Difference between revisions of "Tanh"
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The hyperbolic tangent is defined by the formula | The hyperbolic tangent is defined by the formula | ||
$$\mathrm{tanh}(z)=\dfrac{\mathrm{sinh}(z)}{\mathrm{cosh}(z)},$$ | $$\mathrm{tanh}(z)=\dfrac{\mathrm{sinh}(z)}{\mathrm{cosh}(z)},$$ | ||
where $\mathrm{sinh}$ is the [[sinh|hyperbolic sine]] and $\mathrm{cosh}$ is the [[cosh|hyperbolic cosine]]. | where $\mathrm{sinh}$ is the [[sinh|hyperbolic sine]] and $\mathrm{cosh}$ is the [[cosh|hyperbolic cosine]]. | ||
| − | [ | + | <div align="center"> |
| + | <gallery> | ||
| + | File:Tanhplot.png|Plot of $\mathrm{tanh}$ on $[-5,5]$. | ||
| + | File:Complextanhplot.png|[[Domain coloring]] of $\tanh$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | [[Derivative of tanh]]<br /> | ||
| + | [[Antiderivative of tanh]]<br /> | ||
| + | [[Relationship between tanh and tan]]<br /> | ||
| + | [[Relationship between tan and tanh]]<br /> | ||
| + | [[Relationship between sine, Gudermannian, and tanh]]<br /> | ||
| + | [[Relationship between tanh, inverse Gudermannian, and sin]]<br /> | ||
| + | [[Taylor series for Gudermannian]]<br /> | ||
| + | [[Pythagorean identity for tanh and sech]]<br /> | ||
| + | [[Period of tanh]]<br /> | ||
| + | [[Tanh is odd]]<br /> | ||
| + | [[Tanh of a sum]]<br /> | ||
| + | [[Halving identity for tangent (1)]]<br /> | ||
| + | [[Halving identity for tangent (2)]]<br /> | ||
| + | [[Halving identity for tangent (3)]]<br /> | ||
| + | [[Doubling identity for sinh (2)]]<br /> | ||
| + | |||
| + | =See Also= | ||
| + | [[Arctan]]<br /> | ||
| + | [[Arctanh]]<br /> | ||
| + | [[Tangent]]<br /> | ||
| + | |||
| + | =References= | ||
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Cosh|next=Csch}}: $4.5.3$ | ||
| + | |||
| + | {{:Hyperbolic trigonometric functions footer}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 23:43, 21 October 2017
The hyperbolic tangent is defined by the formula $$\mathrm{tanh}(z)=\dfrac{\mathrm{sinh}(z)}{\mathrm{cosh}(z)},$$ where $\mathrm{sinh}$ is the hyperbolic sine and $\mathrm{cosh}$ is the hyperbolic cosine.
Plot of $\mathrm{tanh}$ on $[-5,5]$.
Domain coloring of $\tanh$.
Properties
Derivative of tanh
Antiderivative of tanh
Relationship between tanh and tan
Relationship between tan and tanh
Relationship between sine, Gudermannian, and tanh
Relationship between tanh, inverse Gudermannian, and sin
Taylor series for Gudermannian
Pythagorean identity for tanh and sech
Period of tanh
Tanh is odd
Tanh of a sum
Halving identity for tangent (1)
Halving identity for tangent (2)
Halving identity for tangent (3)
Doubling identity for sinh (2)
See Also
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.5.3$
