As funcións hiperbólicas son unhas funcións con definicións baseadas na función exponencial , ligada mediante operacións racionais e son análogas ás funcións trigonométricas. [ 1]
Curvas das funcións hiperbólicas sinh , cosh e tanh Curvas das funcións hiperbólicas csch , sech e coth O seno hiperbólico
sinh
(
x
)
=
e
x
−
e
−
x
2
{\displaystyle \sinh(x)={\frac {e^{x}-e^{-x}}{2}}}
O coseno hiperbólico
cosh
(
x
)
=
e
x
+
e
−
x
2
{\displaystyle \cosh(x)={\frac {e^{x}+e^{-x}}{2}}}
A tanxente hiperbólica
tanh
(
x
)
=
sinh
(
x
)
cosh
(
x
)
{\displaystyle \tanh(x)={\frac {\sinh(x)}{\cosh(x)}}}
e outras derivadas:
coth
(
x
)
=
cosh
(
x
)
sinh
(
x
)
{\displaystyle \coth(x)={\frac {\cosh(x)}{\sinh(x)}}}
(cotanxente hiperbólica )
sech
(
x
)
=
1
cosh
(
x
)
{\displaystyle {\mbox{sech}}(x)={\frac {1}{\cosh(x)}}}
(secante hiperbólica )
csch
(
x
)
=
1
sinh
(
x
)
{\displaystyle {\mbox{csch}}(x)={\frac {1}{\sinh(x)}}}
(cosecante hiperbólica ) Relación entre funcións hiperbólicas e funcións circulares [ editar | editar a fonte ] As funcións trigonométricas
s
e
n
(
t
)
{\displaystyle sen(t)}
c
o
s
(
t
)
{\displaystyle cos(t)}
(
x
,
e
)
{\displaystyle (x,e)}
t é o ángulo, medido en radiáns , comprendido entre o semieixe positivo X, e o segmento OP, segundo as seguintes igualdades:
{
x
(
t
)
=
cos
t
y
(
t
)
=
sin
t
{\displaystyle \left\{{\begin{matrix}x(t)=\cos t\\y(t)=\sin t\end{matrix}}\right.}
Tamén pode interpretarse o parámetro t como a lonxitude do arco de circunferencia unitaria comprendido entre o punto
(
1
,
0
)
{\displaystyle (1,0)}
Animación da representación do seno hiperbólico. De modo análogo, pódense definir as funcións hiperbólicas, como as coordenadas cartesianas
(
x
,
e
)
{\displaystyle (x,e)}
hipérbole equilátera , centrada na orixe, cuxa ecuación é
x
2
−
y
2
=
1
{\displaystyle \ x^{2}-y^{2}=1}
onde t é o dobre da área da rexión comprendida entre o semieixe positivo X , e o segmento OP e a hipérbole, segundo as seguintes igualdades:
{
x
(
t
)
=
cosh
t
y
(
t
)
=
sinh
t
{\displaystyle \left\{{\begin{matrix}x(t)=\cosh t\\y(t)=\sinh t\end{matrix}}\right.}
Con todo, tamén pode demostrarse que é válida a seguinte descrición da hipérbole:
x
(
t
)
=
e
t
+
e
−
t
2
{\displaystyle \ x(t)={\frac {e^{t}+e^{-t}}{2}}}
y
(
t
)
=
e
t
−
e
−
t
2
{\displaystyle \ y(t)={\frac {e^{t}-e^{-t}}{2}}}
