Linked Questions

37 votes
8 answers
11k views

How does one motivate the analytic continuation of the Riemann zeta function?

I saw the functional equation and its proof for the Riemann zeta function many times, but usually the books start with, e.g. tricky change of variable of Gamma function or other seemingly unmotivated ...
36min's user avatar
  • 3,758
44 votes
5 answers
3k views

An "analytic continuation" of power series coefficients

Cauchy residue theorem tells us that for a function $$f(z) = \sum_{k \in \mathbb{Z}} a(k) z^k,$$ the coefficient $a(k)$ can be extracted by an integral formula $$a(k) = \frac{1}{2\pi i}\oint f(z) z^{-...
MCH's user avatar
  • 1,304
22 votes
7 answers
5k views

What makes Gaussian distributions special?

I'm looking for as many different arguments or derivations as possible that support the informal claim that Gaussian/Normal distributions are "the most fundamental" among all distributions. ...
48 votes
2 answers
7k 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, (...
Joseph O'Rourke's user avatar
19 votes
6 answers
7k views

Does the derivative of log have a Dirac delta term?

Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics": $\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/1947pqm..book.....D ...
Mikhail Katz's user avatar
  • 15.1k
4 votes
2 answers
1k views

Are umbral moonshine and umbral calculus connected?

In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...
Daigaku no Baku's user avatar
7 votes
3 answers
1k views

Ramanujan's Master Formula: A proof and relation to umbral calculus

The Ramanujan's master theorem states that: $$ \int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s} $$ I found a really strange proof recently on a personal blog: Define $...
FFjet's user avatar
  • 282
12 votes
1 answer
813 views

How are Sheffer polynomials related to Lie theory?

Sheffer polynomials $\{P_n(x)\}$ have generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$. This form reminds me of the Lie group–Lie algebra correspondence. Is there any ...
Andrius Kulikauskas's user avatar
11 votes
2 answers
1k views

What's the matrix of logarithm of derivative operator ($\ln D$)? What is the role of this operator in various math fields?

Babusci and Dattoli, On the logarithm of the derivative operator, arXiv:1105.5978, gives some great results: \begin{align*} (\ln D) 1 & {}= -\ln x -\gamma \\ (\ln D) x^n & {}= x^n (\psi (n+1)-\...
Anixx's user avatar
  • 9,306
11 votes
2 answers
2k views

Green's function of the Ornstein-Uhlenbeck operator

The Ornstein-Uhlenbeck operator $L$ is given by $$ Lu = \Delta u- \frac{1}{2}x\cdot \nabla u. $$ Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ ...
Alexander Volkmann's user avatar
6 votes
1 answer
321 views

Can there be an application of discrete mathematics in PDEs, mainly the ones used in hydrodynamics?

Can there be applications of graph theory, combinatorics etc. in PDEs mainly hydrodynamics? Tried my luck with Google's search engine, didn't show much info. I guess you can try to use these features ...
Alan's user avatar
  • 1,514
7 votes
1 answer
406 views

What makes Gaussian distributions special? Local field version?

This question is inspired by the recent one about Gaussian measures over the reals: What makes Gaussian distributions special? I would be interested in a similar list of characterizations for the ...
Abdelmalek Abdesselam's user avatar
1 vote
1 answer
283 views

What's the name of this semi-group theorem?

I encountered this theorem, that for a bounded linear transform $L$ and a real parameter $t$ and initial data $u_0$, we have $$\frac{d}{dt} \exp(Lt)[u_0] = L \exp(Lt)[u_0].$$ What is the name of this ...
askquestions2's user avatar
1 vote
0 answers
132 views

Generating Hermite polynomial with coefficient recurrance relation algorithm

I am writing a math paper for my numerical analysis class about orthogonal Hermite polynomials. I want to implement the algorithm for generating the "probabilist's Hermite polynomials": $$ \...
russloewe's user avatar