S(x)与C(x)。
菲涅耳积分,常被写作 S(x)和C(x)。以奥古斯丁·菲涅耳为名。
菲涅耳积分可由下面两个级数求得,对所有x均收敛。
![{\displaystyle S(x)=\int _{0}^{x}\sin(t^{2})\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+3}}{(2n+1)!(4n+3)}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5376792e6de305beffeb0f04725fc70ad6ac43a0)
![{\displaystyle C(x)=\int _{0}^{x}\cos(t^{2})\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+1}}{(2n)!(4n+1)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09c6c3332ffea33fdcf84abd87741b075f3bdaab)
羊角螺线[编辑]
估计值[编辑]
用来计算Fresnel integrals的扇形路径
C和S的值当变数趋近于无穷大时,可用复变分析的方法求得。用以下这个函数的路径积分:
![{\displaystyle e^{-z^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/294fe8388f1e37e66d26dab964051307bd12156d)
在复数平面上的一个扇型的边界,其中下边绕着正x轴,上半边是沿着y = x, x ≥ 0的路径,外圈则是一个半径为R,中心在原点的弧形。
当R趋近于无穷大时,路径积分沿弧形的部分将趋近于零[1],而实数轴部分的积分将可由高斯积分
![{\displaystyle \int _{y-axis}^{}e^{-z^{2}}dz=\int _{0}^{\infty }e^{-t^{2}}dt={\frac {\sqrt {\pi }}{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd84442399e56077eb42c6fe2d1427de1a495ba2)
并且经过简单的计算后,第一象限平分线的那条积分便可以变成菲涅耳积分。
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef143b6ef0ada74c2e3c922080fbb9ce25b25971)
![{\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d2f4b9ef355a64629211a614e2fa06e551786ed)
相关公式[编辑]
下列一些包含菲涅耳积分的关系式[2]
![{\displaystyle \int _{0}^{\infty }e^{-at}\sin(t^{2})\mathrm {d} t={\frac {1}{4}}*{\sqrt {2\pi }}*(\cos {\frac {a^{2}}{4}}*(1-2*{\rm {FresnelC}}((1/2)*a*{\sqrt {2}}/{\sqrt {\pi }}))+\sin {\frac {a^{2}}{4}}*(1-2*\mathrm {FresnelS} ((1/2)*a*{\sqrt {2}}/{\sqrt {\pi }})))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e3b54c62e81bd6896b31dc119e807813a6fd56)
![{\displaystyle \int \sin(ax^{2}+2bx+c)\mathrm {d} x={\frac {{\sqrt {2\pi }}*(\cos((b^{2}-a*c)/a)*{\rm {FresnelS}}({\sqrt {2}}(ax+b)/({\sqrt {\pi a}}))-\sin((b^{2}-a*c)/a)*{\rm {FresnelC}}({\sqrt {2}}(ax+b)/({\sqrt {\pi a}})))}{2{\sqrt {a}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a21c4e7d211cf709458ecd7ac39578069cd32af)
![{\displaystyle \int \mathrm {FresnelC} (t)\mathrm {d} t=\mathrm {FresnelC} (t)*t-{\frac {\sin {\frac {\pi t^{2}}{2}}}{\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0501ac7d2203982fccc6c28169bf6d1a4aa4e007)
![{\displaystyle \int \mathrm {FresnelS} (t)\mathrm {d} t=\mathrm {FresnelS} (t)*t+{\frac {\cos {\frac {\pi t^{2}}{2}}}{\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/821cbfce50594af7ba1c36c269d405ba7398fb01)
![{\displaystyle {\frac {\mathrm {d} ~\mathrm {FresnelC} (t)}{\mathrm {d} t}}=\cos {\frac {\pi t^{2}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd4cd95eb659d04e537b4b86999eca524442dda)
![{\displaystyle {\frac {\mathrm {d} ~\mathrm {FresnelS} (t)}{\mathrm {d} t}}=\sin {\frac {\pi t^{2}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4699655dc8ad44a3b4f0643c9d00718bae8be2)
关联条目[编辑]
参考资料[编辑]
- ^ Beatty, Thomas. How to evaluate Fresnel Integrals (PDF). FGCU MATH - SUMMER 2013. [27 July 2013]. (原始内容存档 (PDF)于2015-02-07).
- ^ Abromowitz and Stegun, Handbook of Mathematical Functions,p303-305, 1972 Natinal Bureau of Standards