# Fresnel integrals

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##### 1: 7.2 Definitions
###### §7.2(iii) FresnelIntegrals
$\mathcal{F}\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection.
###### Values at Infinity
$\lim_{x\to\infty}C\left(x\right)=\tfrac{1}{2},$
##### 2: Sidebar 7.SB1: Diffraction from a Straightedge
The intensity distribution follows $|\mathcal{F}\left(x\right)|^{2}$, where $\mathcal{F}$ is the Fresnel integral (See 7.3.4). Fresnel integrals have many applications in optics. …
##### 3: 7.5 Interrelations
###### §7.5 Interrelations
7.5.6 $e^{\pm\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z% \right))=\tfrac{1}{2}(1\pm i)-(C\left(z\right)\pm iS\left(z\right)).$
##### 4: 7.4 Symmetry
###### §7.4 Symmetry
$C\left(-z\right)=-C\left(z\right),$
$S\left(-z\right)=-S\left(z\right),$
$C\left(iz\right)=iC\left(z\right),$
$S\left(iz\right)=-iS\left(z\right).$
##### 7: 7.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the error function $\operatorname{erf}z$; the complementary error functions $\operatorname{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $\mathcal{F}\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)$; the Voigt functions $\mathsf{U}\left(x,t\right)$ and $\mathsf{V}\left(x,t\right)$. Alternative notations are $Q(z)=\tfrac{1}{2}\operatorname{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\Phi(z)=\tfrac{1}{2}\operatorname{erfc}\left(-z/\sqrt{2}\right)$, $\operatorname{Erf}z=\tfrac{1}{2}\sqrt{\pi}\operatorname{erf}z$, $\operatorname{Erfi}z=e^{z^{2}}F\left(z\right)$, $C_{1}(z)=C\left(\sqrt{2/\pi}z\right)$, $S_{1}(z)=S\left(\sqrt{2/\pi}z\right)$, $C_{2}(z)=C\left(\sqrt{2z/\pi}\right)$, $S_{2}(z)=S\left(\sqrt{2z/\pi}\right)$. …
##### 8: 7.20 Mathematical Applications
###### §7.20(ii) Cornu’s Spiral
Let the set $\{x(t),y(t),t\}$ be defined by $x(t)=C\left(t\right)$, $y(t)=S\left(t\right)$, $t\geq 0$. …
7.20.1 $\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{x}e^{-(t-m)^{2}/(2\sigma^{2})}\,% \mathrm{d}t=\frac{1}{2}\operatorname{erfc}\left(\frac{m-x}{\sigma\sqrt{2}}% \right)=Q\left(\frac{m-x}{\sigma}\right)=P\left(\frac{x-m}{\sigma}\right).$
##### 9: 7.13 Zeros
###### §7.13(iii) Zeros of the FresnelIntegrals
Similarly for $S\left(z\right)$. …
##### 10: 7.6 Series Expansions
###### §7.6(i) Power Series
7.6.10 $C\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2n}\left(\tfrac{1}{2}\pi z^{2}% \right),$
7.6.11 $S\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2n+1}\left(\tfrac{1}{2}\pi z^{% 2}\right).$