One interesting article

Recently I've read the article Phys. Rev. B 84, 075145 (2011) devoted to study of Matsubara Green functions of quantum impurity models on the basis of Legendre polynomials. It was shown that Green functions for models of some class have coefficients of decomposition into Legandre polynomial decreasing much faster than coefficients of Fourier series for Green functions.

Some thoughts rose from this article:

1) Is this property true for every model or only for some class of models? If second, what is that class?
2) If it is general property of Green functions then it must be inferred by some general consideration that hasn't been done.

3) Does this property depend on dimension and type of lattice of the model?

4) Observed property of Green functions restricts the class of functions to which the Green functions belong. Is it possible to formulate this restriction strictly mathematically? How could it be presented?

Report for last 10 days

I've derived similar result to that one derived by Dominici. However, my result was derived in very simple and transparent way in contrast to Dominici. Moreover, my result is correct for all values of parameters $a$ and $b$ whereas Dominici's result is correct only for $a+b\leq2\pi$. Now I am intended to prepare an article. But one problem still persist: I can't rigorously prove some mathematical transformation I use. Before preparing the article I have to deal with this problem.

I've suspended publication of the article on infinite series representation of Bessel functions as I still can't prove the possibility of analytic continuation of my series, which holds (verified numerically).

Report for last 10 days.

I've completed my article. But at the last moment I've found out that my infinite series representation for Bessel functions can be analytically continued to the entire complex plane! Now I have to prove it rigorously and rewrite some pieces of article.

Among other I need to prove that following representation for Bessel function $J_1(z)$ where $z\in\mathbb{C}$ is true: 
\begin{equation}
\label{eq:J1}
J_1(z)=\frac{2}{z}\sum_{k=0}^{\infty}\varepsilon_k\big[(-1)^k-\cos\sqrt{z^2+(k\pi)^2}\big]
\end{equation}
where\begin{equation}
\label{eq:eps}
\varepsilon_k=
\begin{cases}
\frac12, & \mbox{if } k=0 \\
1, & \mbox{if } k>0.
\end{cases}
\end{equation}

It is getting irritating

I still can't make my blog to show math formulas typed using latex commands. I decided to use MathJax. I followed different instructions but with no success. Two of them are the most worth of my attention: this one and this one. Besides that it is worth to consult with the quides of MathJax. I will repeat my attempts later.

+++++++++

Solution: Dynamic Views in Templates was changed into Simple and math formulas became shown correctly.

Further improvement is needed: I'd like my math formulas to be numerated within my blog posts. Can it be fixed?

My other open problems

1) Prove that function of complex argument $z=x+\mathrm{i}y$
\begin{equation}
f_1(z)=\frac{\sin{z}}{z}
\end{equation}
obey a condition:
\begin{equation}
|f_1(z)|\leq c_1\mathrm{e}^{\beta_1|y|}
\end{equation}
where $c_1$ and $\beta_1$ are constant. Prove that $\beta_1=1$.

2) Prove that function of complex argument $z=x+\mathrm{i}y$
\begin{equation}
f_2(z)=J_0(z)
\end{equation}
obey a condition:
\begin{equation}
|f_2(z)|\leq c_2\mathrm{e}^{\beta_2|y|}
\end{equation}
where $c_2$ and $\beta_2$ are constant. Prove that $\beta_2=1$.

3)  Prove that function of complex argument $z=x+\mathrm{i}y$
\begin{equation}
f_3(z)=f(a,b,\mu,\nu,m,z)
\end{equation}
obey a condition:
\begin{equation}
|f_3(z)|\leq c_3\mathrm{e}^{\beta_3|y|}
\end{equation}
where $c_3$ and $\beta_3$ are constant. Besides $\beta_3$ have to be found.

(Main function $f_3(z)$ which have to studied will be present after problem 3 is solved)

My open problem

Here is following series expansion for Bessel function of zero order and of the first kind (piece of my result to be published):

\[J_0(x)=4\sum\limits_{n=1}^{\infty}(-1)^n\big[(2n-1)\pi\big]\frac{\psi_n\cos\psi_n-\sin\psi_n}{{\psi_n}^3},\]
where
\[\psi_n=\sqrt{\frac{4x^2}{3}+\big[(2n-1)\pi\big]^2}.\]

Is it uniformly convergent?

BTW
I can prove uniform convergence for most of my series. Prove appeared to be very simple.

Report for 09.05.2012

1) Have finished writing my article. However, uniform convergence with respect to 'x' still remains unproven.
2) Tried do derive Dominici result. I found that Paley–Wiener theorem can help me with it.

Report for 09.03.12

Small report for today

Plans were
1) To study Abel's test for uniform convergence. Try to apply it to my series.



Studied. But Abel's test haven't allowed me to prove uniform convergence. There are a lot of another tests, but I'll try to prove it in another way. I guess that this way will be cumbersome and it will overblow my article. That isn't good.  In any case I have to try.

2) Insert in article my recent results: uniform convergence with respect to parameter 'b' and absolute convergence of my series.
Done




3) Try to derive results of article [Dominici, 2011] with the help of my series.
I can't derive results of this article with the help of my series but there is a good news! It is very possible that I'll receive this result in much simpler way.

Plans for tomorrow:
I have no strength to write plans :)

Well... My name is Andriy Andrusyk. I am a physicist from Ukraine. My blog will outline my scientific activity. I'll try to take notes on daily basis and hope that it will help me to control my performance.
So, I hope in some time my blog will serve as a my scientific portfolio.

Lets start.

I'm about to finish my article where I'll present my series expansions for Bessel function of integer order which were unknown until now. Yesterday I've proved absolute convergence of my series and proved their uniform convergence with respect to parameter 'b'. Besides that, I previously proved that my series are infinitely differentiable with respect to variable 'x'. Unfortunately, I still can't prove uniform convergence of my series with respect to variable 'x'. However, I hope it isn't necessary for publication and hope that my results are enough for being suitable for further studies and for applications.

My plans for several nearest days:

1) To study Abel's test for uniform convergence. Try to apply it to my series.
2) Insert in article my recent results: uniform convergence with respect to parameter 'b' and absolute convergence of my series.
3) Try to derive results of article [Dominici, 2011] with the help of my series. It will prove usefulness of my result. However, if my series fails this test it won't prove their useless. If I'll repeat result of that article - very good, if not - it does't matter.