标签：xi frac 函数 Re rm pi Bessel nu

在这篇文章中，我们将会罗列Bessel函数的一些基本性质。

A. Definition and Basic Properties

We define the Bessel function $J_{\nu}$ of order $\nu$ by its Poisson representation formula $$J_{\nu}(t) = \frac{(t/2)^{\nu}}{\Gamma(\nu + 1/2)\Gamma(1/2)}\int_{-1}^1e^{its}(1 - s^2)^{\nu}\frac{{\rm d}s}{\sqrt{1 - s^2}},$$ where ${\rm Re}\ \nu > -1/2$ and $t \geq 0$. Note that in this case $J_{\nu}(t)$ is a real number.

Here are some basic properties of Bessel functions.

1. (Recurrence formula)$$\frac{{\rm d}}{{\rm d}t}(t^{-\nu}J_{\nu}(t)) = -t^{-\nu}J_{\nu + 1}(t), \qquad {\rm Re} \ \nu > -\frac{1}{2}; $$
2. (Companion recurrence formula) $$\frac{{\rm d}}{{\rm d}t}(t^{\nu}J_{\nu}(t)) = t^{\nu}J_{\nu - 1}(t), \qquad {\rm Re} \ \nu > \frac{1}{2}; $$
3. $J_{\nu}(t)$ satisfies the Bessel equation $$t^2f''(t) + tf'(t) + (t^2 - \nu^2)f(t) = 0;$$
4. If $\nu \in \mathbb{Z}_+$, then we have the following identity, which was taken by Bessel as the definition of $J_{\nu}$ for integer $\nu$: $$J_{\nu}(t) = \frac{1}{2\pi}\int_0^{2\pi}e^{it\sin\theta}e^{-i\nu\theta} \,{\rm d}\theta =  \frac{1}{2\pi}\int_0^{2\pi}\cos(t\sin\theta - \nu\theta) \,{\rm d}\theta;$$
5. For ${\rm Re}\ \nu > -1/2$ we have the following identity: $$J_{\nu}(t) = \frac{(t/2)^{\nu}}{\Gamma(1/2)}\sum\limits_{j = 0}^{\infty}(-1)^j\frac{\Gamma(j + 1/2)}{\Gamma(j + \nu + 1)}\frac{t^{2j}}{(2j)!} = \sum\limits_{j = 0}^{\infty}\frac{(-1)^j}{j!}\frac{(t/2)^{2j + \nu}}{\Gamma(j + \nu + 1)}; $$
6. For ${\rm Re}\ \nu > 1/2$ we have the following identity: $$\frac{{\rm d}}{{\rm d}t}(J_{\nu}(t)) = \frac{1}{2}(J_{\nu - 1}(t) - J_{\nu + 1}(t)).$$

Remark. The property 1, 2, 3 and 4 still hold when $t \in \mathbb{C}$.

Proposition.  Let ${\rm Re} \ \mu > -1/2, {\rm Re} \ \nu > -1$ and $t > 0$. Then the following identity is valid: \begin{equation}\label{1}\int_0^1J_{\mu}(ts)s^{\mu + 1}(1 - s^2)^{\nu} \,{\rm d}s = \frac{\Gamma(\nu + 1)2^{\nu}}{t^{\nu + 1}}J_{\mu + \nu + 1}(t).\end{equation}

B. The Fourier Transform of Surface Measure on $\mathbb{S}^{n - 1}$

Let ${\rm d}\sigma$ denote the surface on $\mathbb{S}^{n - 1}$ for $n \geq 2$. Then the following is true: $$\widehat{{\rm d}\sigma}(\xi) = \int_{\mathbb{S}^{n - 1}}e^{-2\pi i\xi \cdot \theta} \,{\rm d}\theta = \frac{2\pi}{|\xi|^{\frac{n - 2}{2}}}J_{\frac{n - 2}{2}}(2\pi|\xi|).$$ Using polar coordinates, we can also obtain the Fourier transform of a radial function on $\mathbb{R}^n$: $$\widehat{f}(\xi) = \frac{2\pi}{|\xi|^{\frac{n - 2}{2}}}\int_0^{\infty}J_{\frac{n - 2}{2}}(2\pi r|\xi|)r^{\frac{n}{2}} \,{\rm d}r,$$ where $f(x) = f_0(|x|)$.

Example. Consider the radial function $f(x) = \chi_{B_1(0)}(x)$ on $\mathbb{R}^n$. It follows that $$(\chi_{B_1(0)})^{\wedge}(\xi) = \frac{2\pi}{|\xi|^{\frac{n - 2}{2}}}\int_0^1J_{\frac{n - 2}{2}}(2\pi|\xi|r)r^{\frac{n}{2}} \,{\rm d}r = \frac{J_{\frac{n}{2}}(2\pi|\xi|)}{|\xi|^{\frac{n}{2}}},$$ where we use identity \eqref{1}. More generally, for ${\rm Re}\ \lambda > -1$, let \begin{equation*}m_{\lambda}(\xi) = \begin{cases}(1 - |\xi|^2)^{\lambda} \quad &|\xi| \leq 1, \\ 0 \quad &|\xi| > 1. \end{cases}\end{equation*} Then $$m_{\lambda}^{\vee}(x) = \frac{\Gamma(\lambda + 1)}{\pi^{\lambda}}\frac{J_{\frac{n}{2} + \lambda}(2\pi|x|)}{|x|^{\frac{n}{2} + \lambda}}.$$

C. Asymtotics of Bessel Funtions

Let ${\rm Re}\ \nu > -1/2$. We have the following results:

\begin{equation*}J_{\nu}(t) = \begin{cases}\frac{t^{\nu}}{2^{\nu}\Gamma(\nu + 1)} + O(t^{{\rm Re}\ \nu + 1}) \qquad &t \rightarrow 0^+, \\ \sqrt{\frac{2}{t\pi}}\cos\left(t - \frac{\nu\pi}{2} - \frac{\pi}{4}\right) + O(t^{-\frac{3}{2}}) \qquad &t \rightarrow \infty. \\ \end{cases}\end{equation*} In particular, for fixed $\nu$, we have $J_{\nu}(t) = O(t^{-\frac{1}{2}})$ as $t \rightarrow \infty$.

Ref.  Grafakos, L. Classical Fourier Analysis.

苗长兴, 现代调和分析及其应用讲义.

标签：xi,frac,函数,Re,rm,pi,Bessel,nu
来源： https://www.cnblogs.com/tron-math/p/16663112.html

本站声明： 1. iCode9 技术分享网（下文简称本站）提供的所有内容，仅供技术学习、探讨和分享；
2. 关于本站的所有留言、评论、转载及引用，纯属内容发起人的个人观点，与本站观点和立场无关；
3. 关于本站的所有言论和文字，纯属内容发起人的个人观点，与本站观点和立场无关；
4. 本站文章均是网友提供，不完全保证技术分享内容的完整性、准确性、时效性、风险性和版权归属；如您发现该文章侵犯了您的权益，可联系我们第一时间进行删除；
5. 本站为非盈利性的个人网站，所有内容不会用来进行牟利，也不会利用任何形式的广告来间接获益，纯粹是为了广大技术爱好者提供技术内容和技术思想的分享性交流网站。