hyperbolic sine integral

The functionMathworldPlanetmath hyperbolic sine integralDlmfMathworldPlanetmath (in Latin sinus hyperbolicus integralis) from to is defined as

Shix:=0xsinhtt𝑑t, (1)

or alternatively as


It isn’t an elementary functionMathworldPlanetmath.  The equation (1) implies the Taylor seriesMathworldPlanetmath expansion


which converges for all complex values z and thus defines an entire transcendental function.  Using the Taylor expansions, it is easily seen that


connects Shi to the sine integralDlmfDlmfDlmfMathworldPlanetmath function.

Shix satisfies the linear third differential equationMathworldPlanetmath

Title hyperbolic sine integral
Canonical name HyperbolicSineIntegral
Date of creation 2013-03-22 18:27:48
Last modified on 2013-03-22 18:27:48
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Definition
Classification msc 30A99
Synonym Shi
Related topic HyperbolicFunctions
Related topic SineIntegral