§7.5 Interrelations

Notes:: (7.5.1) follows from (7.2.1)–(7.2.3). (7.5.2) follows from (7.2.6)–(7.2.9). (7.5.3) and (7.5.4) follow from (7.2.10) and (7.2.11). (7.5.5) and (7.5.6) follow from (7.2.10), (7.2.11), and (7.5.2). (7.5.8) follows from (7.2.1), (7.2.7), and (7.2.8). (7.5.9) follows from (7.2.2), (7.2.3), (7.5.7), and (7.5.8). (7.5.10) follows from (7.2.2), (7.5.6), and (7.5.8). (7.5.11) follows from (7.5.5). (7.5.12) follows from (7.4.8) and (7.5.11). For (7.5.13) see Olver (1997b, p. 44).
Keywords:: Dawson’s integral, Fresnel integrals, Goodwin–Staton integral, error functions
Permalink:: http://dlmf.nist.gov/7.5

7.5.1 $\mathop{F\/}\nolimits\!\left(z\right)=\tfrac{1}{2}i\sqrt{\pi}\left(e^{{-z^{2}}}-\mathop{w\/}\nolimits\!\left(z\right)\right)=-\tfrac{1}{2}i\sqrt{\pi}e^{{-z^{2}}}\mathop{\mathrm{erf}\/}\nolimits\!\left(iz\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral, $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, $\mathop{w\/}\nolimits\!\left(z\right)$ : complementary error function, $e$ : base of exponential function and $z$ : complex variable
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.5.E1
Encodings:: TeX, pMML, png

7.5.2 $\mathop{C\/}\nolimits\!\left(z\right)+i\mathop{S\/}\nolimits\!\left(z\right)=\tfrac{1}{2}(1+i)-\mathop{\mathcal{F}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral and $z$ : complex variable
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.5.E2
Encodings:: TeX, pMML, png

7.5.3 $\mathop{C\/}\nolimits\!\left(z\right)=\tfrac{1}{2}+\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right)-\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right),$

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 7.3.9
Referenced by:: §7.12(ii), §7.5
Permalink:: http://dlmf.nist.gov/7.5.E3
Encodings:: TeX, pMML, png

7.5.4 $\mathop{S\/}\nolimits\!\left(z\right)=\tfrac{1}{2}-\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right)-\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right).$

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 7.3.10
Referenced by:: §7.12(ii), §7.5
Permalink:: http://dlmf.nist.gov/7.5.E4
Encodings:: TeX, pMML, png

7.5.5 $e^{{-\frac{1}{2}\pi iz^{2}}}\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)+i\mathop{\mathrm{f}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$ : Fresnel integral, $e$ : base of exponential function and $z$ : complex variable
Referenced by:: §7.13(iv), §7.5, §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.5.E5
Encodings:: TeX, pMML, png

7.5.6 $e^{{\pm\frac{1}{2}\pi iz^{2}}}(\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\pm i\mathop{\mathrm{f}\/}\nolimits\!\left(z\right))=\tfrac{1}{2}(1\pm i)-(\mathop{C\/}\nolimits\!\left(z\right)\pm i\mathop{S\/}\nolimits\!\left(z\right)).$

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $e$ : base of exponential function and $z$ : complex variable
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.5.E6
Encodings:: TeX, pMML, png

In (7.5.8)–(7.5.10)

7.5.7 $\zeta=\tfrac{1}{2}\sqrt{\pi}(1\mp i)z,$

Symbols:: $z$ : complex variable and $\zeta$ : change of variable
Referenced by:: §7.12(ii), §7.5
Permalink:: http://dlmf.nist.gov/7.5.E7
Encodings:: TeX, pMML, png

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

7.5.8 $\mathop{C\/}\nolimits\!\left(z\right)\pm i\mathop{S\/}\nolimits\!\left(z\right)=\tfrac{1}{2}(1\pm i)\mathop{\mathrm{erf}\/}\nolimits\zeta.$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{\mathrm{erf}\/}\nolimits z$ : 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

7.5.9 $\mathop{C\/}\nolimits\!\left(z\right)\pm i\mathop{S\/}\nolimits\!\left(z\right)=\tfrac{1}{2}(1\pm i)\left(1-e^{{\pm\frac{1}{2}\pi iz^{2}}}\mathop{w\/}\nolimits\!\left(i\zeta\right)\right).$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{w\/}\nolimits\!\left(z\right)$ : complementary error function, $e$ : base of exponential function, $z$ : complex variable and $\zeta$ : change of variable
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.5.E9
Encodings:: TeX, pMML, png

7.5.10 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\pm i\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}(1\pm i)e^{{\zeta^{2}}}\mathop{\mathrm{erfc}\/}\nolimits\zeta.$

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $e$ : 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

7.5.11 $|\mathop{\mathcal{F}\/}\nolimits\!\left(x\right)|^{2}={\mathop{\mathrm{f}\/}\nolimits^{{2}}}\!\left(x\right)+{\mathop{\mathrm{g}\/}\nolimits^{{2}}}\!\left(x\right),$ $x\geq 0$ ,

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$ : Fresnel integral and $x$ : real variable
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.5.E11
Encodings:: TeX, pMML, png

7.5.12 $|\mathop{\mathcal{F}\/}\nolimits\!\left(x\right)|^{2}=2+{\mathop{\mathrm{f}\/}\nolimits^{{2}}}\!\left(-x\right)+{\mathop{\mathrm{g}\/}\nolimits^{{2}}}\!\left(-x\right)-2\sqrt{2}\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi x^{2}\right)\mathop{\mathrm{f}\/}\nolimits\!\left(-x\right)-2\sqrt{2}\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{4}\pi-\tfrac{1}{2}\pi x^{2}\right)\mathop{\mathrm{g}\/}\nolimits\!\left(-x\right),$ $x\leq 0$ .

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{\cos\/}\nolimits z$ : cosine function and $x$ : real variable
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.5.E12
Encodings:: TeX, pMML, png

See Figure 7.3.4.

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

Symbols:: $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral, $\mathop{G\/}\nolimits\!\left(z\right)$ : Goodwin–Staton integral, $e$ : base of exponential function, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral and $x$ : real variable
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.5.E13
Encodings:: TeX, pMML, png

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