dado que
(
e
t
+
e
−
t
2
)
2
−
(
e
t
−
e
−
t
2
)
2
=
1
{\displaystyle \ \left({\frac {e^{t}+e^{-t}}{2}}\right)^{2}-\left({\frac {e^{t}-e^{-t}}{2}}\right)^{2}=1}
De modo que o coseno hiperbólico e o seno hiperbólico admiten unha representación en termos de funcións exponenciais de variable real:
cosh
(
t
)
=
e
t
+
e
−
t
2
{\displaystyle \ \cosh(t)={\frac {e^{t}+e^{-t}}{2}}}
sinh
(
t
)
=
e
t
−
e
−
t
2
{\displaystyle \ \sinh(t)={\frac {e^{t}-e^{-t}}{2}}}
cosh
2
(
x
)
−
s
i
n
h
2
(
x
)
=
1
{\displaystyle \cosh ^{2}(x)-\,\mathrm {sinh} ^{2}(x)=1\,}
Téñense as seguintes fórmulas moi semellantes ás súas correspondentes trigonométricas[ 2]
cosh
(
x
+
y
)
=
cosh
(
x
)
cosh
(
y
)
+
sinh
(
x
)
sinh
(
y
)
{\displaystyle \cosh(x+y)=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)}
o que leva á seguinte relación:
cosh
(
2
x
)
=
cosh
2
(
x
)
+
s
i
n
h
2
(
x
)
{\displaystyle \cosh(2x)=\cosh ^{2}(x)+\,\mathrm {sinh} ^{2}(x)}
e por outra banda
s
i
n
h
(
x
+
y
)
=
s
i
n
h
(
x
)
cosh
(
y
)
+
s
i
n
h
(
y
)
cosh
(
x
)
{\displaystyle \mathrm {sinh} (x+y)=\mathrm {sinh} (x)\cosh(y)+\mathrm {sinh} (y)\cosh(x)}
que leva a:
s
i
n
h
(
2
x
)
=
2
s
i
n
h
(
x
)
cosh
(
x
)
{\displaystyle \mathrm {sinh} (2x)=2\,\mathrm {sinh} (x)\cosh(x)}
tense estoutra relación
tanh
(
x
+
y
)
=
tanh
(
x
)
+
tanh
(
y
)
1
+
tanh
(
x
)
tanh
(
y
)
{\displaystyle \tanh(x+y)={\frac {\tanh(x)+\tanh(y)}{1+\tanh(x)\tanh(y)}}}
que permite ter
tanh
(
2
x
)
=
2
tanh
x
1
+
tanh
2
x
{\displaystyle \tanh(2x)={\frac {2\tanh x}{1+\tanh ^{2}x}}}
d
d
x
(
cosh
(
x
)
)
=
s
i
n
h
(
x
)
{\displaystyle {\frac {d\ }{dx}}(\cosh(x))=\,\mathrm {sinh} \,(x)}
d
d
x
(
s
i
n
h
(
x
)
)
=
cosh
(
x
)
{\displaystyle {\frac {d\ }{dx}}(\,\mathrm {sinh} \,(x))=\cosh(x)}
d
d
x
(
tanh
(
x
)
)
=
s
e
c
h
2
(
x
)
{\displaystyle {\frac {d\ }{dx}}(\,\tanh(x))=\mathrm {sech} ^{2}(x)}
d
d
x
(
c
o
t
h
(
x
)
)
=
−
c
s
c
h
2
(
x
)
{\displaystyle {\frac {d\ }{dx}}(\mathrm {coth} (x))=-\mathrm {csch} ^{2}(x)}
d
d
x
(
s
e
c
h
(
x
)
)
=
−
s
e
c
h
(
x
)
tanh
(
x
)
{\displaystyle {\frac {d\ }{dx}}(\mathrm {sech} (x))=-\mathrm {sech} (x)\tanh(x)}
d
d
x
(
c
s
c
h
(
x
)
)
=
−
c
s
c
h
(
x
)
c
o
t
h
(
x
)
{\displaystyle {\frac {d\ }{dx}}(\mathrm {csch} (x))=-\mathrm {csch} (x)\mathrm {coth} (x)}
Ademais a integración ao ser a operación inversa da derivación é trivial neste caso.
