Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Bessel Functions of the First Kind of Order n

Bessel Functions of the First Kind of Order $n$

$24.2$: Series Expansion of Bessel Function of the First Kind

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

\(=\)

\(\ds \dfrac {x^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 n + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}\)

\(\ds \)

\(=\)

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

$24.3$: Series Expansion of Bessel Function of the First Kind: Negative Index

\(\ds \map {J_{-n} } x\)

\(=\)

\(\ds \dfrac {x^{-n} } {2^{-n} \, \map \Gamma {1 - n} } \paren {1 - \dfrac {x^2} {2 \paren {2 - 2 n} } + \dfrac {x^4} {2 \times 4 \paren {2 - 2 n} \paren {4 - 2 n} } - \cdots}\)

\(\ds \)

\(=\)

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

$24.4$: Bessel Function of the First Kind of Negative Integer Order

$\map {J_{-n} } x = \paren {-1}^n \map {J_n} x$

for $n \in \Z_{\ge 0}$.