Integral Representation of Bessel Function of the First Kind/Integer Order

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Theorem

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.

Let $n \in \Z$ be an integer.

Then:

$\ds \map {J_n} x = \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta} \rd \theta$


Proof

\(\ds \map \exp {\dfrac x 2 \paren {t - \dfrac 1 t} }\) \(=\) \(\ds \sum_{m \mathop = -\infty}^\infty \map {J_m} x t^m\) Generating Function for Bessel Function of the First Kind of Order n of x
\(\ds \dfrac 1 {2 \pi i} \int_C t^{-n - 1} \map \exp {\dfrac x 2 \paren {t - \dfrac 1 t} } \rd t\) \(=\) \(\ds \dfrac 1 {2 \pi i} \int_C t^{-n - 1} \sum_{n \mathop = - \infty}^\infty \map {J_m} x t^m \rd t\) where $C$ denotes a contour encircling the origin once
\(\ds \) \(=\) \(\ds \sum_{n \mathop = - \infty}^\infty \dfrac {\map {J_m} x} {2 \pi i} \int_C t^{m - n - 1} \rd t\)
\(\ds \) \(=\) \(\ds \dfrac {\map {J_n} x} {2 \pi i} 2 \pi i\) Cauchy's Residue Theorem
\(\ds \) \(=\) \(\ds \map {J_n} x\)


Take $C$ to be the unit circle.

Write $t = e^{-i \theta}$ and let $\theta$ be decreasing from $\pi$ to $-\pi$, so that we integrate along $C$ anticlockwise.

Then:

\(\ds \leadsto \ \ \) \(\ds \map {J_n} x\) \(=\) \(\ds \dfrac 1 {2 \pi i} \int_\pi^{-\pi} e^{-i \theta \paren {-n - 1} } \map \exp {\dfrac x 2 \paren {e^{-i \theta} - e^{i \theta} } } \rd e^{-i \theta}\)
\(\ds \) \(=\) \(\ds \dfrac 1 {2 \pi} \int_{-\pi}^\pi \map \exp {- x i \sin \theta + i n \theta} \rd \theta\) Sine in terms of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \dfrac 1 {2 \pi} \paren {\int_0^\pi \map \exp {i n \theta - i x \sin \theta} \rd \theta + \int_{-\pi}^0 \map \exp {i n \theta - i x \sin \theta} \rd \theta}\)
\(\ds \) \(=\) \(\ds \dfrac 1 {2 \pi} \paren {\int_0^\pi \map \exp {i n \theta - i x \sin \theta} \rd \theta - \int_\pi^0 \map \exp {i n \paren {-\theta} - i x \map \sin {-\theta} } \rd \theta}\) substitution of $-\theta$
\(\ds \) \(=\) \(\ds \dfrac 1 {2 \pi} \paren {\int_0^\pi \map \exp {i n \theta - i x \sin \theta} \rd \theta + \int_0^\pi \map \exp {-\paren {i n \theta - i x \sin \theta} } \rd \theta}\)
\(\ds \) \(=\) \(\ds \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta} \rd \theta\) Cosine in terms of Hyperbolic Cosine

$\blacksquare$


Sources

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