Solve for Roots of Bessel Function of First Kind

I have this Bessel function that I am trying to solve for the roots qn. They are dependent upon alpha, which I was able to figure out. The literature sources I have are from the 80s and give me a simple table with 6 roots and alphas ranging from zero to infinity, but I need to be able to use this in my infinite summation model, so being able to solve for more would be useful.

Any help would be greatly appreaciated. I have downloaded the besselzero function so far, but don't know how to modify that to what I am trying to solve, with my modifications being external to the bessel function.

Thank you

Answer 1

[Walter Roberson]

syms q_n
alpha = 123.456;
eqn = alpha * q_n * besselj(0, q_n) + 2 * besselj(1, q_n) == 0

eqn = 

guess = sqrt(5)

guess = 2.2361

vpasolve(eqn, q_n, guess)

ans = 

2.4115338681268972484128947947434

fplot(lhs(eqn), [-5 5])

[Maressa Schulze]

Is there a way to write this to get multiple q's? I need to have q_1 to q_1000 for my work.

Thank you!

[Walter Roberson]

N = 100;
syms q_n
alpha = 123.456;
eqn = alpha * q_n * besselj(0, q_n) + 2 * besselj(1, q_n) == 0

eqn = 

sols = zeros(N, 1, 'sym');

for K = 1 : N
  guess = K * pi;
  sols(K) = vpasolve(eqn, q_n, guess);
end

vpa(sols,10)

ans =