# auxiliary functions for sine and cosine integrals

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##### 2: 6.4 Analytic Continuation
6.4.6 $\mathrm{f}\left(ze^{\pm\pi i}\right)=\pi e^{\mp iz}-\mathrm{f}\left(z\right),$
6.4.7 $\mathrm{g}\left(ze^{\pm\pi i}\right)=\mp\pi ie^{\mp iz}+\mathrm{g}\left(z% \right).$
##### 3: 6.2 Definitions and Interrelations
###### §6.2(iii) AuxiliaryFunctions
6.2.17 $\mathrm{f}\left(z\right)=\phantom{+}\operatorname{Ci}\left(z\right)\sin z-% \operatorname{si}\left(z\right)\cos z,$
6.2.18 $\mathrm{g}\left(z\right)=-\operatorname{Ci}\left(z\right)\cos z-\operatorname{% si}\left(z\right)\sin z.$
6.2.19 $\operatorname{Si}\left(z\right)=\tfrac{1}{2}\pi-\mathrm{f}\left(z\right)\cos z% -\mathrm{g}\left(z\right)\sin z,$
6.2.20 $\operatorname{Ci}\left(z\right)=\mathrm{f}\left(z\right)\sin z-\mathrm{g}\left% (z\right)\cos z.$
##### 4: 6.11 Relations to Other Functions
6.11.3 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=U\left(1,1,-iz\right).$
##### 5: 6.20 Approximations
• 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).

• 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.

• 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.

• ##### 6: 6.5 Further Interrelations
6.5.7 $\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z\right)=E_{1}\left(\mp iz\right)% e^{\mp iz}.$
##### 8: 6.7 Integral Representations
###### §6.7(iii) AuxiliaryFunctions
6.7.12 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-iz}\int_{z}^{\infty}% \frac{e^{it}}{t}\,\mathrm{d}t,$ $|\operatorname{ph}z|\leq\pi.$
6.7.13 $\mathrm{f}\left(z\right)=\int_{0}^{\infty}\frac{\sin t}{t+z}\,\mathrm{d}t=\int% _{0}^{\infty}\frac{e^{-zt}}{t^{2}+1}\,\mathrm{d}t,$
6.7.15 $\mathrm{f}\left(z\right)=2\int_{0}^{\infty}K_{0}\left(2\sqrt{zt}\right)\cos t% \,\mathrm{d}t,$
6.7.16 $\mathrm{g}\left(z\right)=2\int_{0}^{\infty}K_{0}\left(2\sqrt{zt}\right)\sin t% \,\mathrm{d}t.$
##### 9: 6.12 Asymptotic Expansions
6.12.5 $\mathrm{f}\left(z\right)=\frac{1}{z}\sum_{m=0}^{n-1}(-1)^{m}\frac{(2m)!}{z^{2m% }}+R_{n}^{(\mathrm{f})}(z),$
##### 10: 8.21 Generalized Sine and Cosine Integrals
###### §8.21(vii) AuxiliaryFunctions
8.21.18 $f(a,z)=\operatorname{si}\left(a,z\right)\cos z-\operatorname{ci}\left(a,z% \right)\sin z,$
8.21.19 $g(a,z)=\operatorname{si}\left(a,z\right)\sin z+\operatorname{ci}\left(a,z% \right)\cos z.$
8.21.21 $\operatorname{ci}\left(a,z\right)=-f(a,z)\sin z+g(a,z)\cos z.$