# 菲涅耳積分

S(x)C(x)

## 定義

${\displaystyle S(x)=\int _{0}^{x}\sin(t^{2})\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+3}}{(2n+1)!(4n+3)}},}$
${\displaystyle C(x)=\int _{0}^{x}\cos(t^{2})\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+1}}{(2n)!(4n+1)}}.}$

## 羊角螺线

### 估計值

CS的值當變數趨近於無窮大時，可用複變分析的方法求得。用以下這個函數的路徑積分

${\displaystyle e^{-z^{2}}}$

R趨近於無窮大時，路徑積分沿弧形的部分將趨近於零[1]，而實數軸部分的積分將可由高斯積分

${\displaystyle \int _{y-axis}^{}e^{-z^{2}}dz=\int _{0}^{\infty }e^{-t^{2}}dt={\frac {\sqrt {\pi }}{2}},}$

${\displaystyle \int _{slope}^{}exp(-z^{2})dz=\int _{0}^{\infty }\exp(-t^{2}e^{i\pi /2})e^{i\pi /4}dt=e^{i\pi /4}(\int _{0}^{\infty }\cos(-z^{2})dz+i\int _{0}^{\infty }\sin(-z^{2})dz)}$
${\displaystyle \int _{0}^{\infty }\cos t^{2}\,\mathrm {d} t=\int _{0}^{\infty }\sin t^{2}\,\mathrm {d} t={\frac {\sqrt {2\pi }}{4}}={\sqrt {\frac {\pi }{8}}}.}$

## 相关公式

• ${\displaystyle \int _{0}^{\infty }e^{-at}sin(t^{2})}$${\displaystyle =(1/4)*{\sqrt {(}}2)*{\sqrt {(}}\pi )*(cos((1/4)*a^{2})*(1-2*FresnelC((1/2)*a*{\sqrt {(}}2)/{\sqrt {(}}\pi )))+sin((1/4)*a^{2})*(1-2*FresnelS((1/2)*a*{\sqrt {(}}2)/{\sqrt {(}}Pi))))}$
• ${\displaystyle \int (sin(ax^{2}+2bx+c)dx=}$${\displaystyle {\frac {{\sqrt {(}}2)*{\sqrt {(}}\pi )*(cos((b^{2}-a*c)/a)*FresnelS({\sqrt {(}}2)*(a*x+b)/({\sqrt {(}}\pi )*{\sqrt {(}}a)))-sin((b^{2}-a*c)/a)*FresnelC({\sqrt {(}}2)*(a*x+b)/({\sqrt {(}}\pi )*{\sqrt {(}}a))))}{2{\sqrt {(}}a)}}}$
• ${\displaystyle \int (FresnelC(t)dt=FresnelC(t)*t-{\frac {sin((1/2)*\pi *t^{2})}{\pi }}}$
• ${\displaystyle \int (FesnelS(t)dt=FresnelS(t)*t+{\frac {cos((1/2)*\pi *t^{2})}{\pi }}}$
• ${\displaystyle {\frac {dFresnelC(t)}{dt}}=cos((1/2)*\pi *t^{2})}$
• ${\displaystyle {\frac {dFresnelS(t)}{dt}}=sin((1/2)*\pi *t^{2})}$

## 參考資料

1. ^ Beatty, Thomas. How to evaluate Fresnel Integrals (PDF). FGCU MATH - SUMMER 2013. [27 July 2013].
2. ^ Abromowitz and Stegun, Handbook of Mathematical Functions,p303-305, 1972 Natinal Bureau of Standards