Is there a closed form expression for the first zero of the first Bessel function? $j_{1,1}$ denotes the first zero of the first Bessel function of the first kind.  (That's a lot of firsts!)  It's approximately equal to $3.83$.  My question is, is there any closed form expression for its value?  Even a infinite series or infinite product that yields it would be good.
I ask because this value is used in physics, in the context of diffraction of light through a circular aperture, and students often make the mistake of thinking that the number just pops out of nowhere.
 A: If you want an approximation of the first root without invoking at any time the Bessel functions, what you could consider is an $[2m+1,2n]$ Padé approximation of $J_1(x)$ built at $x=0$. It will be looking like
$$J_1(x)\approx\frac{x \left(\frac12+\sum_{j=1}^m a_j \,x^{2j}\right)}{1+\sum_{k=1}^n b_k\, x^{2k} }$$ The coefficients are "easy" to compute since, using the infinite series given in Geremia's answer, we know exactly all required derivatives of $J_1(x)$ at $x=0$. These approximations are equivalent to Taylor expansions to $O\left(x^{2(m+n+1})\right)$.
In order to keep it simple (solution of a quadratic equation in $x^2$), select $m=2$. The following table gives the equation to be solved,  its first root and its decimal representation.
$$\left(
\begin{array}{cccc}
 n & \text{equation} & \text{root} & \text{value} \\
 1 & x^4-40 x^2+384=0 & 4 & 4.00000 \\
 2 & 11 x^4-552 x^2+5760=0 &  \sqrt{\frac{12}{11} \left(23-\sqrt{89}\right)} &
   3.84698 \\
 3 & 23 x^4-1296 x^2+14080=0 &  \sqrt{\frac{8}{23} \left(81-\sqrt{1501}\right)} &
   3.83382 \\
 4 & 97 x^4-5810 x^2+64400=0 & \sqrt{\frac{5}{97} \left(581-\sqrt{87689}\right)} &
   3.83202 
\end{array}
\right)$$ while the "exact" value is $3.83171$.
Just for the fun of it, plot on the same graph, for $0 \leq x \leq 4$, $J_1(x)$ and $\frac{x(11 x^4-552 x^2+5760) }{ 4 x^4+336 x^2+11520}$
A: The series expansion is:
$$J_{1}\left(x\right)=\sum_{n=0}^\infty\frac{x^{2n+1}}{2^{2n+1} n! (n+1)!}$$
$$=\frac{x}{2} - \frac{x^{3}}{16} + \frac{x^{5}}{384} - \frac{x^{7}}{18432} + …$$
The denominators are Sloan Sequence A002474.
Sloan Sequence A115369 is the first zero.
