Eliminating Value from Bessel Function

Question

How I can get the value for "k" as a function of "r" from equality $J1(kr)=Dr$, where D is constant. $J1(kr)$ is the Bessel function with n=1.

Answer 1

You want to solve the equation $$J_1(k r)=D r$$ Let $x=k r$ to make it $$\frac{J_1(x)}{x}=\frac{D}{k}$$ Assuming that you want the relation between $k$ and $x$ for $x < 3.83171$ (first zero of $J_1(x)$), you can approximate the lhs byt its $[4,4]$ Padé approximant built around $x=0$; it is $$\frac{J_1(x)}{x}\sim \frac {\frac{1}{2}-\frac{23 }{480}x^2+\frac{11 }{11520}x^4 } {1+\frac{7 }{240}x^2+\frac{1}{2880}x^4 }$$ Then, you have an implicit relation between $k$ and $r$