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Encyclopedia > Gudermannian function

The Gudermannian function, named after Christoph Gudermann (1798 - 1852), relates the circular and hyperbolic trigonometric functions without resorting to complex numbers. Christoph Gudermann (March 25, 1798 - September 25, 1852) was born in Vienenburg, Germany. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...


It is defined by

begin{align}{rm{gd}}(x)&=int_0^xfrac{dp}{cosh(p)}, &=arcsinleft(tanh(x)right)=arccosleft(mbox{sech}(x)right), &=arctanleft(sinh(x)right)=mbox{arcsec}left(cosh(x)right), &=mbox{arccot}left(mbox{csch}(x)right)=mbox{arccsc}left(coth(x)right), &=2arctanleft(tanhleft(frac{x}{2}right)right)=2arctan(e^x)-frac{pi}{2}.end{align},!

The following identities also hold:

begin{align}{color{white}dot{{color{black}sin(mbox{gd}(x))}}}&=tanh(x);quadcos(mbox{gd}(x))=mbox{sech}(x); tan(mbox{gd}(x))&=sinh(x);quad;sec(mbox{gd}(x))=cosh(x); cot(mbox{gd}(x))&=mbox{csch}(x);quad,csc(mbox{gd}(x))=coth(x); {}_{color{white}.}tanleft(frac{mbox{gd}(x)}{2}right)&=tanhleft(frac{x}{2}right).end{align},!
(Gudermannian function with its asymptotes marked in gray.)
(Gudermannian function with its asymptotes scriptstyle{y=pmfrac{pi}{2}},! marked in gray.)
(The inverse Gudermannian function.)
(The inverse Gudermannian function.)

  Gudermannian function with asymptotes y = +- pi/2 marked in File links The following pages link to this file: Mercator projection Gudermannian function Categories: GFDL images ... Gudermannian function with asymptotes y = +- pi/2 marked in File links The following pages link to this file: Mercator projection Gudermannian function Categories: GFDL images ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...


The inverse Gudermannian function is given by In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

begin{align} mbox{arcgd}(x)&={rm {gd}}^{-1}(x)=int_0^xfrac{dp}{cos(p)}, &={}mbox{arccosh}(sec(x))=mbox{arctanh}(sin(x)), &={}lnleft(sec(x)(1+sin(x))right), &={}ln(tan(x)+sec(x))=lntanleft(frac{pi}{4}+frac{x}{2}right), &={}frac{1}{2}ln frac{1+sin(x)}{1-sin(x)} .end{align},!

The derivatives of the Gudermannian and its inverse are For a non-technical overview of the subject, see Calculus. ...

frac{d}{dx}mbox{gd}(x)=mbox{sech}(x);quadfrac{d}{dx}mbox{arcgd}(x)=sec(x).,!

See also

In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. ... Mercator world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigatium Emendate (1569) The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator, in 1569. ... In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable t. ... It has been suggested that Tractrice be merged into this article or section. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

References


  Results from FactBites:
 
Gudermannian function - Wikipedia, the free encyclopedia (68 words)
Gudermannian function with its asymptotes y = ±π/2 marked in gray.
The Gudermannian function, named after Christoph Gudermann (1798 - 1852), relates the circular and hyperbolic trigonometric functions without resorting to complex numbers.
The derivatives of the Gudermannian and its inverse are
  More results at FactBites »


 

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