Bessel Function of the First Kind/Instances
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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 $0$
\(\ds \map {J_0} x\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac x 2}^{2 k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - \dfrac {x^2} {2^2} + \dfrac {x^4} {2^2 \times 4^2} - \dfrac {x^6} {2^2 \times 4^2 \times 6^2} + \dotsb\) |