# auxiliary functions for Fresnel integrals

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##### 1: 7.5 Interrelations
###### §7.5 Interrelations
7.5.3 $C\left(z\right)=\tfrac{1}{2}+\mathrm{f}\left(z\right)\sin\left(\tfrac{1}{2}\pi z% ^{2}\right)-\mathrm{g}\left(z\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right),$
7.5.5 $e^{-\frac{1}{2}\pi iz^{2}}\mathcal{F}\left(z\right)=\mathrm{g}\left(z\right)+i% \mathrm{f}\left(z\right).$
7.5.11 $|\mathcal{F}\left(x\right)|^{2}={\mathrm{f}}^{2}\left(x\right)+{\mathrm{g}}^{2% }\left(x\right),$ $x\geq 0$,
##### 2: 7.4 Symmetry
$\mathrm{g}\left(-z\right)=\sqrt{2}\sin\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^% {2}\right)-\mathrm{g}\left(z\right).$
##### 4: 7.10 Derivatives
$\frac{\mathrm{d}\mathrm{g}\left(z\right)}{\mathrm{d}z}=\pi z\mathrm{f}\left(z% \right)-1.$
##### 5: 7.2 Definitions
###### §7.2(iv) AuxiliaryFunctions
7.2.10 $\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right),$
7.2.11 $\mathrm{g}\left(z\right)=\left(\tfrac{1}{2}-C\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-S\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right).$
##### 6: 7.7 Integral Representations
###### §7.7(ii) AuxiliaryFunctions
7.7.12 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-\pi iz^{2}/2}\int_{z}^{% \infty}e^{\pi it^{2}/2}\,\mathrm{d}t.$
##### 8: 7.12 Asymptotic Expansions
###### §7.12(ii) FresnelIntegrals
7.12.2 $\mathrm{f}\left(z\right)\sim\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{{% \left(\tfrac{1}{2}\right)_{2m}}}{(\pi z^{2}/2)^{2m}},$
7.12.3 $\mathrm{g}\left(z\right)\sim\frac{1}{\pi z}\sum_{m=0}^{\infty}(-1)^{m}\frac{{% \left(\tfrac{1}{2}\right)_{2m+1}}}{(\pi z^{2}/2)^{2m+1}},$
7.12.4 $\mathrm{f}\left(z\right)=\frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{{\left(% \tfrac{1}{2}\right)_{2m}}}{(\pi z^{2}/2)^{2m}}+R_{n}^{(\mathrm{f})}(z),$
7.12.5 $\mathrm{g}\left(z\right)=\frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{{\left(% \tfrac{1}{2}\right)_{2m+1}}}{(\pi z^{2}/2)^{2m+1}},+R_{n}^{(\mathrm{g})}(z),$
##### 10: Bibliography F
• V. N. Faddeyeva and N. M. Terent’ev (1961) Tables of Values of the Function $w(z)=e^{-z^{2}}(1+2i\pi^{-1/2}\int_{0}^{z}e^{t^{2}}dt)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.
• H. E. Fettis and J. C. Caslin (1969) A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.
• H. E. Fettis and J. C. Caslin (1973) Table errata; Complex zeros of Fresnel integrals. Math. Comp. 27 (121), pp. 219.
• A. Fresnel (1818) Mémoire sur la diffraction de la lumière. Mém. de l’Académie des Sciences, pp. 247–382.
• Y. Fukui and T. Horiguchi (1992) Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.