# sine integral

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##### 1: 6.2 Definitions and Interrelations
###### §6.2(ii) Sine and Cosine Integrals
$\operatorname{Si}\left(z\right)$ is an odd entire function. …
6.2.17 $\mathrm{f}\left(z\right)=\phantom{+}\operatorname{Ci}\left(z\right)\sin z-% \operatorname{si}\left(z\right)\cos z,$
6.2.19 $\operatorname{Si}\left(z\right)=\tfrac{1}{2}\pi-\mathrm{f}\left(z\right)\cos z% -\mathrm{g}\left(z\right)\sin z,$
$\frac{\mathrm{d}\mathrm{f}\left(z\right)}{\mathrm{d}z}=-\mathrm{g}\left(z% \right),$
##### 2: 8.21 Generalized Sine and Cosine Integrals
###### §8.21 Generalized Sine and Cosine Integrals
$\operatorname{si}\left(0,z\right)=-\operatorname{si}\left(z\right),$
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).
##### 3: 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 $\operatorname{Cin}\left(z\right)$.
##### 5: 6.5 Further Interrelations
###### §6.5 Further Interrelations
6.5.7 $\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z\right)=E_{1}\left(\mp iz\right)% e^{\mp iz}.$
##### 6: 6.14 Integrals
###### §6.14(i) Laplace Transforms
6.14.3 $\int_{0}^{\infty}e^{-at}\operatorname{si}\left(t\right)\,\mathrm{d}t=-\frac{1}% {a}\operatorname{arctan}a,$ $\Re a>0$.
###### §6.14(ii) Other Integrals
6.14.6 $\int_{0}^{\infty}{\operatorname{Ci}}^{2}\left(t\right)\,\mathrm{d}t=\int_{0}^{% \infty}{\operatorname{si}}^{2}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}\pi,$
##### 7: 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.

• ##### 8: 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.

• ##### 9: 6.15 Sums
###### §6.15 Sums
6.15.2 $\sum_{n=1}^{\infty}\frac{\operatorname{si}\left(\pi n\right)}{n}=\tfrac{1}{2}% \pi(\ln\pi-1),$
6.15.4 $\sum_{n=1}^{\infty}(-1)^{n}\frac{\operatorname{si}\left(2\pi n\right)}{n}=\pi(% \tfrac{3}{2}\ln 2-1).$
##### 10: 6.11 Relations to Other Functions
6.11.3 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=U\left(1,1,-iz\right).$