菲涅耳积分
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![用来计算Fresnel integrals的扇形路径](/Images/godic/202501/14/Fresnel_Integral_Contour.svg3407.png")
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。
Fresnel integral
中文百科
![S(x)=\int_0^x \sin(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+3}}{(2n+1)!(4n+3)},](/Images/godic/202501/14/d4744323f845b3bb8f077111e949825e3407.png")
![C(x)=\int_0^x \cos(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+1}}{(2n)!(4n+1)}.](/Images/godic/202501/14/41e7c15fcbffa57803cb7a5041f9add83407.png")
英语百科
Fresnel integral 菲涅耳积分
.svg3407.png")
![Euler spiral (x, y) = (C(t), S(t)). The spiral converges to the centre of the holes in the image as t tends to positive or negative infinity.](/Images/godic/202501/14/Cornu_Spiral.svg3407.png")
![Complex Fresnel integral S(z)](/Images/godic/202501/14/Fresnel-sin3407.png")
Fresnel integrals, S(x) and C(x), are two transcendental functions named after Augustin-Jean Fresnel that are used in optics, which are closely related to the error function (erf). They arise in the description of far field Fresnel diffraction phenomena, and are defined through the following integral representations:
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