平稳相近似

${\displaystyle F(v,x)=\int _{0}^{\pi }cos(v*h(t)dt}$ 

位相：${\displaystyle v*h(t)=v*(t-xsin(t))}$ 

被积分函数：${\displaystyle cos(v(t-xsin(t))}$ 

位相的平稳点由下列微分方程给出：

${\displaystyle {\frac {d}{dt}}(v*(t-xsin(t))=0}$ ,令${\displaystyle v=5,x=35}$ 得平稳相点：

${\displaystyle t_{s}=t=arccos(1/35)=1.54}$ 

将积分F(v,x)分为三段：

${\displaystyle sf:=\int _{0}^{1}cos(5*t-5*x*sin(t))dt+\int _{1}^{2}cos(5*t-5*x*sin(t))dt+\int _{2}^{\pi }cos(5*t-5*x*sin(t))dt}$ 

由图可见，在平稳相点左右区间[1,2]，位相平稳，这个平稳相区间贡献了定积分的主要部分，在平稳相区间之外[0,1]和${\displaystyle [2,\pi ]}$ ，由于被积分函数振荡激烈，正负相消，因此贡献可以忽略不计。这就是平稳相近似的原理^[3]

设有下列积分

${\displaystyle F(v)=\int _{a}^{b}e^{ivh(t)}f(t)dt}$ ,

并设${\displaystyle h(t)}$ 的平稳相点为${\displaystyle t_{k}}$ 

将${\displaystyle h(t)}$ 作泰勒展开：

${\displaystyle h(t)\approx h(t_{k})+{\frac {1}{2}}*(t-t_{k})^{2}h''(t_{k})+\cdots }$ 

于是

${\displaystyle F_{k}(v)\approx e^{ivh(t)}f(t_{k})}$ ${\displaystyle \int _{-\infty }^{\infty }e^{ivs^{2}h''(t_{k})/2}ds}$ 

将所有平稳相点${\displaystyle k=1\cdots N}$ 的贡献相加得^[4]

${\displaystyle F(v)\approx {\sqrt {\frac {2*\pi }{abs(v)}}}}$ ${\displaystyle \sum _{k=1}^{N}}$ ${\displaystyle {\frac {f(t_{k})}{\sqrt {abs(h''(t_{k}))}}}exp{(i(vh(t)+{\frac {\pi }{4}}sgn(vh''(t_{k})))}}$ 

一般情形被积分函数是不是积分区间的周期函数，因此积分区间两头的贡献也必须补入^[4]，

${\displaystyle F(v)\approx {\sqrt {\frac {2*\pi }{abs(v)}}}}$ ${\displaystyle \sum _{k=1}^{N}}$ ${\displaystyle {\frac {f(t_{k})}{\sqrt {abs(h''(t_{k}))}}}exp{(i(vh(t)+{\frac {\pi }{4}}sgn(vh''(t_{k})))}}$ ${\displaystyle +{\frac {i}{v}}({\frac {e^{ivh(a)}f(a)}{h'(a)}}-{\frac {ivh(b)f(b)}{h'(b)}})}$ 

• D.Richards Advanced Mathematical Methods with Maple,Cambridge 2002.
• Frank J. Oliver, NIST Handbook of Mathematical Functions,Cambridge 2010