# §7.5 Interrelations

 7.5.1 $F\left(z\right)=\tfrac{1}{2}i\sqrt{\pi}\left(e^{-z^{2}}-w\left(z\right)\right)% =-\tfrac{1}{2}i\sqrt{\pi}e^{-z^{2}}\operatorname{erf}\left(iz\right).$
 7.5.2 $C\left(z\right)+iS\left(z\right)=\tfrac{1}{2}(1+i)-\mathcal{F}\left(z\right).$ ⓘ Symbols: $C\left(\NVar{z}\right)$: Fresnel integral, $\mathcal{F}\left(\NVar{z}\right)$: Fresnel integral, $S\left(\NVar{z}\right)$: Fresnel integral and $z$: complex variable Referenced by: §7.5 Permalink: http://dlmf.nist.gov/7.5.E2 Encodings: TeX, pMML, png See also: Annotations for 7.5 and 7
 7.5.3 $C\left(z\right)=\tfrac{1}{2}+\mathrm{f}\left(z\right)\sin\left(\tfrac{1}{2}\pi z% ^{2}\right)-\mathrm{g}\left(z\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right),$
 7.5.4 $S\left(z\right)=\tfrac{1}{2}-\mathrm{f}\left(z\right)\cos\left(\tfrac{1}{2}\pi z% ^{2}\right)-\mathrm{g}\left(z\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right).$
 7.5.5 $e^{-\frac{1}{2}\pi iz^{2}}\mathcal{F}\left(z\right)=\mathrm{g}\left(z\right)+i% \mathrm{f}\left(z\right).$
 7.5.6 $e^{\pm\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z% \right))=\tfrac{1}{2}(1\pm i)-(C\left(z\right)\pm iS\left(z\right)).$

In (7.5.8)–(7.5.10)

 7.5.7 $\zeta=\tfrac{1}{2}\sqrt{\pi}(1\mp i)z,$ ⓘ Defines: $\zeta$: change of variable (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable Referenced by: §7.12(ii), §7.5 Permalink: http://dlmf.nist.gov/7.5.E7 Encodings: TeX, pMML, png See also: Annotations for 7.5 and 7

and either all upper signs or all lower signs are taken throughout.

 7.5.8 $C\left(z\right)\pm iS\left(z\right)=\tfrac{1}{2}(1\pm i)\operatorname{erf}\zeta.$ ⓘ Symbols: $C\left(\NVar{z}\right)$: Fresnel integral, $S\left(\NVar{z}\right)$: Fresnel integral, $\operatorname{erf}\NVar{z}$: error function, $z$: complex variable and $\zeta$: change of variable Referenced by: §7.5, §7.5, §7.6(i) Permalink: http://dlmf.nist.gov/7.5.E8 Encodings: TeX, pMML, png See also: Annotations for 7.5 and 7
 7.5.9 $C\left(z\right)\pm iS\left(z\right)=\tfrac{1}{2}(1\pm i)\left(1-e^{\pm\frac{1}% {2}\pi iz^{2}}w\left(i\zeta\right)\right).$
 7.5.10 $\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z\right)=\tfrac{1}{2}(1\pm i)e^{% \zeta^{2}}\operatorname{erfc}\zeta.$ ⓘ Symbols: $\mathrm{f}\left(\NVar{z}\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(\NVar{z}\right)$: auxiliary function for Fresnel integrals, $\operatorname{erfc}\NVar{z}$: complementary error function, $\mathrm{e}$: base of exponential function, $z$: complex variable and $\zeta$: change of variable A&S Ref: 7.3.22 (in different form) Referenced by: §7.12(ii), §7.13(iv), §7.5, §7.5 Permalink: http://dlmf.nist.gov/7.5.E10 Encodings: TeX, pMML, png See also: Annotations for 7.5 and 7
 7.5.11 $|\mathcal{F}\left(x\right)|^{2}={\mathrm{f}^{2}}\left(x\right)+{\mathrm{g}^{2}% }\left(x\right),$ $x\geq 0$,
 7.5.12 $|\mathcal{F}\left(x\right)|^{2}=2+{\mathrm{f}^{2}}\left(-x\right)+{\mathrm{g}^% {2}}\left(-x\right)-2\sqrt{2}\cos\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi x^{2}% \right)\mathrm{f}\left(-x\right)-2\sqrt{2}\cos\left(\tfrac{1}{4}\pi-\tfrac{1}{% 2}\pi x^{2}\right)\mathrm{g}\left(-x\right),$ $x\leq 0$.

See Figure 7.3.4.

 7.5.13 $G\left(x\right)=\sqrt{\pi}F\left(x\right)-\tfrac{1}{2}e^{-x^{2}}\mathrm{Ei}% \left(x^{2}\right),$ $x>0$.

For $\mathrm{Ei}\left(x\right)$ see §6.2(i).