# Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24

## Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 24

Published $1968$.

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## $24 \quad$ Bessel Functions

### Bessel's Differential Equation

$24.1$: Bessel's Differential Equation

### Bessel Functions of the First Kind of Order $n$

 $\displaystyle \map {J_n} x$ $=$ $\displaystyle \dfrac {x^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 n + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}$ $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {n + k + 1} } \paren {\dfrac x 2}^{n + 2 k}$
 $\displaystyle \map {J_{-n} } x$ $=$ $\displaystyle \dfrac {x^{-n} } {2^{-n} \, \map \Gamma {1 - n} } \paren {1 - \dfrac {x^2} {2 \paren {2 - 2 n} } + \dfrac {x^4} {2 \times 4 \paren {2 - 2 n} \paren {4 - 2 n} } - \cdots}$ $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {k + 1 - n} } \paren {\dfrac x 2}^{2 k - n}$
$\map {J_{-n} } x = \paren {-1}^n \map {J_n} x$

### Miscellaneous Results

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