# §7.5 Interrelations

 7.5.1 $\mathop{F\/}\nolimits\!\left(z\right)=\tfrac{1}{2}i\sqrt{\pi}\left(e^{-z^{2}}-% \mathop{w\/}\nolimits\!\left(z\right)\right)=-\tfrac{1}{2}i\sqrt{\pi}e^{-z^{2}% }\mathop{\mathrm{erf}\/}\nolimits\!\left(iz\right).$
 7.5.2 $\mathop{C\/}\nolimits\!\left(z\right)+i\mathop{S\/}\nolimits\!\left(z\right)=% \tfrac{1}{2}(1+i)-\mathop{\mathcal{F}\/}\nolimits\!\left(z\right).$
 7.5.3 $\mathop{C\/}\nolimits\!\left(z\right)=\tfrac{1}{2}+\mathop{\mathrm{f}\/}% \nolimits\!\left(z\right)\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}% \right)-\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits% \!\left(\tfrac{1}{2}\pi z^{2}\right),$
 7.5.4 $\mathop{S\/}\nolimits\!\left(z\right)=\tfrac{1}{2}-\mathop{\mathrm{f}\/}% \nolimits\!\left(z\right)\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}% \right)-\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\mathop{\sin\/}\nolimits% \!\left(\tfrac{1}{2}\pi z^{2}\right).$
 7.5.5 $e^{-\frac{1}{2}\pi iz^{2}}\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)=% \mathop{\mathrm{g}\/}\nolimits\!\left(z\right)+i\mathop{\mathrm{f}\/}\nolimits% \!\left(z\right).$
 7.5.6 $e^{\pm\frac{1}{2}\pi iz^{2}}(\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\pm i% \mathop{\mathrm{f}\/}\nolimits\!\left(z\right))=\tfrac{1}{2}(1\pm i)-(\mathop{% C\/}\nolimits\!\left(z\right)\pm i\mathop{S\/}\nolimits\!\left(z\right)).$

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

 7.5.8 $\mathop{C\/}\nolimits\!\left(z\right)\pm i\mathop{S\/}\nolimits\!\left(z\right% )=\tfrac{1}{2}(1\pm i)\mathop{\mathrm{erf}\/}\nolimits\zeta.$
 7.5.9 $\mathop{C\/}\nolimits\!\left(z\right)\pm i\mathop{S\/}\nolimits\!\left(z\right% )=\tfrac{1}{2}(1\pm i)\left(1-e^{\pm\frac{1}{2}\pi iz^{2}}\mathop{w\/}% \nolimits\!\left(i\zeta\right)\right).$
 7.5.10 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\pm i\mathop{\mathrm{f}\/}% \nolimits\!\left(z\right)=\tfrac{1}{2}(1\pm i)e^{\zeta^{2}}\mathop{\mathrm{% erfc}\/}\nolimits\zeta.$
 7.5.11 $|\mathop{\mathcal{F}\/}\nolimits\!\left(x\right)|^{2}={\mathop{\mathrm{f}\/}% \nolimits^{2}}\!\left(x\right)+{\mathop{\mathrm{g}\/}\nolimits^{2}}\!\left(x% \right),$ $x\geq 0$,
 7.5.12 $|\mathop{\mathcal{F}\/}\nolimits\!\left(x\right)|^{2}=2+{\mathop{\mathrm{f}\/}% \nolimits^{2}}\!\left(-x\right)+{\mathop{\mathrm{g}\/}\nolimits^{2}}\!\left(-x% \right)-2\sqrt{2}\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{4}\pi+\tfrac{1}{2}% \pi x^{2}\right)\mathop{\mathrm{f}\/}\nolimits\!\left(-x\right)-2\sqrt{2}% \mathop{\cos\/}\nolimits\!\left(\tfrac{1}{4}\pi-\tfrac{1}{2}\pi x^{2}\right)% \mathop{\mathrm{g}\/}\nolimits\!\left(-x\right),$ $x\leq 0$.

See Figure 7.3.4.

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

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