Is there any integer $n$ such that the bessel function J_n(1)=0?

Question

Is there any integer $n$ such that $t=1$ is a root of the Bessel function of the first kind $J_n (t)$, i.e. $$J_n(1)=\int_{0}^{\pi}\cos (nx-\sin x)=0\,?$$

Answer 1

According to Uniform Upper and Lower Bounds on the Zeros of Bessel Functions of the First Kind, p 2:

$$ j_{v,k} > v + \frac23 |a_{k-1}|^{3/2}$$

where $j_{v,k}$ is the $k$-th positive zero of $J_v$.

So the answer is no, since the first zero is greater than one.

Answer 2

The Bessel functions have the series expansion

$$ J_n(z) = \sum_{k=0}^\infty \dfrac{(-1)^k (z/2)^{n+2k}}{k! (k+n)!} $$

If $0 < z \le 2 \sqrt{1+n}$ this is a convergent alternating series with positive initial term and terms nonincreasing in absolute value (strictly decreasing after the first), implying that the sum is greater than $0$.

Answer 3

The first root of $J_n$ behaves like $n$ if $n$ is large, and lower order correction terms are known, see this website, for instance: http://dlmf.nist.gov/10.21#ii