Bessel functions (of the first kind) of order $\nu$, $J_{\nu}$, have a power series expansion $$J_{\nu}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{k! \Gamma(k+\nu+1)2^{2k+\nu}}z^{2k+\nu}.$$

Besseljintegerorder.png

$J_n$ where $n=0,1,\ldots,5$ plotted on $[-10,10]$.
Complexbesselj0.png

Domain coloring of $J_0$ in $\mathbb{C}$.
Complexbesselj5.png

Domain coloring of $J_5$ in $\mathbb{C}$.

Properties

REDIRECT Bessel J and Y solve Bessel's differential equation

Theorem: The following formula holds: $$zJ_{\nu}'(z)=\nu J_{\nu}(z) - z J_{\nu+1}(z).$$

Proof: █

Theorem: The following formula holds: $$\dfrac{d}{dz}[z^{-\nu}J_{\nu}(z)] = -z^{-\nu}J_{\nu+1}(z).$$

Proof: █

Relations to other special functions

Theorem

The following formula holds: $$y_n\left( \dfrac{1}{ir} \right) = \left(\dfrac{\pi r}{2} \right)^{\frac{1}{2}} e^{ir} \left[ \dfrac{J_{n +\frac{1}{2}}(r)}{i^{n+1}}+i^nJ_{-n-\frac{1}{2}}(r) \right],$$ where $y_n$ denotes a Bessel polynomial and $J_{\nu}$ denotes the Bessel J.

Proof

References

Theorem

The following formula holds: $$J_{n +\frac{1}{2}}(r) = (2\pi r)^{-\frac{1}{2}} \left[\dfrac{e^{ir}}{i^{n+1}} y_n \left( -\dfrac{1}{ir} \right) + i^{n+1}e^{-ir}y_n\left( \dfrac{1}{ir} \right) \right],$$ where $J_{n+\frac{1}{2}}$ denotes a Bessel J, $\pi$ denotes pi, $i$ denotes the imaginary number, $e^{ir}$ denotes the exponential, and $y_n$ denotes a Bessel polynomial.

Proof

References

Theorem

The following formula holds: $$J_{-n-\frac{1}{2}}(r) = (2 \pi r)^{-\frac{1}{2}} \left[ i^n e^{ir} y_n \left( -\dfrac{1}{ir} \right)+ \dfrac{e^{-ir}}{i^n} y_n\left( \dfrac{1}{ir} \right) \right],$$ where $J_{-n-\frac{1}{2}}$ denotes a Bessel function of the first kind and $y_n$ denotes a Bessel polynomial.