Shi

$\begin{figure}\begin{center}\BoxedEPSF{Shi.epsf}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{ShiReIm.epsf scaled 700}\end{center}\end{figure}$

$\begin{displaymath} \mathop{\rm Shi}(z)=\int_0^z {\sinh t\over t}\,dt. \end{displaymath}$

The function is given by the Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) command SinhIntegral[z].

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Sine and Cosine Integrals.'' §5.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231-233, 1972.