Difference between revisions of "Cosh"
(→Properties) |
|||
| (29 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | The hyperbolic cosine function is defined by | + | __NOTOC__ |
| − | $$\mathrm{cosh}(z)=\dfrac{e^z + e^{-z}}{2}$$ | + | |
| + | The hyperbolic cosine function $\cosh \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by | ||
| + | $$\mathrm{cosh}(z)=\dfrac{e^z + e^{-z}}{2}.$$ | ||
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
| − | File: | + | File:Coshplot.png|Graph of $\cosh$. |
| + | File:Complexcoshplot.png|[[Domain coloring]] of [[analytic continuation]] of $\cosh$. | ||
</gallery> | </gallery> | ||
</div> | </div> | ||
=Properties= | =Properties= | ||
| − | + | [[Derivative of cosh]]<br /> | |
| − | + | [[Pythagorean identity for sinh and cosh]]<br /> | |
| − | + | [[Weierstrass factorization of cosh]]<br /> | |
| + | [[Relationship between cosh and hypergeometric 0F1]]<br /> | ||
| + | [[Relationship between Bessel I sub 1/2 and cosh]]<br /> | ||
| + | [[Relationship between cosh and cos]]<br /> | ||
| + | [[Relationship between cos and cosh]]<br /> | ||
| + | [[Relationship between secant, Gudermannian, and cosh]]<br /> | ||
| + | [[Relationship between cosh, inverse Gudermannian, and sec]]<br /> | ||
| + | [[Period of cosh]]<br /> | ||
| + | [[Sum of cosh and sinh]]<br /> | ||
| + | [[Difference of cosh and sinh]]<br /> | ||
| + | [[Cosh is even]]<br /> | ||
| + | [[Sinh of a sum]]<br /> | ||
| + | [[Cosh of a sum]]<br /> | ||
| + | [[Halving identity for sinh]]<br /> | ||
| + | [[Halving identity for cosh]]<br /> | ||
| + | [[Halving identity for tangent (1)]]<br /> | ||
| + | [[Halving identity for tangent (2)]]<br /> | ||
| + | [[Halving identity for tangent (3)]]<br /> | ||
| + | [[Doubling identity for sinh (1)]]<br /> | ||
| + | [[Doubling identity for cosh (1)]]<br /> | ||
| + | [[Doubling identity for cosh (2)]]<br /> | ||
| + | [[Doubling identity for cosh (3)]]<br /> | ||
| + | |||
| + | =See Also= | ||
| + | [[Arccosh]] | ||
| + | |||
| + | =References= | ||
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Sinh|next=Tanh}}: $4.5.2$ | ||
| + | |||
| + | {{:Hyperbolic trigonometric functions footer}} | ||
| − | + | [[Category:SpecialFunction]] | |
Latest revision as of 23:44, 21 October 2017
The hyperbolic cosine function $\cosh \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by
$$\mathrm{cosh}(z)=\dfrac{e^z + e^{-z}}{2}.$$
Graph of $\cosh$.
Domain coloring of analytic continuation of $\cosh$.
Properties
Derivative of cosh
Pythagorean identity for sinh and cosh
Weierstrass factorization of cosh
Relationship between cosh and hypergeometric 0F1
Relationship between Bessel I sub 1/2 and cosh
Relationship between cosh and cos
Relationship between cos and cosh
Relationship between secant, Gudermannian, and cosh
Relationship between cosh, inverse Gudermannian, and sec
Period of cosh
Sum of cosh and sinh
Difference of cosh and sinh
Cosh is even
Sinh of a sum
Cosh of a sum
Halving identity for sinh
Halving identity for cosh
Halving identity for tangent (1)
Halving identity for tangent (2)
Halving identity for tangent (3)
Doubling identity for sinh (1)
Doubling identity for cosh (1)
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): $4.5.2$
