functions f(ϵ,ℓ;r),h(ϵ,ℓ;r)

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1: 7.4 Symmetry
$\mathrm{f}\left(iz\right)=(1/\sqrt{2})e^{\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{2% }}-i\mathrm{f}\left(z\right),$
$\mathrm{f}\left(-z\right)=\sqrt{2}\cos\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^% {2}\right)-\mathrm{f}\left(z\right),$
2: 7.5 Interrelations
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 $e^{\pm\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z% \right))=\tfrac{1}{2}(1\pm i)-(C\left(z\right)\pm iS\left(z\right)).$
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 $|\mathcal{F}\left(x\right)|^{2}=2+{\mathrm{f}}^{2}\left(-x\right)+{\mathrm{g}}% ^{2}\left(-x\right)-2\sqrt{2}\cos\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi x^{2}% \right)\mathrm{f}\left(-x\right)-2\sqrt{2}\cos\left(\tfrac{1}{4}\pi-\tfrac{1}{% 2}\pi x^{2}\right)\mathrm{g}\left(-x\right),$ $x\leq 0$.
3: 7.10 Derivatives
7.10.1 $\frac{{\mathrm{d}}^{n+1}\operatorname{erf}z}{{\mathrm{d}z}^{n+1}}=(-1)^{n}% \frac{2}{\sqrt{\pi}}H_{n}\left(z\right)e^{-z^{2}},$ $n=0,1,2,\dots$.
$\frac{\mathrm{d}\mathrm{f}\left(z\right)}{\mathrm{d}z}=-\pi z\mathrm{g}\left(z% \right),$
$\frac{\mathrm{d}\mathrm{g}\left(z\right)}{\mathrm{d}z}=\pi z\mathrm{f}\left(z% \right)-1.$
4: 8.22 Mathematical Applications
The so-called terminant function $F_{p}\left(z\right)$, defined by
8.22.1 $F_{p}\left(z\right)=\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left(z\right% )=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right),$
5: 7.24 Approximations
• Hastings (1955) gives several minimax polynomial and rational approximations for $\operatorname{erf}x$, $\operatorname{erfc}x$ and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$.

• Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

• Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi}\operatorname{erf}x$ and $e^{x^{2}}F\left(x\right)$ for $-3\leq x\leq 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi}xe^{x^{2}}\operatorname{erfc}x$ and $2xF\left(x\right)$ for $x\geq 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

• Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ for $x\geq 3$ (15D).

• Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$, $\operatorname{erf}z$, $\operatorname{erfc}z$, $C\left(z\right)$, and $S\left(z\right)$; approximate errors are given for a selection of $z$-values.

7: 7.7 Integral Representations
7.7.12 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-\pi iz^{2}/2}\int_{z}^{% \infty}e^{\pi it^{2}/2}\mathrm{d}t.$
7.7.13 $\mathrm{f}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i% \infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+\tfrac{1}{2}\right)\*\Gamma% \left(s+\tfrac{3}{4}\right)\Gamma\left(\tfrac{1}{4}-s\right)\mathrm{d}s,$
8: 2.11 Remainder Terms; Stokes Phenomenon
2.11.11 $F_{n+p}\left(z\right)=\frac{e^{-z}}{2\pi}\int_{0}^{\infty}\frac{e^{-zt}t^{n+p-% 1}}{1+t}\mathrm{d}t=\frac{\Gamma\left(n+p\right)}{2\pi}\frac{E_{n+p}\left(z% \right)}{z^{n+p-1}}.$
2.11.12 $F_{n+p}\left(z\right)=\frac{e^{-z}}{2\pi}\int_{0}^{\infty}\exp\left(-\rho\left% (te^{i\theta}-\ln t\right)\right)\frac{t^{\alpha-1}}{1+t}\mathrm{d}t.$
Owing to the factor $e^{-\rho}$, that is, $e^{-|z|}$ in (2.11.13), $F_{n+p}\left(z\right)$ is uniformly exponentially small compared with $E_{p}\left(z\right)$. … However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the $F$-functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the $F$-functions. In this context the $F$-functions are called terminants, a name introduced by Dingle (1973). …
9: 7.2 Definitions
$\operatorname{erf}z$, $\operatorname{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection. … $\mathcal{F}\left(z\right)$, $C\left(z\right)$, 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. …
7.2.10 $\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right),$