# sine and cosine integrals

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##### 1: 8.21 Generalized Sine and Cosine Integrals
###### §8.21 Generalized Sine and CosineIntegrals
From here on it is assumed that unless indicated otherwise the functions $\operatorname{si}\left(a,z\right)$, $\operatorname{ci}\left(a,z\right)$, $\operatorname{Si}\left(a,z\right)$, and $\operatorname{Ci}\left(a,z\right)$ have their principal values. …
###### §8.21(v) Special Values
For the corresponding expansions for $\operatorname{si}\left(a,z\right)$ and $\operatorname{ci}\left(a,z\right)$ apply (8.21.20) and (8.21.21).
##### 2: 6.2 Definitions and Interrelations
###### Hyperbolic Analogs of the Sine and CosineIntegrals
6.2.17 $\mathrm{f}\left(z\right)=\phantom{+}\operatorname{Ci}\left(z\right)\sin z-% \operatorname{si}\left(z\right)\cos z,$
6.2.20 $\operatorname{Ci}\left(z\right)=\mathrm{f}\left(z\right)\sin z-\mathrm{g}\left% (z\right)\cos z.$
$\frac{\mathrm{d}\mathrm{f}\left(z\right)}{\mathrm{d}z}=-\mathrm{g}\left(z% \right),$
##### 3: 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 $\mathrm{g}\left(ze^{\pm\pi i}\right)=\mp\pi ie^{\mp iz}+\mathrm{g}\left(z% \right).$
Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_{1}\left(z\right)$, $\operatorname{Ci}\left(z\right)$, $\operatorname{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.
##### 5: 6.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the exponential integrals $\operatorname{Ei}\left(x\right)$, $E_{1}\left(z\right)$, and $\operatorname{Ein}\left(z\right)$; the logarithmic integral $\operatorname{li}\left(x\right)$; the sine integrals $\operatorname{Si}\left(z\right)$ and $\operatorname{si}\left(z\right)$; the cosine integrals $\operatorname{Ci}\left(z\right)$ and $\operatorname{Cin}\left(z\right)$.
##### 6: 6.5 Further Interrelations
###### §6.5 Further Interrelations
6.5.7 $\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z\right)=E_{1}\left(\mp iz\right)% e^{\mp iz}.$
##### 7: 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 and Wimp (1963) covers $\operatorname{Ei}\left(x\right)$ for $x\leq-4$ (20D), and $\operatorname{Si}\left(x\right)$ and $\operatorname{Ci}\left(x\right)$ for $x\geq 4$ (20D).

• Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\operatorname{Ein}\left(ax\right)$, $\operatorname{Si}\left(ax\right)$, and $\operatorname{Cin}\left(ax\right)$ for $-1\leq x\leq 1$, $a\in\mathbb{C}$. The coefficients are given in terms of series of Bessel functions.

• Luke (1969b, pp. 402, 410, and 415–421) gives main diagonal Padé approximations for $\operatorname{Ein}\left(z\right)$, $\operatorname{Si}\left(z\right)$, $\operatorname{Cin}\left(z\right)$ (valid near the origin), and $E_{1}\left(z\right)$ (valid for large $|z|$); approximate errors are given for a selection of $z$-values.

• ##### 8: 6.19 Tables
###### §6.19(ii) Real Variables
• Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\operatorname{Si}\left(x\right)$, $-x^{-2}\operatorname{Cin}\left(x\right)$, $x^{-1}\operatorname{Ein}\left(x\right)$, $-x^{-1}\operatorname{Ein}\left(-x\right)$, $x=0(.01)0.5$; $\operatorname{Si}\left(x\right)$, $\operatorname{Ci}\left(x\right)$, $\operatorname{Ei}\left(x\right)$, $E_{1}\left(x\right)$, $x=0.5(.01)2$; $\operatorname{Si}\left(x\right)$, $\operatorname{Ci}\left(x\right)$, $xe^{-x}\operatorname{Ei}\left(x\right)$, $xe^{x}E_{1}\left(x\right)$, $x=2(.1)10$; $x\mathrm{f}\left(x\right)$, $x^{2}\mathrm{g}\left(x\right)$, $xe^{-x}\operatorname{Ei}\left(x\right)$, $xe^{x}E_{1}\left(x\right)$, $x^{-1}=0(.005)0.1$; $\operatorname{Si}\left(\pi x\right)$, $\operatorname{Cin}\left(\pi x\right)$, $x=0(.1)10$. Accuracy varies but is within the range 8S–11S.

• Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$, $\operatorname{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\operatorname{Ei}\left(x\right)$, $E_{1}\left(x\right)$, $x=[0,100]$, 8S.

• ##### 9: 6.11 Relations to Other Functions
6.11.3 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=U\left(1,1,-iz\right).$
##### 10: 6.14 Integrals
###### §6.14(ii) Other Integrals
6.14.5 $\int_{0}^{\infty}\cos t\operatorname{Ci}\left(t\right)\,\mathrm{d}t=\int_{0}^{% \infty}\sin t\operatorname{si}\left(t\right)\,\mathrm{d}t=-\tfrac{1}{4}\pi,$