Função gudermanniana

Predefinição:Sem fontes A função gudermanniana, chamada assim em homenagem a Christoph Gudermann (1798 - 1852), relaciona as funções trigonométricas e as funções hiperbólicas.

Definição:

   $\begin{array}{cc}\mathrm{g}\mathrm{d}(x)& =\underset{0}{\overset{x}{\int }}\frac{dp}{\cosh (p)},\\ & =\arcsin (\tanh (x))=\arccos (\text{sech}(x)),\\ & =\arctan (\sinh (x))=\text{arcsec}(\cosh (x)),\\ & =\text{arccot}(\text{csch}(x))=\text{arccsc}(\coth (x)),\\ & =2\arctan (\tanh \left(\frac{x}{2}\right))=2\arctan (e^x)-\frac{\pi }{2}.\end{array}$

Identidades envolvendo gd(x) :

  $\begin{array}{cc}\dot{\sin (\text{gd}(x))}& =\tanh (x);\cos (\text{gd}(x))=\text{sech}(x);\\ \tan (\text{gd}(x))& =\sinh (x);\sec (\text{gd}(x))=\cosh (x);\\ \cot (\text{gd}(x))& =\text{csch}(x);\csc (\text{gd}(x))=\coth (x);\\ {}_{.}\tan \left(\frac{\text{gd}(x)}{2}\right)& =\tanh \left(\frac{x}{2}\right).\end{array}$

A função gudermanniana inversa.

   $\begin{array}{cc}\text{arcgd}(x)& ={\mathrm{g}\mathrm{d}}^{-1}(x)=\underset{0}{\overset{x}{\int }}\frac{dp}{\cos (p)},\\ & =\text{arccosh}(\sec (x))=\text{arctanh}(\sin (x)),\\ & =\ln (\sec (x)(1+\sin (x))),\\ & =\ln (\tan (x)+\sec (x))=\ln \tan (\frac{\pi }{4}+\frac{x}{2}),\\ & =\frac{1}{2}\ln \frac{1+\sin (x)}{1-\sin (x)}.\end{array}$

A derivada da função gudermanniana e sua inversa são:

   $\frac{d}{dx}\text{gd}(x)=\text{sech}(x);\frac{d}{dx}\text{arcgd}(x)=\sec (x).$

Ligações externas

• MathWorld