数学备忘录：特殊函数

伽马函数

\[ \begin{gather} \Gamma(x)=\int_0^\infty \mathrm{e}^{-t}t^{x-1}\mathrm{d}t\quad(x>0) \\ \Gamma(x+1)=x\Gamma(x) \\ \Gamma(1)=\Gamma(2)=1 \\ \Gamma(n+1)=n!\quad(n=0,1,2,\cdots) \\ \Gamma\left(\frac12\right)=\sqrt{\pi} \\ \Gamma\left(n+\frac12\right)=\frac{(2n)!}{x^{2n}n!}\sqrt{\pi} \\ \Gamma\left(n+\frac12+1\right)=\frac{(2n+1)!}{x^{2n+1}n!}\sqrt{\pi} \end{gather} \]

• Stirling 公式

\[ \begin{gather} n!=\sqrt{2\pi n}n^n\mathrm{e}^{-n} \end{gather} \]

Bessel 函数

• 第一类 \(\nu\) 阶 Bessel 函数

\[ \begin{gather} J_\nu(x)=\sum_{m=0}^\infty\frac{(-1)^m}{m!\Gamma(m+\nu+1)}\left(\frac{x}2\right)^{2m+\nu} \\ J_\nu(0)=0\quad(\nu>0) \\ J_0(0)=1 \\ J_{1/2}(x)=\sqrt{\frac2{\pi x}}\sin x \\ J_{-1/2}(x)=\sqrt{\frac2{\pi x}}\cos x \end{gather} \]

• 整数阶 Bessel 函数(\(\nu=n=0,1,2,\cdots\))

\[ \begin{gather} J_n(x)=\sum_{m=0}^\infty\frac{(-1)^m}{m!(m+n)!}\left(\frac{x}2\right)^{2m+n} \\ J_{-n}(x)=(-1)^nJ_n(x) \end{gather} \]

• Neumann 函数(第二类 \(\nu\) 阶 Bessel 函数)

\[ \begin{gather} Y_\nu(x)= \begin{cases} \frac{J_\nu(x)\cos\nu\pi-J_{-\nu}(x)}{\sin\nu\pi} & (\nu\neq 整数) \\ \lim_{\alpha\to\nu}\frac{J_\alpha(x)\cos\alpha\pi-J_{-\alpha}(x)}{\sin\alpha\pi} & (\nu=整数) \end{cases} \\ Y_\nu(0^+)\to-\infty \\ Y_n(x)=\frac2{\pi}\left(\ln\frac{x}2+\gamma\right)J_n(x)-\frac1{\pi}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\left(\frac{x}2\right)^{2k-n} \\ -\frac1{\pi}\sum_{k=0}^\infty\frac{(-1)^k}{k!(n+k)!}[\Phi(k)+\Phi(n+k)]\left(\frac{x}2\right)^{2k+n} \\ Y_n(x)=\frac1{\pi}\int_0^\pi\sin(x\sin\theta-n\theta)\mathrm{d}\theta-\frac1{\pi}\int_0^\infty[\mathrm{e}^{nt}+(-1)^n\mathrm{e}^{-nt}]\mathrm{e}^{-x\sinh t}\mathrm{d}t \end{gather} \]

• 整数阶 Bessel 函数的生成函数

\[ \exp\left[\frac{x}2\left(r-\frac1{r}\right)\right]=\sum_{n=-\infty}^\infty J_n(x)r^n \]

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\[ \begin{gather} \frac{\mathrm{d}}{\mathrm{d}x}[x^\nu J_\nu(x)]=x^\nu J_{\nu-1}(x) \\ \frac{\mathrm{d}}{\mathrm{d}x}[x^{-\nu} J_\nu(x)]=-x^{-\nu} J_{\nu+1}(x) \\ \frac{\mathrm{d}}{\mathrm{d}x}[J_0(x)]=-J_{1}(x) \\ \frac{\mathrm{d}}{\mathrm{d}x}[xJ_1(x)]=xJ_{0}(x) \\ J_\nu'(x)=\frac12[J_{\nu-1}(x)-J_{\nu+1}(x)] \\ J_{\nu-1}(x)+J_{\nu+1}(x)=\frac{2\nu}xJ_\nu(x) \\ xJ_{\nu-1}(x)=\nu J_\nu(x)+xJ_\nu'(x) \\ xJ_{\nu+1}(x)=\nu J_\nu(x)-xJ_\nu'(x) \\ \int x^{\nu+1}J_\nu(x)\mathrm{d}x=x^{\nu+1}J_{\nu+1}(x) \\ \int x^{-\nu+1}J_\nu(x)\mathrm{d}x=-x^{-\nu+1}J_{\nu-1}(x) \\ J_{n+\frac12}(x)=(-1)^n\sqrt{\frac2{\pi}}x^{n+\frac12}\left(\frac1{x}\frac{\mathrm{d}}{\mathrm{d}x}\right)^n\left(\frac{\sin x}x\right) \\ J_{-\left(n+\frac12\right)}(x)=\sqrt{\frac2{\pi}}x^{n+\frac12}\left(\frac1{x}\frac{\mathrm{d}}{\mathrm{d}x}\right)^n\left(\frac{\cos x}x\right) \end{gather} \]