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:Sinhplot.png| | + | File:Sinhplot.png|Graph of $\mathrm{sinh}$ on $[-5,5]$. |
| − | File: | + | 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= | =See Also= | ||
| Line 30: | Line 40: | ||
[[Arcsinh]] | [[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.
Graph of $\mathrm{sinh}$ on $[-5,5]$.
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): $4.5.1$
