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Bessel function

2022-08-09 16:13:00 【Chandler_river】

Derivation of Bessel equations
- 2D heat transfer model
Generalized power series solutions of Bessel equations
- $y(x)=\sum_{k=0}^{\infty}a_kx^{s+k}(a_0 \neq 0)$
- Substitute in and out $\left\{\begin{matrix} (s^2-n^2)a_0=0\\ [(s+1)^2-n^2]a_1=0\\ [(s+k)^2-n^2]a_k+a_{k-2}=0(k=2,3,...) \end{matrix}\right.$
- 讨论1 n不为整数(包括0)and half odd,则s1-s2=2nNeither is an integer
  - s1=n
    - $\left\{\begin{matrix} a_1=0\\ a_k=-\frac{a_{k-2}}{k(2n+k)}(k=2,3,...) \\a_{2m}=(-1)^m\frac{a_0}{2^{2m}m!(n+1)(n+2)...(n+m)} \end{matrix}\right.$
    - 因为a_0是任意常数,取
      - $a_{2m}=(-1)^m\frac{1}{2^{n+2m}m!\Gamma(n+m+1)}$
    - Get a particular solution to the equationJn(x)(nFirst-order Bessel functions)
      - $J_n(x)=\sum_{m=0}^\infty a_{2m}x^{n+2m}=\sum_{m=0}^{\infty}(-1)^m\frac{x^{n+2m}}{2^{n+2m}m!\Gamma(n+m+1)}$
      - Discrimination by D'Alembert(lim u(m+1)/um )It can be seen that the series converges absolutely on the real axis
  - s2=-n
    - $\left\{\begin{matrix} a_1=0\\ a_k=-\frac{a_{k-2}}{k(k-2n)}\\ a_{2m}=(-1)^m\frac{a_0}{2^{2m}m!(1-n)(2-n)...(m-n)} \end{matrix}\right.$
    - 取 $a_0=\frac{1}{a^{-n}\Gamma(1-n)}$
    - Get another particular solution to the equation,记作(-nFirst-order Bessel functions)
      - $J_{-n}(x)=\sum_{m=0}^\infty (-1)^m\frac{x^{2m-n}}{2^{2m-n}m!\Gamma(1+m-n)}$
  - 因为n \neq -n 所以Jn(x) 与 J-n(x) 线性无关,The general solution of the equation can be expressed as
    - 如果令,Can get one andJn(x)Linearly independent particular solutions(第二类贝塞尔函数/诺伊曼函数)
      - $Y_n(x)=\frac{J_n(x)cosn\pi-J_{-n}(x)}{sinn\pi}$
- 讨论2 n为整数(包括0) 则s1-s2=2n也为整数 Two special solutions of the equation can also be obtained
  - $J_n(x)=\sum_{m=0}^\infty a_{2m}x^{n+2m}=\sum_{m=0}^{\infty}(-1)^m\frac{x^{n+2m}}{2^{n+2m}m!\Gamma(n+m+1)}$
  - $J_{-n}(x)=\sum_{m=0}^\infty (-1)^m\frac{x^{2m-n}}{2^{2m-n}m!\Gamma(1+m-n)}$
  - 利用Γ函数的递推公式,It can be inferred at this timeJ(-n)(x) 与 J(n)(x) 线性相关
    - 定义
      - 其中nfor an integera不为整数,可推出
      - 与J(n)(x)线性无关的Y(n)(x) $Y_n(x)=\frac{2}{\pi}J_n(x)(ln(\frac{x}{2})+C)-\frac{1}{\pi}\sum_{m=0}^{n-1}\frac{(n-m-1)!}{m!}(\frac{x}{2})^{2m-n}-\frac{1}{\pi}\sum_{m=0}^\infty(-1)^m\frac{1}{m!(n+m)!}(\frac{x}{2})^{n+2m}(\sum_{k=0}^{n+m-1}\frac{1}{1+k}+\sum_{k=0}^{m-1}\frac{1}{1+k})(n=1,2,3...)$
- 讨论3 nWhen it is a semi-odd number Bessel functions can be represented by elementary functions
  - $J_{\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi x}}sinx$
  - $\left\{\begin{matrix} J_{\frac{2m+1}{2}}=(-1)^m\sqrt{\frac{2}{\pi}}x^{m+\frac{1}{2}}(\frac{1}{x}\frac{d}{dx})^m(\frac{sinx}{x})\\ J_{-\frac{2m+1}{2}}=\sqrt{\frac{2}{\pi}}x^{m+\frac{1}{2}}(\frac{1}{x}\frac{d}{dx})^m(\frac{cosx}{x}) \end{matrix}\right.$
Recursion formula for Bessel functions
- $J_n(x)=\sum_{m=0}^{\infty}(-1)^m\frac{x^{n+2m}}{2^{n+2m}m!\Gamma(n+m+1)}$
- $Y_n(x\frac{J_n(x)cosn\pi-J_{-n}(x)}{sinn\pi}$
- $\left\{\begin{matrix} J_{n-1}(x)+J_{n+1}(x)=\frac{2n}{x}J_n(x)\\ J_{n-1}(x)-J_{n+1}(x)=2J'_n(x) \end{matrix}\right.$
- $\left\{\begin{matrix} Y_{n-1}(x)+Y_{n+1}(x)=\frac{2n}{x}Y_n(x)\\ Y_{n-1}(x)-Y_{n+1}(x)=2Y'_n(x) \end{matrix}\right.$

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本文为[Chandler_river]所创，转载请带上原文链接，感谢
https://yzsam.com/2022/221/202208091450462641.html

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