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Bessel function

2022-08-09 16:13:00 【Chandler_river】

  • Derivation of Bessel equations
    • 2D heat transfer model
    • x^2y''+xy'+(x^2-n^2)y=0
  • 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_0=\frac{1}{2^n\Gamma(n+1)}
          • 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J_{-n}(x)(-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 y=AJ_n(x)+BJ_{-n}(x)
        • 如果令A=cotn\pi,B=-cscn\pi,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Y_n(x)=\lim_{a->n}\frac{J_a(x)cosa\pi-J_{-a}(x)}{sina\pi}
          • 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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