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Results tagged with ca.classical-analysis-and-odes
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user 121
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
1
vote
Space of rapidly decreasing functions
My initial interpretation is that your inner product comes with a Gaussian weight. If $g=H\_0$, then $c_n = \delta\_{n,0}$, and for any fixed $t\_0$, the sum is a nonzero constant function, which is …
- 45.7k
4
votes
Accepted
Polynomial series
I don't know what you mean by "polynomial series" since your function doesn't seem to have much to do with polynomials (perhaps you could elaborate?).
$S(x) = -\frac12 \theta(\frac12, \frac{\log x}{\ …
- 45.7k
16
votes
a^b = b^a when a is not equal to b.
This doesn't answer your question, but I thought I'd point out that if we assume $1 < a < b$, the solutions to $a^b = b^a$ can be written in the form $a = \left( \frac{s+1}{s} \right)^s$ and $b = \lef …
- 45.7k
2
votes
"Misbehaved" differential equations
I'm not a numerical analyst, and I didn't really understand your question, but one example that came to mind as I was reading it was oscillatory phenomena at discontinuities. If you model a linear in …
- 45.7k
7
votes
Accepted
Finitely additive translation invariant measure on $\mathcal P(\mathbb R)$
Banach-Tarski poses a problem for existence of measures that are invariant under all rigid motions, not just translation. The existence of finitely additive translation-invariant measures that agree …
- 45.7k
1
vote
Growth of the "cube of square root" function
It's not too hard to put a bound on the size of second differences (since without the truncation, they are bounded above by a constant times $n^{-1/2}$), but getting the bound down to one seems delica …
- 45.7k
2
votes
Is there an nontrivial function whose 'period paralellograms' are Gosper Islands?
The Gosper islands are a fundamental domain for the translation action of the Eisenstein integers $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$, since the shapes can be constructed by deforming a Voronoi decomp …
- 45.7k
5
votes
Why is the gradient normal?
The proof I usually see: Choose an arbitrary unit length tangent vector on the level set, and write it with coordinates. If you take the inner product of this vector with the gradient, the sum you ge …
- 45.7k
10
votes
Accepted
What is the Stirling formula for x(x+1)(x+2)...(x+n-1)?
It looks like you want a formula for the asymptotics of the Pochhammer symbol $(x)_n$ as $n \to \infty$. One such formula is provided about halfway down Wolfram's page:
$$(x)_n \sim \frac{\sqrt{2\pi} …
- 45.7k
5
votes
How do I make the conceptual transition from multivariable calculus to differential forms?
I second the recommendation to at least flip through Gravitation. It has an intimidating size, but easygoing manner. I had a lot of difficulty with Spivak's Calculus on Manifolds (which has essentia …
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40
votes
What is convolution intuitively?
I think one's standards of intuitiveness depend strongly on one's background. Even if a picture seems unintuitive at first, it can be helpful later.
If you're an algebraist, I'd suggest the multipl …