Others

Here I share some particular stuff you might find interesting: program codes, description of my works and some theorems, observations that I am not intenting to publish in journal.

A new integral representation for the Lambert W function

I found an integral representation for the principal branch of the W function. This result is simpler than the similar one found by Poisson long-long time ago.


The p-adic Lambert W function

It is possible to define the p-adic version of the classical Lambert W function. I did this in this short note, where it was proven, among others, that the p-adic W function is not an analytic element, i.e., it is not a (uniform) limit of a sequence of rational functions.


C++ code for the complex Lambert W function

I am planning to write a C/C++ code for the r-Lambert function (see below). To do this, I would need to call the classical Lambert W function from my code. I could not find source code for the complex W, so I wrote one.
Here is the code. Please see the documentation for details how to use the it.

The definition of the function is as follows:
complex <double> LambertW(complex <double> z, int k = 0)
Here z is the argument, k is the branch index. As it is usual, k = 0 belongs to the principal branch.
My code uses Halley's method with appropriate initial points. The initial points are determined by the known logarithmic approximation of W when we are not close to branch cuts. Otherwise (1,1)-Padé approximants or series around the branch cut (z=0) were used. The basic approach is written down (without code snippets) in the work of Corless et. al.: On the Lambert W function, Adv. Comput. Math. 5 (1996), no. 4, 329–359.


The r-Lambert function

I worked on a new generalization of the Lambert W function which helps to solve some transcendental equations arising in physics and combinatorics. I wrote a C code which calculates this function in the real case. You can either build this code on your computer if you have a C compiler, or just copy the code text directly to CompileOnline.com
Special thanks to Keith Briggs, who wrote a code to calculate W.
The r-Lambert function has infinitely many branches over the complex plane, the whole description of the branches is recently done.


A p-adic number library in C++

To refresh and update my object oriented coding knowledge (what I have not used for 10 years or so), I wrote a C++ library for basic p-adic number calculations, using some new features of C++11. The code is deposited on GitHub.


Lower and upper estimation for the Mahler measure of the Bell polynomials

With a tiny effort a rather good estimation can be given for the Mahler measure of the Bell polynomials. The result is contained in this pdf.


The Furstenberg topological space

There is a special topology on the set of integers that enables one to prove the infinitude of primes. This topological space has some non-trivial features. My colleague, Rezső Lovas and I gave a metric completion of this space that, interestingly enough, involves infinitely big 'integers' and involves the so-called factorial number system.
These results are published in the Elemente der Mathematik 69: 1-14.


The above partition relation was proven by Sierpiński. Here I offer another proof that uses a useful analytic theorem.
Another partition relation I was thinking on comes from the book Introduction to Cardinal Arithmetic.



An alternative form of the Uehling potential

The classical Coulomb potential ~1/r describes the electric field at distance r around a point charge placed at the origin. From Quantum Electrodynamics (QED) point of view this corresponds to virtual photon emission from the charge towards its ambient. It is known from QED that virtual photons can emit and absorb virtual electron-positron pairs during their traveling so the Coulomb electric field around a point charge must be corrected accordingly. Considering no other effects, this virtual e-e+ emission and absorption (one-loop correction or vacuum polarization effect) results in a modified potential first described by Uehling in 1935. I noted that the Uehling potential can be written in a different but equivalent form by using the Chi and Shi hyperbolic cosine and sine integrals.


Photography

I like documentary style street photography. If you are interested, you can find some of my photos on Instagram.
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