# Integral cosine

The special function defined, for real $x > 0$, by

$$\mathop{\rm Ci} ( x) = - \int\limits _ { x } ^ \infty \frac{\cos t }{t } \ d t = c + \mathop{\rm ln} x - \int\limits _ { 0 } ^ { x } \frac{1 - \cos t }{t } \ d t ,$$

where $c = 0.5772 \dots$ is the Euler constant. Its graph is:

Figure: i051370a

The graphs of the functions $y = \mathop{\rm ci} ( x)$ and $y = \mathop{\rm si} ( x)$.

Some integrals related to the integral cosine are:

$$\int\limits _ { 0 } ^ \infty e ^ {- p t } \mathop{\rm Ci} ( q t ) d t = - \frac{1}{2p} \mathop{\rm ln} \left ( 1 + \frac{p ^ {2} }{q ^ {2} } \right ) ,$$

$$\int\limits _ { 0 } ^ \infty \cos t \mathop{\rm Ci} ( t) \ d t = - \frac \pi {4} ,\ \int\limits _ { 0 } ^ \infty \mathop{\rm Ci} ^ {2} ( t) d t = \frac \pi {2} ,$$

$$\int\limits _ { 0 } ^ \infty \mathop{\rm Ci} ( t) \mathop{\rm si} ( t) d t = - \mathop{\rm ln} 2 ,$$

where $\mathop{\rm si} ( t)$ is the integral sine minus $\pi / 2$.

For $x$ small:

$$\mathop{\rm Ci} ( x) \approx c + \mathop{\rm ln} x .$$

The asymptotic representation, for $x$ large, is:

$$\mathop{\rm Ci} ( x) = \ \frac{\sin x }{x} P ( x) - \frac{\cos x }{x} Q ( x) ,$$

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

The integral cosine has the series representation:

$$\tag{* } \mathop{\rm Ci} ( x) = c + \mathop{\rm ln} x - \frac{x ^ {2} }{2!2} + \frac{x ^ {4} }{4!4} - \dots +$$

$$+ ( - 1 ) ^ {k} \frac{x ^ {2k} }{( 2 k ) ! 2 k } + \dots .$$

As a function of the complex variable $z$, $\mathop{\rm Ci} ( z)$, defined by (*), is a single-valued analytic function in the $z$- plane with slit along the relative negative real axis $( - \pi < \mathop{\rm arg} z < \pi )$. The value of $\mathop{\rm ln} z$ here is taken to be $\pi < \mathop{\rm Im} \mathop{\rm ln} z < \pi$. The behaviour of $\mathop{\rm Ci} ( z)$ near the slit is determined by the limits

$$\lim\limits _ {\eta \downarrow 0 } \mathop{\rm Ci} ( x \pm i \eta ) = \ \mathop{\rm Ci} ( | z | ) \pm \pi i ,\ x < 0 .$$

The integral cosine is related to the integral exponential function $\mathop{\rm Ei} ( z)$ by

$$\mathop{\rm Ci} ( z) = \frac{1}{2} [ \mathop{\rm Ei} ( i z ) + \mathop{\rm Ei} ( - i z ) ] .$$

One sometimes uses the notation $\mathop{\rm ci} ( x) \equiv \mathop{\rm Ci} ( x)$.

The function $\mathop{\rm Ci}$ is better known as the cosine integral. It can, of course, be defined by the integral (as above) in $\mathbf C \setminus \{ {x \in \mathbf R } : {x \leq 0 } \}$.