hyperbolic sine integral


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

Shix:=0xsinhtt𝑑t, (1)

or alternatively as

Shix:=01sinhtxt𝑑t.

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

Shiz=z+z333!+z555!+z777!+,

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

Shix=iSiix

connects Shi to the sine integralDlmfDlmfDlmfMathworldPlanetmath function.

Shix satisfies the linear third differential equationMathworldPlanetmath

xf′′′(x)+2f′′(x)-xf(x)=0.
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