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
$$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} \mathrm{d}t$$
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$$\mathrm{gd}(x) = \displaystyle\int_0^x \mathrm{sech}(t) \mathrm{d}t,$$
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where $\mathrm{sech}$ denotes the [[sech|hyperbolic secant]].
  
 
<div align="center">
 
<div align="center">
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=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
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[[Derivative of Gudermannian]]<br />
<strong>Theorem:</strong> The following formula holds:
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[[Taylor series for Gudermannian]]<br />
$$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{gd}(x)=\mathrm{sech}(x),$$
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[[Relationship between sine, Gudermannian, and tanh]]<br />
where $\mathrm{gd}$ denotes the [[Gudermannian]] and $\mathrm{sech}$ denotes the [[sech|hyperbolic secant]].
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[[Relationship between cosine, Gudermannian, and sech]]<br />
<div class="mw-collapsible-content">
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[[Relationship between tangent, Gudermannian, and sinh]]<br />
<strong>Proof:</strong>
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[[Relationship between csc, Gudermannian, and coth]]<br />
</div>
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[[Relationship between secant, Gudermannian, and cosh]]<br />
</div>
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[[Relationship between cot, Gudermannian, and csch]]<br />
{{: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}}
 
  
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{{:*-integral functions footer}}
  
<center>{{:*-integral functions footer}}</center>
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[[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.

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

$\ast$-integral functions