Bessel Function of the First Kind of Negative Integer Order

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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

$\blacksquare$


Sources