Bessel J

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

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:

  1. REDIRECT Bessel J and Y solve Bessel's differential equation
  1. REDIRECT Bessel J and Y solve Bessel's differential equation (constant multiple in argument)
  1. 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

  1. 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.

Proof

References

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

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