The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers. Celle-ci donne une relation directe entre les fonctions circulaires et hyperboliques sans faire appel aux nombres complexes.
2.
That is, if t = tan 1 2 φ = tanh 1 2 θ {\displaystyle t=\tan {\tfrac {1}{2}}\varphi =\tanh {\tfrac {1}{2}}\theta } then φ = 2 tan − 1 tanh 1 2 θ ≡ gd θ . {\displaystyle \varphi =2\tan ^{-1}\tanh {\tfrac {1}{2}}\theta \equiv \operatorname {gd} \theta .} where gd(θ) is the Gudermannian function. C'est-à-dire que si t = tan φ 2 = tanh θ 2
