Difference between revisions of "Gudermannian"

From specialfunctionswiki
Jump to: navigation, search
(Properties)
Line 7: Line 7:
 
{{:Relationship between tangent, Gudermannian, and sinh}}
 
{{:Relationship between tangent, Gudermannian, and sinh}}
 
{{:Relationship between csc, Gudermannian, and coth}}
 
{{:Relationship between csc, Gudermannian, and coth}}
 +
{{:Relationship between secant, Gudermannian, and cosh}}
  
 
<center>{{:*-integral functions footer}}</center>
 
<center>{{:*-integral functions footer}}</center>

Revision as of 23:00, 25 August 2015

The Gudermannian $\mathrm{gd}$ is defined for $x \in \mathbb{R}$ by the formula $$\mathrm{gd}(x) = \displaystyle\int_0^x \dfrac{1}{\cosh t} dt$$

Properties

Theorem

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

Proof

References

Theorem

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

Proof

References

Theorem

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

Proof

References

Theorem

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

Proof

References

Theorem

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

Proof

References

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