The Sine Integral - Maple Help

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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)
 > $\mathrm{Shi}\left(\mathrm{π}\right)$
 ${\mathrm{Shi}}{}\left({\mathrm{π}}\right)$ (7)
 > $\mathrm{Chi}\left(1.+I\right)$
 ${0.8821721806}{+}{1.283547193}{}{I}$ (8)
 > $\mathrm{convert}\left(\mathrm{Ci}\left(x\right),\mathrm{Ei}\right)$
 ${-}\frac{{1}}{{2}}{}{\mathrm{Ei}}{}\left({1}{,}{I}{}{x}\right){-}\frac{{1}}{{2}}{}{\mathrm{Ei}}{}\left({1}{,}{-}{I}{}{x}\right){+}\frac{{1}}{{2}}{}{I}{}\left({\mathrm{csgn}}{}\left({x}\right){-}{1}\right){}{\mathrm{csgn}}{}\left({I}{}{x}\right){}{\mathrm{π}}$ (9)
 See Also

References

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

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