# Difference between revisions of "Sinh"

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− | The hyperbolic sine function is defined by | + | __NOTOC__ |

+ | The hyperbolic sine function $\mathrm{sinh} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by | ||

$$\mathrm{sinh}(z)=\dfrac{e^z-e^{-z}}{2}.$$ | $$\mathrm{sinh}(z)=\dfrac{e^z-e^{-z}}{2}.$$ | ||

+ | Since this function is [[one-to-one]], its [[inverse function]] the [[arcsinh|inverse hyperbolic sine]] function is clear. | ||

<div align="center"> | <div align="center"> | ||

<gallery> | <gallery> | ||

− | File: | + | File:Sinhplot.png|Graph of $\mathrm{sinh}$ on $[-5,5]$. |

+ | File:Complexsinhplot.png|[[Domain coloring]] of $\sinh$. | ||

</gallery> | </gallery> | ||

</div> | </div> | ||

=Properties= | =Properties= | ||

− | + | [[Derivative of sinh]]<br /> | |

− | + | [[Pythagorean identity for sinh and cosh]]<br /> | |

− | + | [[Relationship between sinh and hypergeometric 0F1]]<br /> | |

− | < | + | [[Weierstrass factorization of sinh]]<br /> |

− | < | + | [[Taylor series for sinh]]<br /> |

− | + | [[Relationship between Bessel I sub 1/2 and sinh]]<br /> | |

− | < | + | [[Relationship between sin and sinh]]<br /> |

− | < | + | [[Relationship between sinh and sin]]<br /> |

− | </ | + | [[Relationship between tangent, Gudermannian, and sinh]]<br /> |

− | </ | + | [[Relationship between sinh, inverse Gudermannian, and tan]]<br /> |

− | {{ | + | [[Period of sinh]]<br /> |

− | + | [[Sum of cosh and sinh]]<br /> | |

− | + | [[Difference of cosh and sinh]]<br /> | |

− | {{: | + | [[Sinh is odd]]<br /> |

+ | [[Sinh of a sum]]<br /> | ||

+ | [[Cosh of a sum]]<br /> | ||

+ | [[Halving identity for sinh]]<br /> | ||

+ | [[Halving identity for tangent (2)]]<br /> | ||

+ | [[Halving identity for tangent (3)]]<br /> | ||

+ | [[Doubling identity for sinh (1)]]<br /> | ||

+ | [[Doubling identity for sinh (2)]]<br /> | ||

+ | [[Doubling identity for cosh (2)]]<br /> | ||

+ | [[Doubling identity for cosh (3)]]<br /> | ||

+ | |||

+ | =See Also= | ||

+ | [[Sine]]<br /> | ||

+ | [[Arcsinh]] | ||

+ | |||

+ | =References= | ||

+ | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Cosh}}: $4.5.1$ | ||

+ | |||

+ | {{:Hyperbolic trigonometric functions footer}} | ||

− | + | [[Category:SpecialFunction]] |

## Latest revision as of 23:44, 21 October 2017

The hyperbolic sine function $\mathrm{sinh} \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by $$\mathrm{sinh}(z)=\dfrac{e^z-e^{-z}}{2}.$$ Since this function is one-to-one, its inverse function the inverse hyperbolic sine function is clear.

Domain coloring of $\sinh$.

# Properties

Derivative of sinh

Pythagorean identity for sinh and cosh

Relationship between sinh and hypergeometric 0F1

Weierstrass factorization of sinh

Taylor series for sinh

Relationship between Bessel I sub 1/2 and sinh

Relationship between sin and sinh

Relationship between sinh and sin

Relationship between tangent, Gudermannian, and sinh

Relationship between sinh, inverse Gudermannian, and tan

Period of sinh

Sum of cosh and sinh

Difference of cosh and sinh

Sinh is odd

Sinh of a sum

Cosh of a sum

Halving identity for sinh

Halving identity for tangent (2)

Halving identity for tangent (3)

Doubling identity for sinh (1)

Doubling identity for sinh (2)

Doubling identity for cosh (2)

Doubling identity for cosh (3)

# See Also

# References

- 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of mathematical functions*... (previous) ... (next): $4.5.1$

**Hyperbolic trigonometric functions**