Chi - Maple Help

Si

The Sine Integral

Ci

The Cosine Integral

Ssi

The Shifted Sine Integral

Shi

The Hyperbolic Sine Integral

Chi

The Hyperbolic Cosine Integral

 Calling Sequence Si(x) Ci(x) Ssi(x) Shi(x) Chi(x)

Parameters

 x - expression

Description

 • These integrals are defined for all complex x as follows:

$\mathrm{Si}\left(x\right)={\int }_{0}^{x}\frac{\mathrm{sin}\left(t\right)}{t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt$

$\mathrm{Ci}\left(x\right)=\mathrm{\gamma }+\mathrm{ln}\left(x\right)+{\int }_{0}^{x}\frac{\mathrm{cos}\left(t\right)-1}{t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt$

$\mathrm{Ssi}\left(x\right)=\mathrm{Si}\left(x\right)-\frac{\mathrm{\pi }}{2}$

$\mathrm{Shi}\left(x\right)={\int }_{0}^{x}\frac{\mathrm{sinh}\left(t\right)}{t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt$

$\mathrm{Chi}\left(x\right)=\mathrm{\gamma }+\mathrm{ln}\left(x\right)+{\int }_{0}^{x}\frac{\mathrm{cosh}\left(t\right)-1}{t}ⅆt$

 • The functions Si, Ssi, and Shi are entire.  The functions Ci and Chi have a logarithmic singularity at the origin and have a branch cut along the negative real axis.

Examples

 > $\mathrm{Ci}\left(1.\right)$
 ${0.3374039229}$ (1)
 > $\mathrm{Ci}\left(3\right)$
 ${\mathrm{Ci}}{}\left({3}\right)$ (2)
 > $\mathrm{evalf}\left(\right)$
 ${0.1196297860}$ (3)
 > $\mathrm{Si}\left(3.14159+7.6I\right)$
 ${63.60695388}{-}{123.1816272}{}{I}$ (4)
 > $\mathrm{Ssi}\left(12345.67890\right)$
 ${-0.00005756635677}$ (5)
 > $\mathrm{Si}\left(12345.67890\right)$
 ${1.570738760}$ (6)
 > $\mathrm{Shi}\left(\mathrm{\pi }\right)$
 ${\mathrm{Shi}}{}\left({\mathrm{\pi }}\right)$ (7)
 > $\mathrm{Chi}\left(1.+I\right)$
 ${0.8821721806}{+}{1.283547193}{}{I}$ (8)
 > $\mathrm{convert}\left(\mathrm{Ci}\left(x\right),\mathrm{Ei}\right)$
 ${-}\frac{{{\mathrm{Ei}}}_{{1}}{}\left({I}{}{x}\right)}{{2}}{-}\frac{{{\mathrm{Ei}}}_{{1}}{}\left({-I}{}{x}\right)}{{2}}{+}\frac{{I}{}\left({\mathrm{csgn}}{}\left({x}\right){-}{1}\right){}{\mathrm{csgn}}{}\left({I}{}{x}\right){}{\mathrm{\pi }}}{{2}}$ (9)

References

 Abramowitz, M. and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover, 1972.