Bessel Function of the First Kind/Instances

Specific Instances of Bessel Function of the First Kind

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.

Order $0$

\(\ds \map {J_0} x\)

\(=\)

\(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac x 2}^{2 k}\)

\(\ds \)

\(=\)

\(\ds 1 - \dfrac {x^2} {2^2} + \dfrac {x^4} {2^2 \times 4^2} - \dfrac {x^6} {2^2 \times 4^2 \times 6^2} + \dotsb\)