Difference between revisions of "Inverse Gudermannian"

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{{:Relationship between csch, inverse Gudermannian, and cot}}
 
{{:Relationship between csch, inverse Gudermannian, and cot}}
 
{{:Relationship between sech, inverse Gudermannian, and cos}}
 
{{:Relationship between sech, inverse Gudermannian, and cos}}
 
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{{:Relationship between coth, inverse Gudermannian, and csc}}
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<strong>Theorem:</strong> The following formula holds:
 
$$\mathrm{coth}(\mathrm{gd}^{-1}(x))=\csc(x),$$
 
where $\mathrm{coth}$ is the [[coth|hyperbolic cotangent]], $\mathrm{gd}^{-1}$ is the [[inverse Gudermannian]], and $\csc$ is the [[cosecant]].
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
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<center>{{:*-integral functions footer}}</center>
 
<center>{{:*-integral functions footer}}</center>

Revision as of 23:37, 25 August 2015

The inverse Gudermannian $\mathrm{gd}^{-1}$ is the inverse function of the Gudermannian function. It may be defined by the following formula for $x \in \mathbb{R}$: $$\mathrm{gd}^{-1}(x)=\displaystyle\int_0^x \dfrac{1}{\cosh(t)} dt,$$ where $\cosh$ denotes the hyperbolic cosine.

Properties

Theorem

The following formula holds: $$\sinh(\mathrm{gd}^{-1}(x))=\tan(x),$$ where $\sinh$ is the hyperbolic sine, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\tan$ is the tangent.

Proof

References

Theorem

The following formula holds: $$\cosh(\mathrm{gd}^{-1}(x))=\sec(x),$$ where $\cosh$ is the hyperbolic cosine, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\sec$ is the secant.

Proof

References

Theorem

The following formula holds: $$\mathrm{tanh}(\mathrm{gd}^{-1}(x))=\sin(x),$$ where $\mathrm{tanh}$ denotes the hyperbolic tangent, $\mathrm{gd}^{-1}$ denotes the inverse Gudermannian, and $\sin$ denotes sine.

Proof

References

Theorem

The following formula holds: $$\mathrm{csch}(\mathrm{gd}^{-1}(x))=\cot(x),$$ where $\mathrm{csch}$ is the hyperbolic cosecant, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\cot$ is the cotangent.

Proof

References

Theorem

The following formula holds: $$\mathrm{sech}(\mathrm{gd}^{-1}(x))=\cos(x),$$ where $\mathrm{sech}$ is the hyperbolic secant, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\cos$ is the cosine.

Proof

References

Theorem

The following formula holds: $$\mathrm{coth}(\mathrm{gd}^{-1}(x))=\csc(x),$$ where $\mathrm{coth}$ is the hyperbolic cotangent, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\csc$ is the cosecant.

Proof

References

<center>$\ast$-integral functions
</center>