Función hiperbólica

Las funciones hiperbólicas son análogas a las funciones trigonométricas ordinarias o funciones circulares. Estas son:

sinh, cosh y tanh

csch, sech y coth

El seno hiperbólico

$\sinh(x)={\frac {e^{x}-e^{-x}}{2}}$

El coseno hiperbólico

$\cosh(x)={\frac {e^{x}+e^{-x}}{2}}$

La tangente hiperbólica

$\tanh(x)={\frac {\sinh(x)}{\cosh(x)}}$

y otras líneas:

$\coth(x)={\frac {\cosh(x)}{\sinh(x)}}$

(cotangente hiperbólica)

${\mbox{sech}}(x)={\frac {1}{\cosh(x)}}$

(secante hiperbólica)

${\mbox{csch}}(x)={\frac {1}{\sinh(x)}}$

(cosecante hiperbólica)

Relaciones

$\cosh ^{2}(x)-sinh^{2}(x)=1\,$

La derivada de sinh(x) está dada por cosh(x) y la derivada de cosh(x) es sinh(x). El gráfico de la función cosh(x) se denomina catenaria.

Series de Taylor

Es posible expresar las funciones hiperbólicas utilizando una Serie de Taylor:

$\sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}$

$\cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}$

$\tanh x=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{n}x^{2n-1}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}$

$\coth x={\frac {1}{x}}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots ={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}B_{n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi$

$\operatorname {sech} x=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =1+\sum _{n=1}^{\infty }{\frac {(-1)^{n}E_{n}x^{2n}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}$

$\operatorname {csch} x={\frac {1}{x}}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\cdots ={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}2(2^{2n}-1)B_{n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi$

Donde

$B_{n}\,$ es el enésimoNúmero de Bernoulli y

$E_{n}\,$ es el enésimo Número de Euler

Inversas de las funciones hiperbólicas

Las funciones recíprocas de las funciones hiperbólicas son:

${\mbox{arcsinh}}(x)=\ln(x+{\sqrt {x^{2}+1}})$

${\mbox{arccosh}}(x)=\ln(x\pm {\sqrt {x^{2}-1}})$

${\mbox{arctanh}}(x)=\ln \left({\sqrt {\frac {1-x^{2}}{1-x}}}\right)={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)$

${\mbox{arccoth}}(x)=\ln \left({\frac {\sqrt {x^{2}-1}}{x-1}}\right)={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)$

${\mbox{arcsech}}(x)=\ln \left({\frac {1\pm {\sqrt {1-x^{2}}}}{x}}\right)$

${\mbox{arccsch}}(x)=\ln \left({\frac {1\pm {\sqrt {1+x^{2}}}}{x}}\right)$

Las series de Taylor de las funciones inversas de las funciones hiperbólicas vienen dadas por:

$\operatorname {asinh} (x)=x-\left({\frac {1}{2}}\right){\frac {x^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{5}}{5}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{7}}{7}}+\cdots =$

$\operatorname {asinh} (x)=\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{2n+1}}{(2n+1)}},\left|x\right|<1$

$\operatorname {acosh} (x)=\ln 2-(\left({\frac {1}{2}}\right){\frac {x^{-2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-6}}{6}}+\cdots )=$

$\operatorname {acosh} (x)=\ln 2-\sum _{n=1}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-2n}}{(2n)}},x>1$

$\operatorname {atanh} (x)=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+{\frac {x^{7}}{7}}+\cdots =$

$\operatorname {atanh} (x)=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)}},\left|x\right|<1$

$\operatorname {acsch} (x)=\operatorname {asinh} (x^{-1})=x^{-1}-\left({\frac {1}{2}}\right){\frac {x^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-5}}{5}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-7}}{7}}+\cdots =$

$\operatorname {acsch} (x)=\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-(2n+1)}}{(2n+1)}},\left|x\right|<1$

$\operatorname {asech} (x)=\operatorname {acosh} (x^{-1})=\ln 2-(\left({\frac {1}{2}}\right){\frac {x^{2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{6}}{6}}+\cdots )=$

$\operatorname {asech} (x)=\ln 2-\sum _{n=1}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{2n}}{(2n)}},0<x\leq 1$

$\operatorname {acoth} (x)=\operatorname {atanh} (x^{-1})=x^{-1}+{\frac {x^{-3}}{3}}+{\frac {x^{-5}}{5}}+{\frac {x^{-7}}{7}}+\cdots =$

$\operatorname {acoth} (x)=\sum _{n=0}^{\infty }{\frac {x^{-(2n+1)}}{(2n+1)}},\left|x\right|>1$

Véase también

• Trigonometría
• Funciones trigonométricas
• Logaritmo natural
• Número e