Difference between revisions of "Gudermannian"
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| − | + | [[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 /> | |
Revision as of 05:58, 6 June 2016
The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} \mathrm{d}t$$
Graph of $\mathrm{gd}$.
Domain coloring of $\mathrm{gd}$.
Properties
Theorem: The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{gd}(x)=\mathrm{sech}(x),$$ where $\mathrm{gd}$ denotes the Gudermannian and $\mathrm{sech}$ denotes the hyperbolic secant.
Proof: █
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
