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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