User:Sure Beae: Difference between revisions

My website is https://foil.town/

I’m interested in maths, FreeBSD/HardenedBSD, cybersecurity, networking, and biology. Specifically, discrete calculus, and its relations/intersections with Riemann-Liouville fractional calculus, analytic number theory, polypi (offset polygamma) function (see A generalized polygamma function by Oliver Espinosa and Victor H. Moll for a related function with nice analytic properties), the generalized Bernoulli numbers, and so on.

Here is a graph based on the YouTube video How to Extend the Sum of Any* Function

https://www.desmos.com/calculator/xjzu4jqghl

Here is the exact version that isn’t using the difference between two divergent summations

https://www.desmos.com/calculator/y2i9qmoqjn

Here is a graph of the famous Faulhaber’s formula, which is misnamed as Faulhaber never wrote or interacted with it directly.

https://www.desmos.com/calculator/ywv6hjiq0l

Hasse’s Newton-like approx

https://www.desmos.com/calculator/eyz2xo1fa8

A fun game I would play when I was in school, doing complex mappings using the zeros of 3D trig-based functions, here’s 1/z as example

https://www.desmos.com/3d/weknvqvovz

The easiest way to contact me on other platforms is ‘sure’ on IRC or @bord:chat.foil.town on matrix.

Notes to myself

https://www.desmos.com/calculator/ppmuwcuhaw

Reflection formula

B x = 2 τ − x Π ( x ) cos ⁡ ( τ x 4 ) B 1 − x 1 − x {\displaystyle B_{x}=2\tau ^{-x}\Pi \left(x\right)\cos \left({\frac {\tau x}{4}}\right){\frac {B_{1-x}}{1-x}}}

Euler Product

∂ ∂ x ( ∇ − 1 x s ) | x = 0 = { − s ∏ p 1 1 − p s − 1 for  s < 0 − 2 τ − s Π ( s ) cos ⁡ ( τ s 4 ) ∏ p

1 1 − p − s for  s > 1 {\displaystyle {\frac {\partial }{\partial x}}\left(\nabla ^{-1}x^{s}\right){\bigg |}_{x=0}={\begin{cases}-s\prod _{p}{\frac {1}{1-p^{s-1}}}&{\text{for }}s<0\\-2\tau ^{-s}\Pi \left(s\right)\cos \left({\frac {\tau s}{4}}\right)\prod _{p}^{}{\frac {1}{1-p^{-s}}}&{\text{for }}s>1\end{cases}}}