# SHI Hyperbolic Sine Integral

## SHI.1 Introduction |
top up back next into bottom |

Let be a complex variable of .The function Hyperbolic Sine Integral (noted ) is defined by the following third order differential equation

SHI.1.1 |

Although is a singularity of SHI.1.1, the initial conditions can be given by

SHI.1.2 |

Related function: Hyperbolic Cosine Integral

## SHI.2 Series and asymptotic expansions |
top up back next into bottom |

## SHI.2.1 Asymptotic expansion at |
top up back next into bottom |

## SHI.2.1.2 General form |
top up back next into bottom |

#### SHI.2.1.2.1 Auxiliary function

The auxiliary function has the exact form#### SHI.2.1.2.2 Auxiliary function

The coefficients of satisfy the following recurrence#### SHI.2.1.2.3 Auxiliary function

The coefficients of satisfy the following recurrence
## SHI.2.2 Asymptotic expansion at |
top up back next into bottom |

## SHI.2.2.2 General form |
top up back next into bottom |

SHI.2.2.2.1 |

SHI.2.2.2.2 |

SHI.2.2.2.3 |

SHI.2.2.2.4 |