Linked Questions

58 votes
7 answers
9k views

Why is the Gaussian so pervasive in mathematics?

This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as ...
Randy Qian's user avatar
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
51 votes
6 answers
12k views

What does Mellin inversion "really mean"?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
Frank Thorne's user avatar
  • 7,199
16 votes
2 answers
2k views

Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?

Consider the operator $\frac D{e^D-1}$ which we will call "shadow": $$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } e^{-iwx}\frac{-iw}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^...
Anixx's user avatar
  • 9,306
6 votes
3 answers
2k views

A question on fractional derivatives

I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange. I just wanted to ask if there is a notion of ...
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
16 votes
1 answer
2k views

What are some of the earliest examples of analytic continuation?

I'm wondering how Riemann knew that $\zeta(z)$ could be extended to a larger domain. In particular, who was the first person to explicitly extend the domain of a complex valued function and what was ...
Mustafa Said's user avatar
  • 3,679
9 votes
2 answers
1k views

Newton series and Fourier transform - is there an analogy?

Fourier expansion for a function: $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$ Newton series expansion of a function: $$f(x)...
Anixx's user avatar
  • 9,306
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
3 votes
4 answers
2k views

History of the Sampling Theorem

In January, 1949, Shannon publishes the paper Communication in the Presence of Noise, Proc. IRE, Vol. 37, no. 1, pp. 10-21, available here, which establishes the Information Theory. In this paper, the ...
Papiro's user avatar
  • 1,568
7 votes
1 answer
2k views

Pochhammer symbol of a differential, and hypergeometric polynomials

I have a minor result which I'm sure has come up somewhere before but I can't seem to find it. Consider a confluent hypergeometric function of the form $$\newcommand{\ff}{{}_1F_1} \ff(b+k;b;z)\...
Emilio Pisanty's user avatar
1 vote
1 answer
458 views

Deriving the functional equation for $\zeta(s)$ from summing the powers of the zeros required to count the integers

When counting the number of integers $n(x)$ below a certain non-integer number $x$, the following series could be used: $$n(x) = x-\frac12 + \sum_{n=1}^{\infty} \left(\frac{e^{x \mu_n}} {\mu_n}+\frac{...
Agno's user avatar
  • 4,179