Which representation and method I can use to find approximately first 4 roots of $J_0$ with pen and paper? How can I do it?
The same question about maximal and minimal points and values?
Background:
Here you see the graph of Bessel functions $J_n(x)$
I want to justify this sketch for $J_0(x)$, i.e. I want to
- Find roots, max and min points, maximal and minimal values approximately.
- The same for its derivative.
- Prove, that the limit is 0.
- Prove, that the asymptotic behavior $\simeq \frac{1}{\sqrt{z}}$
Second item of the list is needed, because I know that $J_0'(x) = -J_1(x)$, so I will be able to justify the graph for $J_1$ at the same time. $3,4$ are basic results, so no problem here.
For the zero's, a long time ago, one of my PhD students used $$x_n=p+\frac{1}{8p}-\frac{31}{384 p^3}+\cdots \qquad \text{where} \qquad p= \left(n-\frac{1}{4}\right)\pi$$ which works pretty well as shown below $$\left( \begin{array}{ccc} n & \text{estimate} & \text{exact} \\ 1 & 2.403074548 & 2.404825558 \\ 2 & 5.520037754 & 5.520078112 \\ 3 & 8.653723235 & 8.653727913 \\ 4 & 11.79153341 & 11.79153444 \\ 5 & 14.93091739 & 14.93091771 \\ 6 & 18.07106384 & 18.07106397 \\ 7 & 21.21163657 & 21.21163663 \\ 8 & 24.35247150 & 24.35247153 \\ 9 & 27.49347912 & 27.49347913 \\ 10 & 30.63460646 & 30.63460647 \end{array} \right)$$
You can approximate the location of the extrema at the mid point, that is to say at $\frac {x_n+x_{n+1}} 2$.