# Bessel Function of the First Kind of Negative Integer Order

## Theorem

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$, where $n$ is a positive integer.

Then:

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

## Proof

 $\displaystyle \map {J_{-n} } x$ $=$ $\displaystyle \dfrac 1 \pi \int_0^\pi \map \cos {-n \theta - x \sin \theta} \rd \theta$ Integral Representation of Bessel Function of the First Kind/Integer Order $\displaystyle$ $=$ $\displaystyle \dfrac 1 \pi \int_0^\pi \map \cos {-n \paren {\pi - \theta} - x \sin \paren {\pi - \theta} } \rd \paren {\pi - \theta}$ substitution of $\pi - \theta$ $\displaystyle$ $=$ $\displaystyle -\dfrac 1 \pi \int_\pi^0 \map \cos {n \theta - x \sin \theta - n \pi} \rd \theta$ Sine of Supplementary Angle $\displaystyle$ $=$ $\displaystyle \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta - n \pi} \rd \theta$ Reversal of Limits of Definite Integral $\displaystyle$ $=$ $\displaystyle \paren {-1}^{-n} \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta} \rd \theta$ Cosine of Angle plus Integer Multiple of Pi $\displaystyle$ $=$ $\displaystyle \paren {-1}^n \map {J_n} x$ Integral Representation of Bessel Function of the First Kind/Integer Order

$\blacksquare$