Difference between revisions of "Bessel J"

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The Bessel functions of the first kind, $J_{\nu}$, have a power series expansion
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__NOTOC__
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$$J_{\nu}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{k! \Gamma(k+\nu+1)2^{2k+\nu}}z^{2k+\nu}.$$
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The Bessel functions of the first kind, $J_{\nu} \colon \mathbb{C} \rightarrow \mathbb{C}$ are defined by
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$$J_{\nu}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{k! \Gamma(k+\nu+1)2^{2k+\nu}}z^{2k+\nu},$$
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where $\Gamma$ denotes the [[gamma]] function.
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
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File:Besseljintegerorder.png|$J_n$ where $n=0,1,\ldots,5$ plotted on $[-10,10]$.
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File:Besseljn=0plot.png|Graph of $J_0$.
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File:Complexbesselj0.png|[[Domain coloring]] of $J_0$ in $\mathbb{C}$.
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File:Multiplebesseljplot.png|Graph of various Bessel functions.
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File:Complexbesselj5.png|[[Domain coloring]] of $J_5$ in $\mathbb{C}$.
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File:Complexbesselj0plot.png|[[Domain coloring]] of $J_0$.
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File:Complexbesselj0.5plot.png|[[Domain coloring]] of $J_{\frac{1}{2}}$.
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File:Complexbesselj5plot.png|[[Domain coloring]] of $J_5$.
 
File:Page 359Abramowitz-Stegun(Bessel functions).jpg|Bessel functions from [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/ Abramowitz&Stegun]
 
File:Page 359Abramowitz-Stegun(Bessel functions).jpg|Bessel functions from [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/ Abramowitz&Stegun]
 
</gallery>
 
</gallery>
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=Properties=
 
=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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[[Bessel polynomial in terms of Bessel functions]]<br />
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<strong>Theorem:</strong> 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]].
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[[Bessel at n+1/2 in terms of Bessel polynomial]]<br />
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<div class="mw-collapsible-content">
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[[Bessel at -n-1/2 in terms of Bessel polynomial]]<br />
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<strong>Proof:</strong> █
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[[Relationship between Bessel J and hypergeometric 0F1]]<br />
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</div>
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[[Relationship between Bessel I and Bessel J]]<br />
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</div>
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[[Relationship between Anger function and Bessel J]]<br />
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[[Derivative of Bessel J with respect to its order]]<br />
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{{:Bessel J sub nu and Y sub nu solve Bessel's differential equation}}
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[[Integral of monomial times Bessel J]]<br />
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[[Integral of Bessel J for Re(nu) greater than -1]]<br />
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{{:Bessel J sub nu and Y sub nu solve Bessel's differential equation (constant multiple in argument)}}
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[[Integral of Bessel J for nu=2n]]<br />
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[[Integral of Bessel J for nu=2n+1]]<br />
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Integral of Bessel J for nu=n+1]]<br />
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<strong>Theorem:</strong> The following formula holds:
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[[Integral of Bessel J for nu=1]]<br />
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$$zJ_{\nu}'(z)=\nu J_{\nu}(z) - z J_{\nu+1}(z).$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Theorem:</strong> The following formula holds:
 
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$$\dfrac{d}{dz}[z^{-\nu}J_{\nu}(z)] = -z^{-\nu}J_{\nu+1}(z).$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed">
 
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<strong>Theorem:</strong> (Generating function) The following formula holds:
 
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$$\exp \left( \dfrac{1}{2} z \left( t-\dfrac{1}{t} \right) \right) = \displaystyle\sum_{k=-\infty}^{\infty} t^k J_k(z).$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed">
 
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<strong>Theorem:</strong> The following formula holds for $n\in\mathbb{Z}$:
 
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$$J_n(z)=\dfrac{1}{\pi} \displaystyle\int_0^{\pi} \cos(n\xi-x\sin(\xi))d\xi.$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed">
 
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<strong>Theorem:</strong> The following formula holds for $n>-\frac{1}{2}$:
 
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$$J_n(z)=\dfrac{\left(\frac{z}{2}\right)^n}{\sqrt{\pi}\Gamma(n+\frac{1}{2})} \displaystyle\int_{-1}^1 (1-t^2)^{n-\frac{1}{2}}e^{izt}dt.$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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=Relations to other special functions=
 
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{{:Bessel polynomial in terms of Bessel functions}}
 
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{{:Bessel at n+1/2 in terms of Bessel polynomial}}
 
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{{:Bessel at -n-1/2 in terms of Bessel polynomial}}
 
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{{:Relationship between Bessel J sub nu and hypergeometric 0F1}}
 
  
 
=Videos=
 
=Videos=
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[https://www.youtube.com/watch?v=p-_MnYBLkcA Orthogonal Properties of Bessel Function, Orthogonal Properties of Bessel Equation]
 
[https://www.youtube.com/watch?v=p-_MnYBLkcA Orthogonal Properties of Bessel Function, Orthogonal Properties of Bessel Equation]
  
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=Links=
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=External Links=
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[http://www.johndcook.com/blog/2013/08/03/addition-formulas-for-bessel-functions/ Addition formulas for Bessel functions]<br />
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[http://www.johndcook.com/blog/2013/08/03/addition-formulas-for-bessel-functions/ Addition formulas for Bessel functions by John D. Cook]<br />
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[http://www.johndcook.com/Bessel_functions.html Relations between Bessel functions by John D. Cook]
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[http://www.johndcook.com/Bessel_functions.html Relations between Bessel functions by John D. Cook]<br />
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[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0021%7CLOG_0023 Bessel's functions of the second order - C.V. Coates]<br />
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=References=
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* {{BookReference|A Treatise on Bessel Functions|1895|Andrew Gray|author2=G.B. Mathews|prev=findme|next=findme}}: Ch. 2 $10$
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* {{BookReference|Higher Transcendental Functions Volume II|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=findme|next=findme}}: $\S 7.2.1 (2)$
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=findme}}: $9.1.10$
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{{:Bessel functions footer}}
  
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<center>{{:Bessel functions footer}}</center>
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[[Category:SpecialFunction]]

Latest revision as of 05:41, 4 March 2018

The Bessel functions of the first kind, $J_{\nu} \colon \mathbb{C} \rightarrow \mathbb{C}$ are defined by $$J_{\nu}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{k! \Gamma(k+\nu+1)2^{2k+\nu}}z^{2k+\nu},$$ where $\Gamma$ denotes the gamma function.

Properties

Bessel polynomial in terms of Bessel functions
Bessel at n+1/2 in terms of Bessel polynomial
Bessel at -n-1/2 in terms of Bessel polynomial
Relationship between Bessel J and hypergeometric 0F1
Relationship between Bessel I and Bessel J
Relationship between Anger function and Bessel J
Derivative of Bessel J with respect to its order
Integral of monomial times Bessel J
Integral of Bessel J for Re(nu) greater than -1
Integral of Bessel J for nu=2n
Integral of Bessel J for nu=2n+1
Integral of Bessel J for nu=n+1
Integral of Bessel J for nu=1

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

External Links

Addition formulas for Bessel functions by John D. Cook
Relations between Bessel functions by John D. Cook
Bessel's functions of the second order - C.V. Coates

References


Bessel functions

Anger function

Bessel-Clifford

Bessel $J_{\nu}$

Bessel $Y_{\nu}$

Bessel $I_{\nu}$

Modified Bessel $K_{\nu}$

Spherical Bessel $j_{\nu}$

Spherical Bessel $y_{\nu}$
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