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

Graph of $J_0$.
Complexbesselj0.png

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

Domain coloring of $J_5$ in $\mathbb{C}$.
Bessel functions from Abramowitz&Stegun

Properties

Theorem: If $n \in \mathbb{Z}$, then $J_{-n}(x)=(-1)^nJ_n(x)$. Moreover this means that $J_n$ and $J_{-n}$ are linearly dependent.

Proof: █

REDIRECT Bessel J and Y solve Bessel's differential equation

REDIRECT Bessel J and Y solve Bessel's differential equation (constant multiple in argument)

REDIRECT Bessel J and Y solve Bessel's differential equation (monomial multiple outside,weighted monomial in argument)

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: █

Theorem: (Generating function) The following formula holds: $$\exp \left( \dfrac{1}{2} z \left( t-\dfrac{1}{t} \right) \right) = \displaystyle\sum_{k=-\infty}^{\infty} t^k J_k(z).$$

Proof: █

Theorem: The following formula holds for $n\in\mathbb{Z}$: $$J_n(z)=\dfrac{1}{\pi} \displaystyle\int_0^{\pi} \cos(n\xi-x\sin(\xi))d\xi.$$

Proof: █

Theorem: The following formula holds for $n>-\frac{1}{2}$: $$J_n(z)=\dfrac{\left(\frac{z}{2}\right)^n}{\sqrt{\pi}\Gamma(n+\frac{1}{2})} \displaystyle\int_{-1}^1 (1-t^2)^{n-\frac{1}{2}}e^{izt}dt.$$

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.

Proof

References

Theorem

The following formula holds: $$J_{\nu}(z) = \left( \dfrac{z}{2} \right)^{\nu} \dfrac{1}{\Gamma(\nu+1)} {}_0F_1 \left(-;\nu+1;-\dfrac{z^2}{4} \right),$$ where $J_{\nu}$ denotes the Bessel function of the first kind, $\Gamma$ denotes the gamma function and ${}_0F_1$ denotes the hypergeometric 0F1.

Proof

References

REDIRECT Relationship between Bessel I and Bessel J

Theorem

The following formula holds for integer $n$: $$\mathbf{J}_n(z)=J_n(z),$$ where $\mathbf{J}_n$ denotes an Anger function and $J_n$ denotes a Bessel function of the first kind.