Definitions Hyperbolic function

-


sinh, cosh , tanh



csch, sech , coth



(a) cosh(x) average of e , e



(b) sinh(x) half difference of e , e


there various equivalent ways defining hyperbolic functions. may defined in terms of exponential function:



hyperbolic sine:








sinh
⁡
x
=




e

x


−

e

−
x



2


=




e

2
x


−
1


2

e

x





=



1
−

e

−
2
x




2

e

−
x





.


{\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.}






hyperbolic cosine:








cosh
⁡
x
=




e

x


+

e

−
x



2


=




e

2
x


+
1


2

e

x





=



1
+

e

−
2
x




2

e

−
x





.


{\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.}






hyperbolic tangent:








tanh
⁡
x
=



sinh
⁡
x


cosh
⁡
x



=




e

x


−

e

−
x





e

x


+

e

−
x





=


{\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}=}






=




e

2
x


−
1



e

2
x


+
1



=



1
−

e

−
2
x




1
+

e

−
2
x





.


{\displaystyle ={\frac {e^{2x}-1}{e^{2x}+1}}={\frac {1-e^{-2x}}{1+e^{-2x}}}.}






hyperbolic cotangent:








coth
⁡
x
=



cosh
⁡
x


sinh
⁡
x



=




e

x


+

e

−
x





e

x


−

e

−
x





=


{\displaystyle \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}=}






=




e

2
x


+
1



e

2
x


−
1



=



1
+

e

−
2
x




1
−

e

−
2
x





,

x
≠
0.


{\displaystyle ={\frac {e^{2x}+1}{e^{2x}-1}}={\frac {1+e^{-2x}}{1-e^{-2x}}},\qquad x\neq 0.}






hyperbolic secant:








sech
⁡
x
=


1

cosh
⁡
x



=


2


e

x


+

e

−
x





=


{\displaystyle \operatorname {sech} x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}=}






=



2

e

x





e

2
x


+
1



=



2

e

−
x




1
+

e

−
2
x





.


{\displaystyle ={\frac {2e^{x}}{e^{2x}+1}}={\frac {2e^{-x}}{1+e^{-2x}}}.}






hyperbolic cosecant:








csch
⁡
x
=


1

sinh
⁡
x



=


2


e

x


−

e

−
x





=


{\displaystyle \operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}=}






=



2

e

x





e

2
x


−
1



=



2

e

−
x




1
−

e

−
2
x





,

x
≠
0.


{\displaystyle ={\frac {2e^{x}}{e^{2x}-1}}={\frac {2e^{-x}}{1-e^{-2x}}},\qquad x\neq 0.}





the hyperbolic functions may defined solutions of differential equations: hyperbolic sine , cosine unique solution (s, c) of system












c
′

(
x
)



=
s
(
x
)





s
′

(
x
)



=
c
(
x
)






{\displaystyle {\begin{aligned}c (x)&=s(x)\\s (x)&=c(x)\end{aligned}}}



such s(0) = 0 , c(0) = 1.


they unique solution of equation




f
″

(
x
)
=
f
(
x
)
,


{\displaystyle f (x)=f(x),}

such



f
(
0
)
=
1
,

f
′

(
0
)
=
0
,


{\displaystyle f(0)=1,f (0)=0,}

hyperbolic cosine, ,



f
(
0
)
=
0
,

f
′

(
0
)
=
1
,


{\displaystyle f(0)=0,f (0)=1,}

hyperbolic sine.


hyperbolic functions may deduced trigonometric functions complex arguments:



hyperbolic sine:








sinh
⁡
x
=
−
i
sin
⁡
(
i
x
)


{\displaystyle \sinh x=-i\sin(ix)}






hyperbolic cosine:








cosh
⁡
x
=
cos
⁡
(
i
x
)


{\displaystyle \cosh x=\cos(ix)}






hyperbolic tangent:








tanh
⁡
x
=
−
i
tan
⁡
(
i
x
)


{\displaystyle \tanh x=-i\tan(ix)}






hyperbolic cotangent:








coth
⁡
x
=
i
cot
⁡
(
i
x
)


{\displaystyle \coth x=i\cot(ix)}






hyperbolic secant:








sech
⁡
x
=
sec
⁡
(
i
x
)


{\displaystyle \operatorname {sech} x=\sec(ix)}






hyperbolic cosecant:








csch
⁡
x
=
i
csc
⁡
(
i
x
)


{\displaystyle \operatorname {csch} x=i\csc(ix)}





where imaginary unit property




i

2


=
−
1.


{\displaystyle i^{2}=-1.}


the complex forms in definitions above derive euler s formula.







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