# Integral sine

The special function defined for real $x$ by

$$\operatorname{Si}(x)=\int\limits_0^x\frac{\sin t}{t}dt.$$

For $x>0$ one has

$$\operatorname{Si}(x)=\frac\pi2-\int\limits_x^\infty\frac{\sin t}{t}dt.$$

One sometimes uses the notation

$$\operatorname{si}(x)=-\int\limits_x^\infty\frac{\sin t}{t}dt\equiv\operatorname{Si}(x)-\frac\pi2.$$

Some particular values are:

$$\operatorname{Si}(0)=0,\quad\operatorname{Si}(\infty)=\frac\pi2,\quad\operatorname{si}(\infty)=0.$$

Some special relations:

$$\operatorname{Si}(-x)=-\operatorname{Si}(x);\quad\operatorname{si}(x)+\operatorname{si}(-x)=-\pi;$$

$$\int\limits_0^\infty\operatorname{si}^2(t)dt=\frac\pi2;\quad\int\limits_0^\infty e^{-pt}\operatorname{si}(qt)dt=-\frac1p\arctan\frac pq;$$

$$\int\limits_0^\infty\sin t\operatorname{si}(t)dt=-\frac\pi4;\quad\int\limits_0^\infty\operatorname{Ci}(t)\operatorname{si}(t)dt=-\ln2,$$

where $\operatorname{Ci}(t)$ is the integral cosine. For $x$ small,

$$\operatorname{Si}(x)\approx x.$$

The asymptotic representation for large $x$ is

$$\operatorname{Si}(x)=\frac\pi2-\frac{\cos x}{x}P(x)-\frac{\sin x}{x}Q(x),$$

where

$$P(x)\sim\sum_{k=0}^\infty\frac{(-1)^k(2k)!}{x^{2k}},$$

$$Q(x)\sim\sum_{k=0}^\infty\frac{(-1)^k(2k+1)!}{x^{2k+1}}.$$

The integral sine has the series representation

$$\operatorname{Si}(x)=x-\frac{x^3}{3!3}+\ldots+(-1)^k\frac{x^{2k+1}}{(2k+1)!(2k+1)}+\ldots.\tag{*}$$

As a function of the complex variable $z$, $\operatorname{Si}(z)$, defined by (*), is an entire function of $z$ in the $z$-plane.

The integral sine is related to the integral exponential function $\operatorname{Ei}(z)$ by

$$\operatorname{si}(z)=\frac{1}{2i}[\operatorname{Ei}(iz)-\operatorname{Ei}(-iz)].$$