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.
I've proved uniform convergence with respect to $x$ of more general series expansion for Bessel function $J_0(bx)$:
ReplyDelete\begin{equation}
J_0(bx)=\frac{2}{\sqrt{1-b^2}}\sum\limits_{k=1}^{\infty}(k\pi){\sin(k\pi\sqrt{1-b^2})}\frac{\sin{\varphi_k}-{\varphi_k}\cos{\varphi_k}}{{\varphi_k}^3},\quad b\in(0,1)
\end{equation}
where
\begin{equation}
\varphi_k=\sqrt{x^2+(k\pi)^2}
\end{equation}
To be more specific, series $\sum\limits_{k=1}^{\infty}(k\pi){\sin(k\pi\sqrt{1-b^2})}\frac{\sin{\varphi_k}-{\varphi_k}\cos{\varphi_k}}{{\varphi_k}^3}$ is proved to be uniformly convergent with respect to $x$.
Series expansion for $J_0(x)$ presented above can be derived from the $J_0(bx)$ presented here by setting $b=\sqrt{3}/2$ and scaling variable $x$ by $1/b$.
It is worth to note that series expansion for $J_0(bx)$ is conditionally convergent
PS
This series expansion for $J_0(bx)$ is published for the first time.