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Statements
gd
(
x
)
=
2
arctan
(
e
x
)
-
1
2
π
=
arcsin
(
tanh
x
)
=
arccsc
(
coth
x
)
=
arccos
(
sech
x
)
=
arcsec
(
cosh
x
)
=
arctan
(
sinh
x
)
=
arccot
(
csch
x
)
.
Gudermannian
𝑥
2
superscript
𝑒
𝑥
1
2
𝜋
𝑥
hyperbolic-cotangent
𝑥
𝑥
𝑥
𝑥
𝑥
{\displaystyle{\displaystyle\operatorname{gd}\left(x\right)=2\operatorname{%
arctan}\left(e^{x}\right)-\tfrac{1}{2}\pi\\
=\operatorname{arcsin}\left(\tanh x\right)=\operatorname{arccsc}\left(\coth x%
\right)\\
=\operatorname{arccos}\left(\operatorname{sech}x\right)=\operatorname{arcsec}%
\left(\cosh x\right)\\
=\operatorname{arctan}\left(\sinh x\right)=\operatorname{arccot}\left(%
\operatorname{csch}x\right).}}
gd
x
Gudermannian
𝑥
{\displaystyle{\displaystyle\operatorname{gd}\NVar{x}}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
csch
z
𝑧
{\displaystyle{\displaystyle\operatorname{csch}\NVar{z}}}
cosh
z
𝑧
{\displaystyle{\displaystyle\cosh\NVar{z}}}
coth
z
hyperbolic-cotangent
𝑧
{\displaystyle{\displaystyle\coth\NVar{z}}}
sech
z
𝑧
{\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
sinh
z
𝑧
{\displaystyle{\displaystyle\sinh\NVar{z}}}
tanh
z
𝑧
{\displaystyle{\displaystyle\tanh\NVar{z}}}
arccsc
z
𝑧
{\displaystyle{\displaystyle\operatorname{arccsc}\NVar{z}}}
arccos
z
𝑧
{\displaystyle{\displaystyle\operatorname{arccos}\NVar{z}}}
arccot
z
𝑧
{\displaystyle{\displaystyle\operatorname{arccot}\NVar{z}}}
arcsec
z
𝑧
{\displaystyle{\displaystyle\operatorname{arcsec}\NVar{z}}}
arcsin
z
𝑧
{\displaystyle{\displaystyle\operatorname{arcsin}\NVar{z}}}
arctan
z
𝑧
{\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
x
𝑥
{\displaystyle{\displaystyle x}}
Identifiers
