Presentation of the momentum operator $\hat p$ in $x$-space derived from canonical commutation relation: $[\hat x, \hat p]=\mathrm{i}$.

Here I show how to derive the explicit presentation of the momentum operator $\hat p$ in $x$-space: $\hat p = -\mathrm{i}\frac{d}{dx}$ out of the commutation relation $[\hat x, \hat p]=\mathrm{i}$. I wonder, why all textbooks on quantum mechanics omit this issue. This fact is implicitly kept in mind but never is presented explicitly. The prove of this fact is simple.

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.