A derivada de
s
i
n
h
(
x
)
{\displaystyle sinh(x)}
c
o
s
h
(
x
)
{\displaystyle cosh(x)}
c
o
s
h
(
x
)
{\displaystyle cosh(x)}
s
i
n
h
(
x
)
{\displaystyle sinh(x)}
c
o
s
h
(
x
)
{\displaystyle cosh(x)}
catenaria .
Inversas das funcións hiperbólicas e derivadas [ editar | editar a fonte ] As funcións recíprocas das funcións hiperbólicas e as súas derivadas son:[ 3]
arg sinh
(
x
)
=
ln
(
x
+
x
2
+
1
)
d
d
x
(
arg sinh
(
x
)
)
=
1
x
2
+
1
arg cosh
(
x
)
=
ln
(
x
+
x
2
−
1
)
;
x
≥
1
d
d
x
(
arg cosh(x)
)
=
1
x
2
−
1
;
x
>
1
arg tanh
(
x
)
=
1
2
ln
(
1
+
x
1
−
x
)
;
|
x
|
<
1
d
d
x
(
arg tanh(x)
)
=
1
1
−
x
2
;
|
x
|
<
1
arg coth
(
x
)
=
1
2
ln
(
x
+
1
x
−
1
)
;
|
x
|
>
1
d
d
x
(
arg coth(x)
)
=
1
1
−
x
2
;
|
x
|
>
1
arg sech
(
x
)
=
ln
(
1
x
+
1
−
x
2
x
)
;
0
<
x
≤
1
d
d
x
(
arg sech(x)
)
=
−
1
x
1
−
x
2
;
0
<
x
<
1
arg csch
(
x
)
=
ln
(
1
x
+
1
+
x
2
|
x
|
)
;
x
≠
0
d
d
x
(
arg csch(x)
)
=
−
1
|
x
|
1
+
x
2
;
x
≠
0
{\displaystyle {\begin{aligned}{\mbox{arg sinh}}(x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)&{\frac {d}{dx}}({\mbox{arg sinh}}(x))&={\frac {1}{\sqrt {x^{2}+1}}}\\{\mbox{arg cosh}}(x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right);x\geq 1&{\frac {d}{dx}}({\mbox{arg cosh(x)}})&={\frac {1}{\sqrt {x^{2}-1}}};x>1\\{\mbox{arg tanh}}(x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right);\left|x\right|<1&{\frac {d}{dx}}({\mbox{arg tanh(x)}})&={\frac {1}{1-x^{2}}};\left|x\right|<1\\{\mbox{arg coth}}(x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right);\left|x\right|>1&{\frac {d}{dx}}({\mbox{arg coth(x)}})&={\frac {1}{1-x^{2}}};\left|x\right|>1\\{\mbox{arg sech}}(x)&=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1-x^{2}}}{x}}\right);0<x\leq 1&{\frac {d}{dx}}({\mbox{arg sech(x)}})&={\frac {-1}{x{\sqrt {1-x^{2}}}}};0<x<1\\{\mbox{arg csch}}(x)&=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1+x^{2}}}{\left|x\right|}}\right);x\neq 0&{\frac {d}{dx}}({\mbox{arg csch(x)}})&={\frac {-1}{\left|x\right|{\sqrt {1+x^{2}}}}};x\neq 0\end{aligned}}}
As series de Taylor das funcións inversas das funcións hiperbólicas veñen dadas por:
arg sinh
(
x
)
=
x
−
(
1
2
)
x
3
3
+
(
1
⋅
3
2
⋅
4
)
x
5
5
−
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
7
7
+
⋯
=
{\displaystyle {\mbox{arg sinh}}(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 =}
arg sinh
(
x
)
=
∑
n
=
0
∞
(
(
−
1
)
n
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
2
n
+
1
(
2
n
+
1
)
,
|
x
|
<
1
{\displaystyle {\mbox{arg sinh}}(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}
arg cosh
(
x
)
=
ln
2
x
−
(
(
1
2
)
x
−
2
2
+
(
1
⋅
3
2
⋅
4
)
x
−
4
4
+
