Difference between revisions of "Gudermannian"
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The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula | The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula | ||
| − | $$\mathrm{gd}(x) = \displaystyle\int_0^x \ | + | $$\mathrm{gd}(x) = \displaystyle\int_0^x \mathrm{sech}(t) \mathrm{d}t,$$ |
| + | where $\mathrm{sech}$ denotes the [[sech|hyperbolic secant]]. | ||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Gudermannianplot.png|Graph of $\mathrm{gd}$. | ||
| + | File:Domcolgudermannian.png|[[Domain coloring]] of $\mathrm{gd}$. | ||
| + | </gallery> | ||
| + | </div> | ||
=Properties= | =Properties= | ||
| − | + | [[Derivative of Gudermannian]]<br /> | |
| + | [[Taylor series for Gudermannian]]<br /> | ||
| + | [[Relationship between sine, Gudermannian, and tanh]]<br /> | ||
| + | [[Relationship between cosine, Gudermannian, and sech]]<br /> | ||
| + | [[Relationship between tangent, Gudermannian, and sinh]]<br /> | ||
| + | [[Relationship between csc, Gudermannian, and coth]]<br /> | ||
| + | [[Relationship between secant, Gudermannian, and cosh]]<br /> | ||
| + | [[Relationship between cot, Gudermannian, and csch]]<br /> | ||
| + | |||
| + | {{:*-integral functions footer}} | ||
| − | + | [[Category:SpecialFunction]] | |
Latest revision as of 22:07, 19 September 2016
The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula $$\mathrm{gd}(x) = \displaystyle\int_0^x \mathrm{sech}(t) \mathrm{d}t,$$ where $\mathrm{sech}$ denotes the hyperbolic secant.
Graph of $\mathrm{gd}$.
Domain coloring of $\mathrm{gd}$.
Properties
Derivative of Gudermannian
Taylor series for Gudermannian
Relationship between sine, Gudermannian, and tanh
Relationship between cosine, Gudermannian, and sech
Relationship between tangent, Gudermannian, and sinh
Relationship between csc, Gudermannian, and coth
Relationship between secant, Gudermannian, and cosh
Relationship between cot, Gudermannian, and csch
