Bessel J

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

• Besseljintegerorder.png

  $J_n$ where $n=0,1,\ldots,5$

• Complexbesselj0.png

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

• Complexbesselj5.png

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

Bessel functions of the second kind are defined via the formula $$Y_{\nu}(z)=\dfrac{J_{\nu}(z)\cos(\nu \pi)-J_{-\nu}(z)}{\sin(\nu \pi)}.$$

Properties

Theorem: Bessel functions $J_{\nu}$ and $Y_{\nu}$ are independent solutions of the second-order differential equation $$z^2 y + zy' + (z^2-\nu^2)y=0.$$

Proof: █

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: █

Videos

Bessel Equation and Bessel functions
Mod-1 Lec-6 Bessel Functions and Their Properties-I
Bessel's Equation by Free Academy
Taylor Series, Bessel, single Variable Calculus, Coursera.org
Ordinary Differential Equations Lecture 7—Bessel functions and the unit step function
Laplace transform of Bessel function order zero
Laplace transform: Integral over Bessel function is one
Orthogonal Properties of Bessel Function, Orthogonal Properties of Bessel Equation

Links

Addition formulas for Bessel functions
Relations between Bessel functions by John D. Cook