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
−
6
6
+
⋯
)
=
{\displaystyle {\mbox{arg cosh}}(x)=\ln 2x-(\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 )=}
arg cosh
(
x
)
=
ln
2
x
−
∑
n
=
1
∞
(
(
−
1
)
n
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
−
2
n
(
2
n
)
,
x
>
1
{\displaystyle {\mbox{arg cosh}}(x)=\ln 2x-\sum _{n=1}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-2n}}{(2n)}},x>1}
arg tanh
(
x
)
=
x
+
x
3
3
+
x
5
5
+
x
7
7
+
⋯
=
{\displaystyle {\mbox{arg tanh}}(x)=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+{\frac {x^{7}}{7}}+\cdots =}
arg tanh
(
x
)
=
∑
n
=
0
∞
x
2
n
+
1
(
2
n
+
1
)
,
|
x
|
<
1
{\displaystyle {\mbox{arg tanh}}(x)=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)}},\left|x\right|<1}
arg csch
(
x
)
=
arg sinh
(
x
−
1
)
=
x
−
1
−
(
1
2
)
x
−
3
3
+
(
1
⋅
3
2
⋅
4
)
x
−
5
5
−
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
−
7
7
+
⋯
=
{\displaystyle {\mbox{arg csch}}(x)={\mbox{arg sinh}}(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 =}
arg csch
(
x
)
=
∑
n
=
0
∞
(
(
−
1
)
n
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
−
(
2
n
+
1
)
(
2
n
+
1
)
,
|
x
|
>
1
{\displaystyle {\mbox{arg csch}}(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}
arg sech
(
x
)
=
arg cosh
(
x
−
1
)
=
ln
2
−
(
(
1
2
)
x
2
2
+
(
1
⋅
3
2
⋅
4
)
x
4
4
+
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
6
6
+
⋯
)
=
{\displaystyle {\mbox{arg sech}}(x)={\mbox{arg cosh}}(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 )=}
arg sech
(
x
)
=
ln
2
−
∑
n
=
1
∞
(
(
−
1
)
n
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
2
n
(
2
n
)
,
0
<
x
≤
1
{\displaystyle {\mbox{arg sech}}(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}
arg coth
(
x
)
=
arg tanh
(
x
−
1
)
=
x
−
1
+
x
−
3
3
+
x
−
5
5
+
x
−
7
7
+
⋯
=
{\displaystyle {\mbox{arg coth}}(x)={\mbox{arg tanh}}(x^{-1})=x^{-1}+{\frac {x^{-3}}{3}}+{\frac {x^{-5}}{5}}+{\frac {x^{-7}}{7}}+\cdots =}
arg coth
(
x
)
=
∑
n
=
0
∞
x
−
(
2
n
+
1
)
(
2
n
+
1
)
,
|
x
|
>
1
{\displaystyle {\mbox{arg coth}}(x)=\sum _{n=0}^{\infty }{\frac {x^{-(2n+1)}}{(2n+1)}},\left|x\right|>1}
Da relación do coseno e o seno hiperbólico pódense derivar as seguintes relacións:
e
x
=
cosh
x
+
sinh
x
{\displaystyle e^{x}=\cosh x+\sinh x\!}
e
e
−
x
=
cosh
x
−
sinh
x
.
{\displaystyle e^{-x}=\cosh x-\sinh x.\!}
Estas expresións son análogas ás que están en termos de senos e cosenos, baseadas na fórmula de Euler, como suma de exponenciais complexas.
↑ Cálculo de Granville
↑ Manual de Matemáticas para Ingenieros y estudiantes . Mir. ↑ Cálculo con Geometría Analítica . Prenttice-Hall Hispanoamericana S.A. ISBN 0-13-111807-2 .
