# 渐近展开

## 形式定义

${\displaystyle f(z)-\sum _{n=0}^{m}a_{n}\phi _{n}(z)=o(\phi _{m}(z)),\quad z\rightarrow z_{0},\forall m\in \mathbb {Z} _{0}^{+}}$

${\displaystyle \sum _{n=0}^{\infty }a_{n}\phi _{n}(z)}$

f(z) 在 z=z0 点处的渐近级数。记作

${\displaystyle f(z)\sim \sum _{n=0}^{\infty }a_{n}\phi _{n}(z),\quad z\rightarrow z_{0}}$

## 渐近展开的唯一性

${\displaystyle a_{m}=\lim _{z\rightarrow z_{0}}{\frac {1}{\phi _{m}(z)}}\left[f(z)-\sum _{n=0}^{m-1}a_{n}\phi _{n}(z)\right]}$

${\displaystyle \phi _{\infty }(z)=o(\phi _{m}(z)),\quad z\rightarrow z_{0},\forall m\in \mathbb {Z} _{0}^{+}}$

## 渐近幂级数

${\displaystyle \phi _{n}(z)={\begin{cases}z^{-n}&z_{0}=\infty \\(z-z_{0})^{n}&{\text{otherwise}}\end{cases}}}$

### 一些例子

${\displaystyle {\frac {e^{z}}{z^{z}{\sqrt {2\pi z}}}}\Gamma (z+1)\sim 1+{\frac {1}{12z}}+{\frac {1}{288z^{2}}}-{\frac {139}{51840z^{3}}}-\cdots ,\quad z\rightarrow \infty ,|\arg z|<\pi }$
${\displaystyle \exp \left[i\left({\frac {\pi }{4}}-{\frac {\nu \pi }{2}}-z\right)\right]{\sqrt {\frac {\pi z}{2}}}H_{\nu }^{(1)}(z)\sim \sum _{k=0}^{\infty }{\frac {i^{k}\prod _{n=1}^{k}(4\nu ^{2}-(2n-1)^{2})}{2^{3k}k!}}z^{-k},\quad z\rightarrow \infty ,-\pi <\arg z<2\pi }$

${\displaystyle z^{a}U(a,b,z)\sim \sum _{n=0}^{\infty }(a)^{(n)}(a-b+1)^{(n)}(-z)^{-n},\quad z\rightarrow \infty ,|\arg z|<{\frac {3\pi }{2}}}$

## 参考

1. ^ 吴崇试. 4. 数学物理方法（第二版）. 北京大学出版社. [2003]. ISBN 9787301068199.
• Hardy, G. H., Divergent Series, Oxford University Press, 1949.