Can we have positive reals $x,y,z$ with $$ x^{\left( y^z \right)} = y^{\left( z^x \right)} = z^{\left( x^y \right)} $$ in cyclic permutation, other than the line $x=y=z$? I put this at…
Will Jagy
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13
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2 answers

Sets of divergence of Fourier series

Carleson theorem (later extended by Hunt) states that given an $L^2$ function $f:{\mathbb R}/{\mathbb Z}\to{\mathbb C}$, the set of points $x$ where the Fourier series $$\lim_{n\to\infty}\sum_{k=-n}^n\hat f(k)e^{2\pi ik x}$$ does not converge to…
Andrés E. Caicedo
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Is there a reason why integrals are so much easier to evaluate than sums?

One possibility is the integration by substitution formula. But this doesn't explain why $\int_1^{\infty}x^{-3}dx$ is easy to evaluate (and rational) whilst $\sum_1^{\infty}n^{-3}$ is unknown (and seemingly not a simple combination of well known…
teil
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