グーデルマン関数
(逆三角関数を参照。)
以下の恒等式が成り立つ。
sin  gd  x ̇ = tanh  x ; csc  gd  x = coth  x ; cos  gd  x = sech  x ; sec  gd  x = cosh  x ; tan  gd  x = sinh  x ; cot  gd  x = csch x ; . tan  1 2 gd x = tanh  1 2 x . {\displaystyle {\begin{aligned}{\color {white}{\dot {\color {black}\sin \operatorname {gd} x}}}&=\tanh x;\quad \csc \operatorname {gd} x=\coth x;\\\cos \operatorname {gd} x&=\operatorname {sech} x;\quad \,\sec \operatorname {gd} x=\cosh x;\\\tan \operatorname {gd} x&=\sinh x;\quad \,\cot \operatorname {gd} x=\operatorname {csch} \,x;\\{}_{\color {white}.}\tan {\tfrac {1}{2}}\operatorname {gd} \,x&=\tanh {\tfrac {1}{2}}x.\end{aligned}}}