# auxiliary functions

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##### 2: 7.10 Derivatives
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$\frac{\mathrm{d}\mathrm{g}\left(z\right)}{\mathrm{d}z}=\pi z\mathrm{f}\left(z% \right)-1.$
##### 3: 7.5 Interrelations
###### §7.5 Interrelations
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7.5.3 $C\left(z\right)=\tfrac{1}{2}+\mathrm{f}\left(z\right)\sin\left(\tfrac{1}{2}\pi z% ^{2}\right)-\mathrm{g}\left(z\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right),$
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7.5.5 $e^{-\frac{1}{2}\pi iz^{2}}\mathcal{F}\left(z\right)=\mathrm{g}\left(z\right)+i% \mathrm{f}\left(z\right).$
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7.5.11 $|\mathcal{F}\left(x\right)|^{2}={\mathrm{f}}^{2}\left(x\right)+{\mathrm{g}}^{2% }\left(x\right),$ $x\geq 0$,
##### 4: 6.4 Analytic Continuation
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6.4.6 $\mathrm{f}\left(ze^{\pm\pi i}\right)=\pi e^{\mp iz}-\mathrm{f}\left(z\right),$
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6.4.7 $\mathrm{g}\left(ze^{\pm\pi i}\right)=\mp\pi ie^{\mp iz}+\mathrm{g}\left(z% \right).$
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##### 5: 7.4 Symmetry
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$\mathrm{g}\left(-z\right)=\sqrt{2}\sin\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^% {2}\right)-\mathrm{g}\left(z\right).$
##### 6: 7.2 Definitions
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###### §7.2(iv) AuxiliaryFunctions
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7.2.10 $\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right),$
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7.2.11 $\mathrm{g}\left(z\right)=\left(\tfrac{1}{2}-C\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-S\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right).$
##### 7: 6.11 Relations to Other Functions
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6.11.3 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=U\left(1,1,-iz\right).$
##### 8: 7.24 Approximations
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###### §7.24(i) Approximations in Terms of Elementary Functions
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• 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 $\mathrm{g}\left(x\right)$.

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• 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).

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##### 10: 6.20 Approximations
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• Hastings (1955) gives several minimax polynomial and rational approximations for $E_{1}\left(x\right)+\ln x$, $xe^{x}E_{1}\left(x\right)$, and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$. These are included in Abramowitz and Stegun (1964, Ch. 5).

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• MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$, with accuracies up to 20S.

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• Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function13.2(i)) from which Chebyshev expansions near infinity for $E_{1}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If $|\operatorname{ph}z|<\pi$ the scheme can be used in backward direction.