Difference between revisions of "Bessel J"
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| − | a | + | Bessel functions (of the first kind) of order $\nu$, $J_{\nu}$, have a power series expansion |
| + | $$J_{\nu}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{k! \Gamma(k+\nu+1)2^{2k+\nu}}x^{2k+\nu}.$$ | ||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Besseljintegerorder.png|$J_n$ where $n=0,1,\ldots,5$ | ||
| + | File:Complexbesselj0.png|[[Domain coloring]] of $J_0$ in $\mathbb{C}$. | ||
| + | File:Complexbesselj5.png|Domain coloring of $J_5$ in $\mathbb{C}$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | 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= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> Bessel functions $J_{\nu}$ and $Y_{\nu}$ are independent solutions of the second-order differential equation | ||
| + | $$z^2 y'' + zy' + (z^2-\nu^2)y=0.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$zJ_{\nu}'(z)=\nu J_{\nu}(z) - z J_{\nu+1}(z).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\dfrac{d}{dz}[z^{-\nu}J_{\nu}(z)] = -z^{-\nu}J_{\nu+1}(z).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | =Videos= | ||
| + | [https://www.youtube.com/watch?v=__fdGscBZjI Bessel Equation and Bessel functions] <br /> | ||
| + | [https://www.youtube.com/watch?v=6n5QyYMe9U0&noredirect=1 Mod-1 Lec-6 Bessel Functions and Their Properties-I]<br /> | ||
| + | [https://www.youtube.com/watch?v=Wu7cp7qGjBo Bessel's Equation by Free Academy]<br /> | ||
| + | [https://www.youtube.com/watch?v=q_jh7cBIxZ8&noredirect=1 Taylor Series, Bessel, single Variable Calculus, Coursera.org]<br /> | ||
| + | [https://www.youtube.com/watch?v=l2jqD_g5lgQ&noredirect=1 Ordinary Differential Equations Lecture 7—Bessel functions and the unit step function]<br /> | ||
| + | [https://www.youtube.com/watch?v=II_JUVmSDa8&noredirect=1 Laplace transform of Bessel function order zero]<br /> | ||
| + | [https://www.youtube.com/watch?v=1tgpAyGXhus Laplace transform: Integral over Bessel function is one]<br /> | ||
| + | [https://www.youtube.com/watch?v=p-_MnYBLkcA Orthogonal Properties of Bessel Function, Orthogonal Properties of Bessel Equation] | ||
| + | |||
| + | =Links= | ||
| + | [http://www.johndcook.com/blog/2013/08/03/addition-formulas-for-bessel-functions/ Addition formulas for Bessel functions]<br /> | ||
| + | [http://www.johndcook.com/Bessel_functions.html Relations between Bessel functions by John D. Cook] | ||
Revision as of 01:31, 17 March 2015
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
