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
| \(\ds \map {J_{-n} } x\) | \(=\) | \(\ds \dfrac 1 \pi \int_0^\pi \map \cos {-n \theta - x \sin \theta} \rd \theta\) | Integral Representation of Bessel Function of the First Kind/Integer Order | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 \pi \int_0^\pi \map \cos {-n \paren {\pi - \theta} - x \sin \paren {\pi - \theta} } \rd \paren {\pi - \theta}\) | substitution of $\pi - \theta$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\dfrac 1 \pi \int_\pi^0 \map \cos {n \theta - x \sin \theta - n \pi} \rd \theta\) | Sine of Supplementary Angle | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta - n \pi} \rd \theta\) | Reversal of Limits of Definite Integral | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1}^{-n} \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta} \rd \theta\) | Cosine of Angle plus Integer Multiple of Pi | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1}^n \map {J_n} x\) | Integral Representation of Bessel Function of the First Kind/Integer Order |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {II}$. Bessel functions: $1$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 24$: Bessel Functions: Bessel Function of the First Kind of Order $n$: $24.4$