- \(\Gamma\) Gamma function - Wikipedia For \(\Re(z)>0\) the \(n\)th derivative of the gamma function is:
\[ \frac{d^n}{d z^n} \Gamma(z)=\int_0^{\infty} t^{z-1} e^{-t}(\ln t)^n d t \] 2. Polygamma function[^2] \[\psi(z)=\frac{\mathrm{d}}{\mathrm{d} z} \ln \Gamma(z)=\frac{\Gamma^{\prime}(z)}{\Gamma(z)}\] hence:
\[ \psi^{(m)}(z):=\frac{\mathrm{d}^m}{\mathrm{~d} z^m} \psi(z)=\frac{\mathrm{d}^{m+1}}{\mathrm{~d} z^{m+1}} \ln \Gamma(z) \] Thus \[ \Gamma^{\prime}(z)=\psi^{(0)}(z){\Gamma(z)} \] 3. Polygamma function's series representation
\[ \psi^{(m)}(z)=(-1)^{m+1} m ! \zeta(m+1, z) \]
- Hurwitz zeta function[^3]
\[ \zeta(s,a)= \sum_{n=0}^{\infty} \frac{1}{(n+a)^s} \]
Beseel asymptotic forms[^4] \[ J_\nu(x) \sim(x / 2)^\nu, \quad x \rightarrow 0+, \] and \[ J_\nu(x) \sim(2 / \pi x)^{1 / 2} \cos (x-y \nu / 2-\pi / 4), \quad x \rightarrow+\infty \]
贝塞尔函数相关
\[\begin{aligned} \mathrm{J}_0(R) &=\frac{1}{2 \pi \mathrm{i}} \int^{(0+)} \mathrm{e}^{\frac{r_1}{2}\left(u \mathrm{e}^{\mathrm{i} \theta}-\frac{1}{u \mathrm{e}^{i \theta}}\right)} \mathrm{e}^{-\frac{r_2}{2}\left(u-\frac{1}{u}\right)} u^{-1} \mathrm{~d} u \\ &=\frac{1}{2 \pi \mathrm{i}} \sum_{m=-\infty}^{\infty} \mathrm{J}_m\left(r_1\right) \mathrm{e}^{\mathrm{i} m \theta} \int^{(0+)} \mathrm{e}^{-\frac{r_2}{2}\left(u-u^{-1}\right)} u^{m-1} \mathrm{~d} u \\ &=\sum_{m=-\infty}^{\infty} \mathrm{J}_m\left(r_1\right) \mathrm{J}_{-m}\left(-r_2\right) \mathrm{e}^{\mathrm{i} m \theta} \\ &=\sum_{m=-\infty}^{\infty} \mathrm{J}_m\left(r_1\right) \mathrm{J}_m\left(r_2\right) \mathrm{e}^{\mathrm{i} m \theta} ; \end{aligned}\]
- 关于\(J_n(z)\)的不等式
\[\left|J_n(z)\right| \leqslant\left|\frac{z}{2}\right|^n \sum_{k=0}^{\infty} \frac{1}{k !(n+k) !}\left|\frac{z}{2}\right|^{2 k} \leqslant \frac{1}{n !}\left|\frac{z}{2}\right|^n \sum_{k=0}^{\infty} \frac{1}{k !(n+1)^k}\left|\frac{z}{2}\right|^{2 k}\]